Formal reduction and integration of systems of nonlinear differential equations [PhD Thesis ed.]

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 THESE presentee par

Gerard Ei henm uller pour obtenir le grade de DOCTEUR de l'Universite Joseph Fourier (Arr^ete ministeriel du 30 mars 1992)

(spe ialite : Mathematiques Appliquees)

Redu tion et Integration symbolique des systemes d'equations dierentielles non-lineaires Formal redu tion and integration of systems of nonlinear dierential equations

Date de soutenan e : le 11 de embre 2000 Composition du Jury : L: BRENIG

Universite Libre de Bruxelles

G: CHEN J: DELLA DORA J: THOMANN E: TOURNIER

Universite de Lille Institut National Polyte hnique de Grenoble Universite Louis Pasteur de Strassbourg Universite Joseph Fourier

(president et rapporteur) (rapporteur) (examinateur) (examinateur) (examinateur)

These preparee au sein du Laboratoire LMC-IMAG et nan ee par la Gottlieb Daimlerund Karl Benz-Stiftung

Remer iements D'abord je voudrais remer ier Evelyne Tournier qui a a

epte de diriger ette these et qui m'a fait de ouvrir le al ul formel. Je la remer ie egalement pour m'avoir permis de parti iper au projet europeen CATHODE. Je tiens a remer ier mon o-dire teur de these Jean Della Dora d'avoir a

epte d'assurer l'en adrement de e travail. Je lui suis tres re onnaissant pour les orientations qu'il a suggerees et pour m'avoir fait de ouvrir les systemes dynamiques. Je suis parti ulierement heureux que Leon Brenig ait a

epte d'^etre rapporteur de ma these et president du jury. Je le remer ie ainsi que Guoting Chen, rapporteur et spe ialiste des formes normales, de leur le ture tres attentive et de leurs nombreuses suggestions. Je remer ie egalement Jean Thomann qui a a

epte d'examiner ma these. Sans le support nan ier de la Gottlieb Daimler- und Karl Benz-Stiftung ette these n'aurait pas ete possible. En parti ulier Petra Jung, Horst Nienstadt et Gisbert Freiherr zu Putlitz m'ont beu oup aide. Qu'ils en soient remer ies. Mes remer iements vont aussi a tous mes ollegues de l'equipe MOSAIC et aux an iens ollegues de l'equipe Cal ul Formel ave lesquels j'ai eu le plaisir de ollaborer ou bien simplement de passer d'agreables moments au ours de es trois annees. Il y a les membres permanents Fran oise Jung, Claire Di Cres enzo, Claudine Chay, Moulay Barkatou, Valerie Perrier, Mi hele Benois, Rodney Coleman, Dominique Duval, MarieLauren e Mazure et Gilles Villard et les non-permanents Mihaela, Voi hita, Aude, Fred, Claude-Pierre, Olivier, Ines, Yann, Josselin, Lu , Loi , E khard, Tes et Rene. Je voudrais egalement remer ier Sylvie, Sophie, Steph, Je, Cyril et tous mes amis, kayakistes ou pas. Parmis les nombreuses personnes ren ontrees au ours de ma these, je voudrais egalement remer ier Jose Cano, Fuensanta Aro a et Fouad Zinoun. En n je voudrais remer ier Daniela et ma famille qui m'ont beau oup soutenu et qui se rejouissent ave moi de l'a

omplissement de e travail.

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Contents Introdu tion

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Introdu tion fran aise

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Resume par hapitre (en Fran ais)

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1 Formal Solutions for Dynami al Systems 1.1 Flows, Ve tor Fields and Dierential Equations 1.2 Linearization of ve tor elds . . . . . . . . . . . 1.3 Equivalen e of ve tor elds . . . . . . . . . . . 1.4 Time transformations . . . . . . . . . . . . . . 1.5 Convergen e and Formal Solutions . . . . . . .

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2 The Newton diagram 2.1 The support and the Newton diagram . . 2.2 Power transformations . . . . . . . . . . . 2.3 Power transformations as dieomorphisms 2.3.1 Power transformations in Rn . . . 2.3.2 Power transformations in C n . . . 2.4 Cones . . . . . . . . . . . . . . . . . . . .

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3 Normal Forms 3.1 The Poin are-Dula normal form . . . . . . . . . . . . . . . . 3.2 The Jordan form . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Computation of normal forms by Poin are Transformations . 3.3.1 The homologi al equation . . . . . . . . . . . . . . . . 3.3.2 The matrix representation of the homologi al operator 3.4 Computation of normal forms by Lie transformation . . . . . 3.4.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 A tion on a ve tor eld . . . . . . . . . . . . . . . . . 3.4.3 The ow of a non-singular ve tor eld . . . . . . . . . 3.4.4 Organization of the omputations . . . . . . . . . . . . 3.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

4 Resolution of singularities by blowing-up

4.1 Two-dimensional polar blowing-up . . . . . . . . . . . . 4.2 Quasi-homogeneous dire tional blowing up . . . . . . . . 4.2.1 Quasi-homogeneous ve tor elds . . . . . . . . . 4.2.2 The ee t on the Newton diagram . . . . . . . . 4.2.3 Constru tion of blowing-ups via adjoint matri es 4.2.4 The ex eptional divisor . . . . . . . . . . . . . . 4.3 Su

essive blowing-up . . . . . . . . . . . . . . . . . . .

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5 Classi ation

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6 Regular Points

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7 Two dimensional elementary singular points

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7.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.2 Systems with real oeÆ ients . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8 Two dimensional nonelementary singular points

8.1 The Verti es . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 The Poin are-Dula Normal Form for Class V 8.1.2 The Se tors . . . . . . . . . . . . . . . . . . . 8.2 The Edges . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Redu tion of the Singularity . . . . . . . . . . 8.2.2 Subse tors and Re ursion . . . . . . . . . . .

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9 Three- and higher-dimensional elementary singular points

9.1 Integration of n-dimensional normal forms for m = 1 . . . . . . . . . . . . 9.2 Integration of real n-dimensional normal forms for m = 1 . . . . . . . . . 9.3 Redu tion of three-dimensional normal forms for m=2 . . . . . . . . . . . 9.3.1 The Choi e of the Matrix A . . . . . . . . . . . . . . . . . . . . . . 9.3.2 The Classi ation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Real three-dimensional systems . . . . . . . . . . . . . . . . . . . . 9.4 Redu tion of n-dimensional normal forms for m > 2 . . . . . . . . . . . . 9.4.1 Conditions for the hoi e of the matrix A . . . . . . . . . . . . . . 9.4.2 The resonant plane lies entirely within N n . . . . . . . . . . . . . . 9.4.3 The set M \ N n ontains m linearly independent ve tors . . . . . . 9.4.4 The set M \ N n ontains less than m linearly independent ve tors 9.5 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Three-dimensional nonelementary singular points 10.1 10.2 10.3 10.4 10.5

The verti es . . . . . . . . . . . . . . . . . . . . . . . . . The edges . . . . . . . . . . . . . . . . . . . . . . . . . . The fa es . . . . . . . . . . . . . . . . . . . . . . . . . . The se tors . . . . . . . . . . . . . . . . . . . . . . . . . The virtual Newton diagram . . . . . . . . . . . . . . . 10.5.1 The onstru tion of the virtual Newton diagram

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Contents

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10.6 Examples for the redu tion of three-dimensional nilpotent systems by blowingups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 10.7 Higher-dimensional nonelementary singular points . . . . . . . . . . . . . . 139 11 The FRIDAY pa kage

11.1 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Using the pa kage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Introdu ing examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Integration of n-dimensional systems in the neighbourhood of a regular point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Computation of n-dimensional normal forms for non-nilpotent singular ve tor elds . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Integration of two-dimensional elementary singular points . . . . . 11.3.4 Integration of two-dimensional nonelementary singular points . . . 11.3.5 Integration and Redu tion of three-dimensional elementary singular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Con lusion

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Con lusion fran

aise

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Bibliography

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Introdu tion Dynami al systems are present everywhere in s ien e. Many models of natural pro esses yield dynami al systems. A dynami al system is a system whose state hanges with time. There exist two main types of dynami al systems: dis rete dynami al systems that are represented by dieren e equations and ontinuous dynami al systems that are represented by dierential equations. The state of a system an be des ribed by a number of variables that we will entralize in the n-dimensional ve tor = ( 1 ). The variable denotes time. We are interested in ontinuous dynami al systems that an be des ribed by a system of autonomous dierential equations _= ( ) (1) X

x ; : : : ; xn

X

t

F X

where = ( 1 ) is a ve tor. The omponents 2 1 ( C ) are de ned on the open onvex subset . The fa t that we onsider systems implies that is greater than 1. In parti ular the ases = 2 and = 3 will be studied very losely. The solutions of dynami al systems are given by their ow that is denoted by . The velo ity of the ow is given by the ve tor eld . In general the ow  is approximated by numeri al algorithms. But those methods are not very pre ise in the neighbourhood of singularities and they do not allow the study of systems ontaining parameters. On the other hand most dierential equations (even simple ones) have no expli it solution. Therefore in this thesis we will employ another approa h proposed by H. Poin are to solve this dilemma. The expli it analyti study of a dierential equation is repla ed by qualitative studies. To study the qualitative behaviour of dynami al systems means to lassify them into equivalen e lasses of similar behaviour. This lassi ation is realized by lo al dieomorphisms. That means that those studies are stri tly lo al. They are only valid in the neighbourhood of a point or another obje t. In two dimensions and for some higher dimensional problems the results of those onsiderations an be used to approximate algebrai solutions. Dynami al systems des ribed by equations as (1) often arise from modeling problems in s ien e. The variable denotes involved physi al quantities that hange with time. Those hanges are des ribed by a system of dierential equations. F

f ; : : : ; fn

fi

M

C

M;

n

n

n

n

F

X

Example 1 (Planar Pendulum)

Newton's third law

F

=

ma

9

10

Introdu tion

x l

mg

Figure 1: Planar pendulum, see example 1.

des ribes the behaviour of many physi al systems. In absen e of fri tion we nd the relations F = m g sin x 2 a = t 2 (l x) for a planar pendulum (see gure 1). The variable m denotes the mass of the pendulum , l its length and g the gravitational onstant. As the a

eleration a is the se ond derivative of displa ement x this is a se ond order dierential equation g x + sin x = 0 l that an be rewritten as a rst order system



x_ = y y_ = gl sin(x)

in the variables x and the new variable y. This thesis is split into three parts. The rst part introdu es some main tools that will be used in the algorithms for the redu tion of two- and higher-dimensional systems. Those algorithms are des ribed in the se ond part. The third part of this thesis deals with the implementation of the algorithms and the programming aspe ts. We will also give some examples for the use of the Maple pa kage that has been implemented by the author. We will give a more pre ise overview for ea h hapter of this thesis. First part: The integration of dynami al systems

Formal solutions for dynami al systems This hapter introdu es some basi notations and de nitions for dynami al systems as they an be found in many referen es on dynami al systems (see for example the works

Introdu tion

11

from K. Alligood, T. Sauer and J. Yorke [39℄, F. Verhulst [26℄, J. Hale and H. Kor ak [33℄, S. Chow and J. Hale [14℄, J. Gu kenheimer and P. Holmes [36℄, D. Arrowsmith and C. Pla e [17℄ and J. Hubbard and B. West [34℄). Systems of dierential equations and ve tor elds an be used to represent the same dynami s. Therefore the theory of dynami al systems largely uses the notations of ve tor elds. Their solutions are alled global or lo al ows. In general global ows an't be

al ulated. That is why the omputation of lo al ows and lo al studies of ve tor elds or dierential equations is the main intention in the theory of dynami al systems. If the linear part of a system exists and if the linearized ve tor eld is equivalent or onjugated to the initial one a lot of qualitative attributes of ve tor elds an be derived from the linearized one. Therefore de nitions for equivalen e of ve tor elds using dieomorphisms play a very important role in qualitaive studies of ve tor elds and dierential equations. Transformations that are no equivalen e transformations, so alled time transformations, are also frequently used to nd solution urves for dierential equations. The transformations do not yield equivalent ve tor elds and we an no longer retransform solutions of time transformed equations to solution of the initial dierential equations. Therefore the urves obtained from an integration of the new system an no longer be onsidered as equivalent to the ow of the initial one. That is why we introdu e the notation of solution

urves that are parametrizations of the ows of the initial system. In general normal form onstru tions that are introdu ed in hapter 3 yield diverging series. For this reason all al ulated solutions might also be divergent. That is why all onsiderations are purely formal. We work in the ring of formal power series. The implemented algorithms work with trun ations that are polynomials as formal power series

an not be handeled in omputations.

The Newton diagram Many transformations applied to a ve tor eld an be interpreted geometri ally and the geometri aspe ts of ve tor elds an be used to nd transformations that redu e and simplify the onsidered ve tor eld. The most important tool for the geometri interpretation of transformation is the Newton diagram or the Newton polygon. The Newton polygon is also used dire tly to nd solutions for dierential equations (see for example works from J. Della Dora and F. Jung [19℄, F. Beringer and F. Jung [6℄ and J. Cano [37℄). For the algorithms proposed here however the Newton diagram and the support of a onsidered system will be used to

al ulate matri es that de ne power transformations. Power transformations are a very powerful tool for handling systems of dierential equations. They an be used to redu e singularities of systems having a nilpotent linear part as we will see in hapter 4 or to integrate systems that are in normal form (see

hapter 7 and 9). Power transformation manipulate the exponents of the on erned system. The geometri interpretation of this manipulation is very simple as it indu es an aÆne transformation on the support of the on erned system. Some results on erning power transformations are given in works from A. Bruno ([9℄ and [1℄). To validate the use of power transformations we have to prove that power transformations are dieomorphisms. But some power transformations are not inje tive on the whole

12

Introdu tion

de nition set. Therefore the de nition set is limited su h that the transformations are inje tive on this set. Their surje tivity is guranteed by a onstru tion that makes them "pie ewise surje tive". In se tion 4 and 8 we will handle systems that also have negative exponents. They an be redu ed by power transformations as their support lies within a one. Therefore the intention of geometri manipulations of ve tor eld often onsists in manipulating those

ones. Normal forms

As mentioned before some of the most important properties of a ve tor eld an be dedu ed from its linear part if it exists. In ertain ases there exist dieomorphisms that transform a onsidered ve tor eld into a linear one. Though in general it is not possible to linearize a given ve tor eld entirely we an nd dieomorphisms that redu e the ve tor eld to a "simpler" ve tor eld. However the de nition of "simpler" is not unique. The omplexity of a ve tor eld usually depends on the number and the properties of the nonlinear terms. A "simpler" ve tor eld usually has fewer nonlinear terms or its nonlinear terms have spe ial properties. When the system an no longer be redu ed is said to be in normal form. There exist many dierent approa hes to normal forms. We use the Poin are-Dula normal form as it an be integrated for two-dimensional systems ( hapter 7). The Poin areDula normal form is mainly due to H. Poin are [47℄, H. Dula [21℄ and G. Birkho [7℄. It

an be al ulated using the adjoint representation method that is due to G. Iooss [35℄ or more eÆ iently using Lie theory as it has been proposed by K. T. Chen [40℄ or A. Deprit [3℄. Re ently a lot of work has been done in normal form theory. The rst step in al ulating the Poin are-Dula normal form is the al ulation of the Jordan form of the matrix representing its linear part. This is far from being trivial (see I. Gil [29℄ and M. Griesbre ht [31℄). But it an be avoided by using the Frobenius form of a matrix (see works from G. Chen [13℄). R. Cushman and A. Sanders [16℄ propose an algorithm that an be used for the al ulation of ve tor elds with nilpotent but non-vanishing linear part. The

ase of normal forms for Lotka-Voltera systems have been studied by S. Louies and L. Brenig [50℄. Algorithms that use Carleman-linearizations to ompute normal forms have been proposed by L. Stolovit h [56℄ and G. Chen [13℄. Resolution of singularities by blowing-up

The Poin are-Dula normal form theorem an no longer be applied if the linear part of the

onsidered ve tor eld is nilpotent. In this ase blowing-up is used. Blowing-up involves

hanges of oordinates (polar oordinates or power transformations) whi h expand or blow-up the singularity of the ve tor eld into a set on whi h a nite number of simpler singularities o

ur. Dire tional blowing-up was rst introdu ed for plane algebrai urves by O. Zariski [66℄ and for two-dimensional dierential equation by A. Seidenberg [53℄. Sin e then many others have worked on this subje t. For our onstru tions we will use quasihomogeneous blowing-up that is given by power transformations and that have rst been used by A.

Introdu tion

13

Bruno [9℄ and more re ently by M. Brunella and M. Miari [8℄. They use unimodular matri es and work mainly with two-dimensional problems. We extend blowing-up onstru tion to power transformations de ned by any invertible matrix as unimodular matri es are not suÆ ient for problems appearing in dimension 3. It will be shown in hapter 2 that those matri es de ne dieomorphisms in a subset of the on erned neighbourhood and that their use is therefore allowed. Se ond part: The Algorithms

Classi ation The proposed algorithms handle the redu tion of dynami al systems and the omputation of solution urves for several ases. The ase of a simple or regular point, the ase of an elementary singular point (non-nilpotent linear part) and the ase of a nonelementary singular point (nilpotent linear part) are treated separately. The rst step in any al ulation is to lassify a given system to allow to handle it with the appropriate methods.

Regular points In the neighbourhood of a regular point any system an be redu ed to a system with a very simple form that is a kind of normal form. It an easily be integrated. The problem of al ulating this normal form an be redu ed to the problem of al ulating the ow of a system of dierential equations near a simple point. The hange of oordinates is omputed using Taylor series. This method is des ribed in many referen es treating Lie theory and ve tor elds (W. Groebner [62℄, W. Groebner and H.Knapp [63℄ and P. Olver[46℄).

Two-dimensional elementary singular points A singular point is alled elementary singular point if the onsidered system has a nonnilpotent Ja obian matrix there. Then the Poin are-Dula theorem an be applied to

al ulate the normal form of the onsidered system. The two dimensional Poin are-Dula normal form is integrable as it has been shown by A. Bruno [9℄. A spe ial ase is represented by systems with purely imaginary eigenvalues. They yield periodi solutions that are best represented in polar oordinates. This also allows us to obtain real solution urves if the initial system is real.

Two-dimensional nonelementary singular points In the ase of a nonelementary singular point there exist two methods to nd solution

urves. Blowing-ups that are introdu ed in hapter 4 redu e the omplexity of the onsidered singularity. They yield several new systems instead of only one initial system. The new systems are treated re ursively be applying the entire algorithm (starting with the lassi ation). Another method that is to use time transformations to ompute nonnilpotent systems that have their support in a one. An appropiate power transformation redu es this system to a non-nilpotent system with integer exponents. Solution urves for this system an easily be al ulated via normal forms as in the ase of an elementary singular point.

14

Introdu tion

Both methods are ontrolled by the Newton diagram. The edges are used to ompute the matri es de ning the power transformations for the blowing-ups and the verti es de ne the time transformations. These methods yield many solutions. Therefore a entral point of this part of the algorithm is the use of se tors. The se tors de ne the domains of the on erned neighbourhood where the al ulated solution urves are valid. They allow a very eÆ ient handling of the solutions.

Three and higher dimensional elementary singular points The algorithms des ribed in this hapter have so far only been treated very super ially by A. Bruno [1℄. Using these works as a starting point we propose a more omplete study of the ase of three- and higher-dimensional elementary singular points. In a rst step the Poin are-Dula normal form is al ulated. This normal form an be redu ed to a system from whi h we an split a system of lower dimension. The power transformation used for the redu tion has to verify very stri t onditions. Problems an o

ur if some of those ve tors have negative oordinates. These problems arise from the higher dimension of the resonant plane and its position in the spa e of exponents. We propose a lassi ation of three-dimensional normal forms that allows the redu tion and integration of any three-dimensional normal form. The virtual Newton diagram allows to generalize the results obtained from the intense study of three-dimensional normal forms to higher dimensional systems.

Three and higher dimensional nonelementary singular points Three and higher dimensional systems with nonelementary singular points an also be treated by blowing-ups. However there still remain many problems as it has been shown for 3 dimensions by X. Gomez-Mont and I. Luengo [30℄. Three dimensional systems an not always be redu ed entirely by a nite number of su

essive blowing-ups. Nevertheless we give some examples that use 3 dimensional blowing-ups. For these examples the orre tness of the onstru tions have been proved by F. Cano and D. Cerveau [10℄. We propose a onstru tion of blowing-ups that is ontrolled by the Newton diagram or the virtual Newton diagram and that onsiders a de nition of se tors that is dierent to the de nition given by A. Bruno [9℄. It is strongly onne ted to the ones that ontain the support of the initial system. Third part: Implementation

The algorithms des ribed in the previous part have been implemented in the FRIDAY pa kage.

Maple

1

The FRIDAY pa kage This hapter gives a des ription of the pa kage and a large number of examples for its use. The FRIDAY pa kage is organized in modules a

ording to the lassi ation of dynami al 1

FRIDAY stands for Formal Redu tion and Integration of Dynami al Autonomous Systems

Introdu tion

15

systems. However only a few pro edures are visible to the user. The main pro edure

an be used to integrate any two-dimensional and a large number of three-dimensional dynami al systems. Besides this pro edure the modules that ompute normal forms an be used separately. In this thesis we will show how tools as normal forms and power transformations an be used for the formal redu tion and integration of dynami al systems. For two-dimensional systems and three-dimensional systems with elementary singular points the proposed algorithms have been implemented and tested. However there still remain theoreti al and pra ti al problems espe ially in the eld of three- and higher dimensional systems with nonelementary singular points.

Introdu tion Les systemes dynamiques sont presents partout dans la s ien e. Ils proviennent de nombreux modeles simulant des phenomenes naturels. Il y a deux types prin ipaux de systemes dynamiques: les systemes dis rets qui sont representes par des equations aux dieren es et les systemes ontinus qui sont ara terises par des systemes d'equations dierentielles. L'etat d'un systeme peut ^etre de rit par un nombre de variables qui sont reunies dans le ve teur = ( 1 enote le temps. Dans ette these nous nous n ). La variable d interessons aux systemes dynamiques ontinus qui peuvent ^etre representes par un systeme d'equations dierentielles autonomes. Ces systemes seront notes _= ( ) (2) ou = ( 1 etudions des systemes n ) est un ve teur de dimension . Le fait que nous  implique que 1. Nous allons en parti ulier etudier les as = 2 et = 3. Les solutions des systemes dynamiques sont donnees par leur ot . La vitesse du

ot est de nie par le hamp de ve teurs . En general le ot  est appro he par des algorithmes numeriques. Mais es methodes ne sont pas tres pre ises dans le voisinage des singularites et elles ne permettent pas l'utilisation de parametres. Par ontre la plupart des equations dierentielles (m^emes les plus simples) n'ont pas de solutions expli ites. Dans ette these nous allons utiliser une autre appro he, propose par Poin are. L'etude expli ite analytique d'une equation dierentielle est rempla ee par une analyse qualitative. Etudier le omportement qualitatif des systemes dynamiques signi e les repartir dans des lasses d'equivalen e representant des systemes ayant le m^eme omportement. Cette

lassi ation est realisee a l'aide de dieomorphismes lo aux. Cela veut dire que l'analyse qualitative utilisant ette appro he fournit des resultats qui ne sont valables que lo alement dans le voisinage d'un point. Dans le as de systemes dynamiques de dimension deux ette analyse peut ^etre utilisee pour appro her des solutions algebriques. Des systemes dynamiques de la forme (2) sont souvent issus de la modelisation de problemes s ienti ques. La variable represente les quantites physiques on ernees qui hangent au ours du temps . Ces hangements sont ara terises par un systeme d'equations dierentielles. X

x ; ::: ;x

t

X

F

F X

f ; ::: ;f

n

n >

n

n

F

X

t

Exemple 1 (Pendule) Le omportement de nombreux mod eles en me anique est de rit par la troisieme loi universelle de la me anique F

En l'absen e de fri tion, les equations F a

=

ma :

= 2 sin = t 2 ( ) mg

lx

17

x

18

Resume par hapitre

x l

mg

Figure 2: Le pendule de l'exemple 1.

de nissent le omportement d'un pendule (voir gure 2). La variable m represente la masse du pendule, l sa longueur et g est la onstante gravitationelle. Nous pouvons de rire es relations par une equation dierentielle d'ordre deux x +

g sin x = 0 l

ar l'a

eleration a est egale a la deuxieme derivee du depla ement x. Ce i de nit un systeme d'ordre un en deux dimensions



x_ = y y_ = gl sin(x)

que nous obtenons en introduisant la variable y.

Dans ette these nous avons montre omment des outils omme les formes normales et les transformation quasi-monomiales peuvent ^etre utilises pour la redu tion et l'integration formelle des systemes dynamiques. Pour les systemes en dimension deux et les systemes non nilpotents en dimension trois les algorithmes proposes ont ete implantes et testes. Neanmoins, de nombreux problemes theoriques et pratiques restent a resoudre, surtout dans le domaine des systemes nilpotents.

Resume par hapitre Cette these est divisee en trois parties. Dans la premiere partie nous allons introduire des outils essentiels qui seront utilises pour la redu tion des systemes onsideres. Dans la deuxieme partie nous allons de rire les aspe ts algorithmiques de es redu tions. L'implantation en de es algorithmes est le sujet de la troisieme partie. Nous allons de rire les aspe ts de programmation et quelques exemples qui illustrent l'utilisation du logi iel implante. Nous detaillons i-dessous le plan de haque hapitre.

Maple

Premiere partie: Integration des systemes dynamiques Les solutions formelles des systemes dynamiques

Dans e hapitre nous introduisons les notions et les de nitions de base que nous utiliserons

onstamment par la suite. Elles peuvent egalement ^etre trouvees dans de nombreuses referen es sur les systemes dynamiques (voir par exemple les travaux de K. Alligood, T. Sauer et J. Yorke [39℄, F. Verhulst [26℄, J. Hale et H. Kor ak [33℄, S. Chow et J. Hale [14℄, J. Gu kenheimer et P. Holmes [36℄, D. Arrowsmith et C. Pla e [17℄ et J. Hubbard et B. West [34℄). Les systemes dynamiques et les hamps de ve teurs peuvent ^etre utilises pour representer la m^eme dynamique. Par onsequent la theorie des systemes dynamiques fait largement appel a la notion de hamps de ve teurs. En general les ots globaux des hamps de ve teurs ne peuvent pas ^etre al ules. Pour ette raison le al ul des ots lo aux et l'etude lo ale sont les buts prin ipaux de la theorie des systemes dynamiques. Si la partie lineaire d'un hamps de ve teurs existe dans le voisinage d'un point singulier elle- i peut ^etre equivalente ou onjuge au hamp de ve teurs non lineaire asso ie. Cela signi e que de nombreuses ara teristiques du hamp de ve teurs non lineaire peuvent ^etre deduits du

hamp linearise. Pour ette raison la notion d'equivalen e de hamps de ve teurs est tres importante dans l'analyse qualitative des hamps de ve teurs et des equations dierentielles. Nous allons souvent utiliser des hangements de temps. Ces transformations ne sont pas des transformations d'equivalen e. Par onsequent les hamps de ve teurs issus d'un

hangement de temps ne sont pas equivalents au hamp de ve teurs initial. Les ots al ules pour des systemes transformes par un hangement de temps ne peuvent en general pas ^etre transformes en des solutions du systeme initial. Pour ette raison nous introduisons la notion de ourbes de solutions. Ces ourbes representent des parametrisations du ot du systeme initial. Pourtant, dans ertains as, il existe une relation d'equivalen e entre le systeme de 19

20

Resume par hapitre

depart et le systeme transforme par un hangement de temps. Dans le as des systemes hamiltoniens les deux systemes ont les m^emes integrales premieres. Si nous onsiderons uniquement les systemes en deux dimensions, les deux systemes peuvent ^etre onsideres

omme provenant de la m^eme equation dierentielle s alaire. Ces relations peuvent egalement ^etre utilisees pour veri er les resultats al ules. Le al ul des formes normales que nous allons introduire au hapitre 3 donne souvent des series divergentes. C'est pourquoi nous nous pla erons souvent dans l'anneau des series formelles. Toutes les onsiderations seront alors purement formelles.

Le diagramme de Newton De nombreuses transformations que nous allons utiliser pour la redu tion des hamps de ve teurs peuvent ^etre interpretees geometriquement. De plus, les aspe ts geometriques des

hamps de ve teurs peuvent ^etre utilises pour trouver des transformations qui reduisent le hamp de ve teurs omme on le souhaite. L'outil le plus important pour l'analyse geometrique est le diagramme ou le polygone de Newton. Il peut aussi ^etre utilise dire tement pour al uler des solutions d'equations dierentielles (voir par exemple J. Della Dora and F. Jung [19℄, F. Beringer and F. Jung [6℄ and J. Cano [37℄). Neanmoins, pour les algorithmes que nous allons proposer i i le diagramme de Newton et le support d'un systeme vont ^etre utilises pour de nir des tranformations quasi-monomiales. Ces transformations sont un outil tres puissant pour manipuler des systemes dynamiques. Nous allons les utiliser pour reduire les singularites des systemes nilpotents (voir hapitre 4) et pour integrer des systemes qui sont sous forme normale (voir les hapitres 7 et 9). Les transformations quasi-monomiales manipulent les exposants des systemes on ernes. L'interpretation geometrique de es manipulations peut ^etre de rite par l'eet d'une transformation aÆne sur les exposants. Quelques resultats sur les transformations quasi-monomiales peuvent ^etre trouves dans les travaux de A. Bruno ([9℄ et [1℄). Pour valider l'utilisation des transformations quasi-monomiales nous devons prouver que e sont des dieomorphismes. Neanmoins ertaines transformations ne sont pas inje tives sur l'ensemble du domaine de de nition. Pour ette raison nous allons limiter e domaine de sorte que la transformation devienne inje tive. La surje tivite des transformations quasi-monomiales est assuree par une onstru tion qui les rend surje tives par mor eaux. Dans les hapitres 8 et 9 nous allons travailler sur des systemes ave des exposants negatifs. Ces systemes ne peuvent ^etre manipules que par e que leur support est in lus dans un ^one onvexe. Pour ette raison la manipulation des hamps de ve teurs est souvent fortement liee a la manipulations de ^ones dans l'espa e des exposants.

Les formes normales Comme nous venons de le mentionner, de nombreuses ara teristiques d'un hamp de ve teurs peuvent ^etre deduites de sa partie lineaire si elle- i existe. Cependant, dans

ertains as le hamp de ve teurs linearise n'est pas onjuge au hamps de ve teurs initial

ar il n'existe au un dieomorphisme permettant de lineariser le hamps de ve teurs. M^eme si ette linearisation n'est pas possible nous pouvons trouver des dieomorphismes qui simpli ent le hamp de ve teurs onsidere. Mais la de nition de e que "simple" veut

Resume par hapitre

21

dire n'est pas unique. Nous dirons qu'un hamp de ve teurs est "simple" lorsque sa partie lineaire peut ^etre de rite par une matri e sous forme de Jordan et que sa partie non-lineaire ne ontient que des termes resonnants. Un hamps de ve teurs simpli e est appele forme normale du hamp de ve teur de depart. Il existent de nombreuses appro hes aux formes normales. Nous allons utiliser la forme normale de Poin are-Dula ar elle peut ^etre integree en dimension deux (voir hapitre 7). Cette forme normale est due a H. Poin are [47℄, H. Dula [21℄ et G. Birkho [7℄. Elle peut ^etre al ulee en utilisant la methode de la representation adjointe due a G. Iooss [35℄ ou plus eÆ a ement en utilisant la theorie des transformations de Lie (voir K. T. Chen [40℄ et A. Deprit [3℄). Re emment, beau oup de rapports ont ete publies sur le sujet des formes normales. La premiere etape du al ul de la forme normale de Poin are-Dula est le al ul de la forme de Jordan de la matri e representant la partie lineaire. A ause de la

omplexite de la representation des nombres algebriques e i n'est pas un probleme trivial. La forme de Jordan peut ^etre al ulee a partir de la forme de Frobenius d'une matri e (voir I. Gil [29℄ et M. Griesbre ht [31℄). Un algorithme de al ul de formes normales pour des systemes ave une partie lineaire nilpotente a ete propose par R. Cushman et A. Sanders [16℄. Les formes normales peuvent aussi ^etre al ulees en utilisant les linearisations de Carleman (L. Stolovit h [56℄ et G. Chen [13℄).

Resolution de singularites par e latements Nous ne pouvons plus appliquer le theoreme de Poin are-Dula si la partie lineaire du systeme on erne est nilpotente. Dans e as nous allons utiliser des e latements. Ces e latements sont de nis par des hangements de variables qui deploient ou e latent la singularite. Ce pro ede nous donne un nombre ni de singularites plus simples. Des hangements de variables qui permettent d'e later une singularite sont par exemple l'introdu tion de oordonnees polaires ou ertaines transformations quasi-monomiales. Les e latemets dire tionnels ont ete introduits par O. Zariski [66℄ pour les ourbes algebriques et par A. Seidenberg [53℄ pour les systemes d'equations dierentielles de dimension deux. Depuis, de nombreux travaux ont ete realises sur e sujet. Pour nos

onstru tions nous allons utiliser les e latements quasi-homogenes qui sont de nis par des transformations quasi-momomiales et qui ont ete introduits par A. Bruno [9℄. Plus re emment, M.Brunella et M. Miari [8℄ ont travaille dans e domaine. Ils utilisent des transformations de nies par des matri es unimodulaires et travaillent essentiellement en dimension 2. Nous etendons la onstru tion des e latements a l'utilisation de toute matri e inversible. Ce i est ne essaire ar l'utilisation des matri es unimodulaires n'est pas appropriee a de nombreux problemes en dimension superieure. Nous pouvons utiliser

es matri es uniquement gr^a e aux resultats du hapitre 2 ou nous demontrons que es matri es de nissent des dieomorphismes.

Deuxieme partie: Les algorithmes La lassi ation Les algorithmes que nous allons proposer reduisent et integrent des systemes dynamiques pour dierents as. Ces as doivent ^etre traites separement. Don , la premiere etape de

22

Resume par hapitre

tous les al uls ee tues est la lassi ation des systemes on ernes. Nous allons distinguer le as d'un point regulier, le as d'un point singulier elementaire et le as d'un point singulier non elementaire, e qui nous permet de traiter les systemes onsideres ave des methodes onvenables. Les points reguliers

Dans le voisinage d'un point regulier nous pouvons reduire haque systeme a une forme tres simple que nous appellerons "forme normale". Elle peut fa ilement ^etre integree. Nous allons ramener le probleme du al ul de ette forme normale au probleme du al ul du ot d'un hamp de ve teurs au voisinage d'un point regulier. Le hangement de variables qui permet de mettre le systeme initial sous forme normale peut ^etre al ule par des series de Taylor. Cette methode peut ^etre trouvee dans de nombreuses referen es sur la theorie des transformations de Lie (par exemple dans W. Groebner [62℄, W. Groebner et H.Knapp [63℄ et P. Olver[46℄). Les points singuliers elementaires en dimension deux

Nous appelons un point singulier "point singulier elementaire" si la matri e ja obienne du systeme onsidere est non nilpotente. Dans e as nous pouvons appliquer le theoreme de Poin are-Dula pour al uler la forme normale de e systeme. En deux dimensions toute forme normale peut ^etre integree. Les systemes reels dont la matri e ja obienne a des valeurs propres imaginaires pures representent un as parti ulier. Les solutions de es systemes sont des solutions periodiques. La meilleure fa on de les representer est d'introduire des oordonnees polaires. Ce i nous permet egalement d'obtenir des solutions reelles a ondition que le systeme de depart soit reel. Les points singuliers non elementaires en dimension deux

Dans le as d'un point singulier non elementaire, nous allons utiliser des e latements et des hangements de temps pour reduire les systemes on ernes. Les al uls sont organises d'apres les fa es du diagramme de Newton. Pour haque sommet un hangement de temps permet d'obtenir un systeme non nilpotent dont le support est in lus dans un ^one onvexe. Comme les exposants du systeme obtenu sont negatifs nous devons appliquer une transformation quasi-monomiale qui nous fournit un systeme dont les exposants sont des entiers positifs. Ce systeme peut ^etre traite omme un systeme au voisinage d'un point singulier elementaire. Pour haque ar^ete du diagramme de Newton nous utilisons les e latements, deja introduit au hapitre 4. Ces e latements nous fournissent plusieurs systemes au lieu d'un seul systeme de depart. Ceux- i peuvent ^etre traites re ursivement. Ces methodes fournissent de nombreuses solutions. Un point entral de l'algorithme est don de de nir des domaines du voisinage etudie ou es solutions sont valables. Nous allons appeler es domaines "se teurs" et les al uler de fa on a e qu'ils re ouvrent entierement un voisinage du point etudie.

Resume par hapitre

23

Les points singuliers elementaires en dimension n

Les algorithmes que nous de rivons dans e hapitre n'ont jusqu'a present ete etudies que d'une maniere super ielle par A. Bruno [9℄. Nous utilisons ses travaux omme point de depart pour proposer une etude plus omplete. Dans une premiere etape la forme normale de Poin are-Dula du systeme on erne est al ulee. Celle- i peut ^etre transformee en un systeme que nous pouvons diviser en un systeme de dimension reduite et un systeme integrable. Neanmoins, la transformation quasi-monomiale utilisee dans e but doit veri er des onditions tres stri tes. Pour les systemes de dimension trois, nous proposons une lassi ation des formes normales qui permet la redu tion et l'integration de toute forme normale. Les resultats obtenus gr^a e a une etude approfondie des systemes en dimension trois peuvent ^etre generalises aux systemes de dimension superieure. Cependant, si la dimension du systeme onsidere est superieure a trois, de nombreux problemes peuvent appara^itre a ause de la dimension roissante du plan resonnant. Si la dimension du plan resonnant depasse deux, le ^one ontenant les exposants de la forme normale peut ^etre de ni par un nombre de ve teurs trop important et es ve teurs peuvent avoir des oordonnees negatives. Pour resoudre es problemes, nous proposons une

onstru tion omplementaire, le diagramme de Newton virtuel. Elle permet de onstruire un ensemble de ^ones qui possedent une stru ture plus reguliere. Cet ensemble de ^ones de nit des e latements qui permettent d'obtenir des systemes de dimension reduite et de

ouvrir entierement le voisinage etudie par des se teurs. Les points singuliers non elementaires en dimension n

Les e latements permettent egalement de traiter des systemes nilpotents de dimension trois et superieure. Neanmoins ertains systemes ne peuvent ^etre entierement reduits

omme l'ont demontre X. Gomez-Mont et I. Luengo [30℄. Seuls quelques as pre is somme les systemes non di ritiques peuvent ^etre reduits par ette methode (voir F. Cano et D. Cerveau [10℄). Pour es systemes nous proposons une onstru tion qui utilise des e latements ontrolees par le diagramme de Newton et son extension, le diagramme de Newton virtuel. Cette methode permet de de nir des se teurs qui ouvrent entierement le voisinage on erne. La de nition des se teurs obtenus par ette methode est dierente de elle proposee par A. Bruno. Sa onstru tion est fortement liee a la manipulation de

^ones dans l'espa e des exposants.

Troisieme partie: Le logi iel Les algorithmes de rits dans la partie pre edente ont ete implantes en pa kage FRIDAY2 .

Maple dans le

Le logi iel FRIDAY

Dans e hapitre nous donnons une des ription du pa kage FRIDAY et de nombreux exemples pour son utilisation. Ce pa kage est organise en modules orrespondant a la 2

FRIDAY est un a ronyme pour Formal Redu tion and Integration of Dynami al Autonomous Systems

24

Resume par hapitre

lassi ation des systemes dynamiques. Cependant, l'utilisateur n'a a

es qu'a ertaines pro edures de ontr^ole. La pro edure prin ipale peut ^etre utilisee pour l'integration d'un systeme quel onque en dimension deux et pour de nombreux systemes en trois dimensions. A part ette pro edure e sont surtout les modules al ulant les formes normales qui peuvent ^etre utilises separement. Dans ette these nous avons montre omment des outils omme les formes normales et les transformations quasi-monomiales peuvent ^etre utilises pour la redu tion et l'integration formelle des systemes dynamiques. Pour les systemes en dimension deux et les systemes non nilpotents en dimension trois les algorithmes proposes ont ete implantes et testes. Neanmoins, de nombreux problemes theoriques et pratiques restent a resoudre, surtout dans le domaine des systemes nilpotents.

Chapter 1

Formal Solutions for Dynami al Systems This hapter introdu es some de nitions that are basi for the developement of the theory in the following hapters. Consider a system of autonomous dierential equations of the form X = F (X ) (1.1) t where F = (f1 ; f2 ; : : : ; fn ), fi 2 C 1(M; C n ) and X = (x1 ; x2 ; : : : ; xn ) are ve tors of dimension n. The system is alled autonomous be ause the right hand side of equation (1.1) does not depend on the independent variable t that usually stands for time. We are looking for solutions represented by the dependent variable X (t). The results and notations presented in this hapter have been subje t of many publi ations. For example in the works from K. Alligood, T. Sauer and J. Yorke [39℄, F. Verhulst [26℄, J. Hale and H. Kor ak [33℄, S. Chow and J. Hale [14℄, J. Gu kenheimer and P. Holmes [36℄, D. Arrowsmith and C. Pla e [17℄ and J. Hubbard and B. West [34℄.

1.1 Flows, Ve tor Fields and Dierential Equations Let M be a onvex open subset of C n or Rn .

De nition 1 (global ow) A global ow on M is a ontinuously dierentiable fun tion  : R  M ! M su h that 8X 2 M 1. (0; X ) = X 2. (t; (s; X )) = (t + s; X ), 8t; s 2 R.

The ow is alled global be ause it is de ned for all t 2 R. It an be related to dierential equations by the de nition of ve tor elds.

De nition 2 (ve tor eld) A ve tor eld asso iated to a ow  is a fun tion F : M ! Rn , F 2 C 0 , de ned on the open subset M , that asso iates a ve tor in R n to any point in M su h that 8X 2 M : F (X )

=

d dt

(t; X )jt=0 = lim!0 25

 (; X )

(0; X )





:

26

1.2. Linearization of ve tor elds X (t) = (t; X0 ) is a solution of the initial value problem X_

= F (X ); X (0) = X0 . The existen e and the uniqueness of a lo al ow representing a solution is guaranteed by the following theorem. Theorem 1 (existen e and uniqueness) Let M be an open subset of Rn or C n and F : M ! Rn or C n be a ontinuously dierentiable map and let X0 2 M . Then there is some onstant > 0 and a unique solution X (t) = (t; X0 ) : ( ; ) ! M of the initial value problem X_ = F (X ); X (0) = X0 :

From de nition 2 it follows that every ow orresponds to autonomous dierential equation (1.1). The opposite is not true be ause in general the solutions of (1.1) an not be extended inde nitely in time. But for every autonomous dierential equation there an be found lo al ows de ned on a subset of R  M . De nition 3 (lo al ow) Let A be an open subset of R  M . A lo al ow is a ontinuously dierentiable fun tion  : A ! M su h that 1. f0g  M 2. 3.

AR M 8X 2 M : A \ (R  fX g) is onvex (0; X ) = X and 8t; s 2 R; 8X 2 M : (t; (s; X )) = (s + t; X ) if this expression makes sense.

As it an be dedu ed from above ows, ve tor elds and autonomous dierential equations an be used to represent the same dynami s.

1.2 Linearization of ve tor elds Solutions of dierential equations or ve tor elds are parti ulary interesting in the neighbourhood of isolated singular points as the behaviour of the solutions an be quite omplex there. De nition 4 (singular point) A point X0 2 M is alled singular point of a ve tor eld F if F (X0 ) = 0. It is an isolated singular point if X0 has a non-empty neighbourhood

su h that F (X ) 6= 0 8X 2 X0 .

A singular point is often alled equilibrium point or singularity whereas all other points are alled simple, regular or ordinary points. Any system (1.1) an be linearized in a point X0 . That means that instead of studying the behaviour of the nonlinear system we an study the linear system X t

= DF (X0 )X

(1.2)

where DF (X0 ) denotes the ja obian matrix of F in X0 . As linear systems are well known this is the simplest way to obtain information about a dierential equation (1.1).

Chapter 1. Formal Solutions for Dynami al Systems

27

The point X0 is alled a nonelementary singular point if X0 is a singular point and all eigenvalues of DF (X0 ) are zero. In this ase the matrix DF (X0 ) and the linear system (1.2) are alled nilpotent. If X0 is a singular point but DF (X0 ) is non-nilpotent we all X0 an elementary singular point of the system F . For the al ulation of solutions of dierential equations an important question arises. The question is if the linearized system (1.2) and the orresponding non-linear system (1.1) have lo ally the same ow stru ture. Under pre ise onditions the Hartman-Grobman theorem gives a positive answer to this question.

Theorem 2 (The Hartman-Grobman theorem) If none of the eigenvalues of the matrix DF (X0 ) has a zero real part then the ve tor eld F is topologi ally onjugated to the linearized ve tor eld DF (X0 )X in a neighbourhood of X0 .

The theorem gives no result for ve tor elds if the matrix DF (X0 ) has zero eigenvalues. In those ases ertain nonlinear parts of the ve tor eld have a determining role. If a nonlinear system is topologi ally onjugated to a linear one there exists a homeomorphism that linearizes the nonlinear system. But homeomorphisms are not ne essarily smooth and do therefore not preserve very well the qualitative behaviour of a on erned system. For this reason we have to use a stronger de nition of onjuga y and equivalen e that is based on dieomorphisms.

1.3 Equivalen e of ve tor elds To study the qualitative behaviour of ve tor elds or systems of dierential equations means to lassify them into equivalen e lasses of similar behaviour and to des ribe the

hara teristi s of those lasses. This lassi ation is done via dieomorphisms. Systems that an be transformed into ea h other by dieomorphisms are alled equivalent. Let M and V be open subsets of Rn or C n .

De nition 5 (dieomorphisms) Let M and V be open sets on E . A map H : M ! V is a C k -dieomorphism if

1. H is of lass C k and 2. H is invertible and its inverse is also of lass C k . C k -dieomorphisms an be used to de ne equivalen e or onjuga y relations.

De nition 6 (C k equivalen e of ve tor elds) Let F and F~ be two ve tor elds de-

ned respe tively on M and V , that are two open subsets of E .  and denote their lo al

ows. F and F~ are alled lo ally C k -equivalent if there exists a C k -dieomorphism H : V M whi h takes the ow of F~ to the ow  of F and preserves the orientation of the

ows. If in addition to this, the parametrization of the ows is preserved the systems are

alled C k - onjugated.

!

28

1.4. Time transformations

Applying a dieomorphism H : X 7! H (X ) to a dierential equation (1.1) yields ( ) X_ = F (H (X ))

DH X

and a new ve tor eld

~ = DH

F

1

(F Æ H )

(1.3)

that is onjugated to the ve tor eld F . Equation (1.3) de nes the a tion of a dieomorphism H on a ve tor eld F . F~ is often denoted H  F and the operator  is alled pull-ba k. A dieomorphism H su h that the relation (1.3) is veri ed an be found if the ve tor elds F and F~ are onjugated.

As it is mu h simpler to study ve tor elds in the neighbourhood of the origin than in the neighbourhood of a point X0 we will often use dieomorphisms given by a translation

Example 2 (Translations)

( ) = X + X0 :

H X

H

transforms any ve tor eld

F

into the onjugated ve tor eld H

The ve tor eld F~ is similar to DH = id.

F

 F = DH

ex ept that

1

(F Æ H ) :

X0

has been transformed to the origin as

In the Hartman-Grobman theorem the notation of topologi al onjuga y is used. Topologi ally onjugated means C 0 - onjugated and refers to C 0 -dieomorphisms that are homeomorphisms.

Remark 1

Transformations de ned by dieomorphisms are the main tool for the redu tion of ve tor elds. All normalizing transformations and power transformations that are used in the following are dieomorphisms. 1.4

Time transformations

A dynami al system des ribes the one-parameter evolution of several dynami variables. The parameter in whi h they evolve is alled time t. At ertain steps of the proposed algorithms we will apply transformations to t. These transformations are alled time transformations. In ontrast to previously introdu ed transformations by C k -dieomorphisms time transformations do not preserve the orientation of ows. The resulting ve tor elds are therefore not equivalent to the initial ve tor elds. For some ve tor elds there exists a fa torization ( ) = h(X )F~ (X )

F X

with the fa tor h(X ). It is mu h simpler to study the ve tor eld F~ that results from a division of F by h instead of studying F . F~ an be seen as a result of an appli ation of a

Chapter 1. Formal Solutions for Dynami al Systems

29

time transformation to F . If we substitute h(X )t by  t~ the dierential equation (1.1) is transformed to a new equation X ~ (1.4) ~ = F (X ) : t

The hange of variables  t~ = h(X )t is equivalent to a hange of variables t~ = (t)

where is the solution of the dierential equation  t~ = h(X ) : t

(1.5)

The solution X (t) of the initial system an easily be al ulated from the solutions X (t~) of equation (1.4) if is known. Example 3 (Time transformations)



that an be transformed to the system

(

Consider the 2 dimensional system dx dt dy dt

dx dt~ dy dt~

= x2 = xy

(1.6)

=x =y

(1.7)

by the time hange given by  t~ = ht with h = x. The solutions of (1.7) are given by X (t~) = (a et~; b et~). Solving (1.5) yields

(t) =

and the solution of the system (1.6) is



ln( a(t + ))



b 1 1 ; : t+ a t+

Observe that the behaviour of the solutions (x(t); y(t)) for t ! 1; ; 1 is dierent from the behaviour of (x(t~); y(t~)) for t~ ! 1; ; 1. As a onsequen e the dedu tion of some qualitative hara teristi s su h as stability of the solutions of F from the behaviour of X (t~) is impossible if is not known. The time hange t~ = (t) alters the parametrization of the solution urves and it is easy to see that the orientation of the ow is not always preserved. F and F~ are therefore neither equivalent nor onjugated. The only ommon hara teristi between the solutions of F and F~ is that they are dierent parametrizations of the same urve. However we an de ne an equivalen e relation between the ve tor eld F and the ve tor eld F~ in the ase of Hamiltonian ve tor elds and if we treat two-dimensional systems. Those equivalen e relations are important for the veri ation of omputed solution urves.

30

1.4. Time transformations

For two dimensional systems al ulating solutions ( ) for   ( ) = ( ) (1.8)

an be redu ed to the problem of nding parametrized solutions for the s alar dierential equation = (( )) (1.9) X t

X

f x; y

t

g x; y

dy

g x; y

dx

f x; y

:

This equation an be parametrized as in equation (1.8) or as  ( ) ( ) ~= ( ) f x;y

X

h x;y

t

g x;y

(1.10)

h(x;y )

whi h is equivalent to the appli ation of a time hange with ~ = ( ) to equation (1.8). We an therefore say that the systems (1.8) and (1.10) are equivalent in the sense that their solutions are both parametrized solution urves for the s alar equation (1.9). Another equivalen e relation between and ~ an be given if we onsider Hamiltonian systems of dierential equations. A Hamiltonian system is hara terized by its energy fun tion . The dependent variables are given by = ( ) where and are ve tors. A Hamiltonian system is given by  _ = (1.11) _= A major property of Hamiltonian systems is that they own rst integrals. t

F

h x; y  t

F

H

X

Y; Z

Y

Z

H

Y

Z H

Z

Y

:

De nition 7 ( rst integral) let U be open and nonempty. A real valued map

 : ! R; U

X

7! ( )  2 X ;

C

1

that is not onstant on any open subset of R is alled a rst integral of a dierential equation (1.1) if the fun tion  is onstant along any solution X (t) with initial value X (0) = X0 . n

( ( )) = ( 0 ) X t

for all t for whi h X (t) is de ned.

X

(1.12)

It is obvious that the Energy fun tion is a rst integral of the Hamiltonian system (1.11) as ( ( ) ( )) = 0 for any ( ( ) ( )) solving equation (1.11). Applying a time hange to a Hamiltonian ve tor eld yields a ve tor eld that is equivalent to the initial one in the sense that it has the same rst integrals. This is due to the fa t that the ondition (1.12) does not depend on the parametrization of the urve ( ). Most systems of dierential equations however do not possess rst integrals. For this reason this on lusion an not be generalized to non-Hamiltonian systems. H

H Y

t ;Z t

t

Y

X t

t ;Z t

Chapter 1. Formal Solutions for Dynami al Systems

31

Figure 1.1: The level urves of H and the solution urves of the dynami al system (1.13) are identi but the level urves have no dire tion. See example 4. Example 4 (pendulum)

s alar equation

The pendulum equation is either given by the se ond order g x + sin(x) = 0 l

or by the rst order system



x_ = y : y_ = gl sin(x)

(1.13)

The points (k; 0) with k 2 Z are singularities of (1.13). In hapter 7 we will approximate solution urves in the neighbourhood or these singularities. The total energy of the system (kineti plus potential energy) is given by H (x; y ) = (1=2)ml2 y 2 + mgl(1

os(x)) :

H is onstant along the solutions (x(t); y (t)) of the system and along any parametrization (x(t~); y(t~)) with t~ = (t) of those urves. In ontrast to the real solutions (x(t); y(t)) it makes no sense to give a dire tion to the urves (x(t~); y(t~)), that denote the lines where

the total energy of the system is onstant, as the sense of parametrization might have been inversed by the time transformation. The urves (x(t~); y(t~)) are alled level urves of H . Level urves and the solutions of equation (1.13) are sket hed in gure 1.1 for g=l = 1=2.

Time transformations will be extensively used in the following. Therefore all al ulated solution urves an only be interpreted as level urves of the energy fun tion if we treat Hamiltonian systems or parametrized solutions of the asso iated s alar dierential equations if we deal with 2-dimensional systems. Those properties are used to verify omputed solution urves. That also means that for 3 dimensional systems the veri ation of

al ulated results is mu h more diÆ ult.

32 1.5

1.5. Convergen e and Formal Solutions Convergen e and Formal Solutions

Normal form al ulations, that will be introdu ed in hapter 3, often yield diverging series. Therefore the ring of onvergent power series kfX g is not suÆ ient for our al ulations. The following al ulations and onsiderations are purely formal. We work in the ring of formal power series k[[X ℄℄, that extends the ring of polynomials k[X ℄, as it also admits in nite sums without presuming that they are onverging. The aspe ts of onvergen e will not be onsidered here though onditions for the onvergen e of normal form transformations have been given by H. Poin are [47℄, H. Dula [21℄, C. Siegel [54℄, A. Bruno and S. Wal her [2℄ and others. We will mainly work with systems of rst order dierential equation of dimension 2 and 3. First order means that only the rst derivative o

urs in the equation. However higher order dierential equations an always be transformed to a system of order one. As we will see with some restri tions the algorithm an also be used for problems that admit parameters. The systems of dimension two are very well known due to a large number of publi ations in this domain. Three dimensional problems have so far not been studied extensively using the approa h proposed here. So far there didn't exist any programms for integrating a large number of 3 dimensional systems of dierential equations.

Chapter 2

The Newton diagram In this hapter power transformations are introdu ed. Those transformations a t on the exponents of the monomials of a given system _ = F (X )

X

:

(2.1)

Their ee t on the exponents of a given system an be interpreted geometri ally for a better illustration of the a tion of those transformations. Further those geometri aspe ts

an be used to nd power transformations that manipulate the exponents of a system in an appropriate way. Therefore some geometri notations as the Newton diagram and the support of a system are needed. In hapter 4 we will use the Newton diagram to al ulate matri es that de ne power transformations as it has been done by A.Bruno [9℄ and more re ently by M. Brunella and M. Miari [8℄. Se tion 2.2 is losely related to the work of A. Bruno [1℄ who states theorem 3. In his approa h he on entrates on the use of unimodular matri es. However unimodular matri es are not suÆ ient to solve all problems on erning the integration of two- and higher-dimensional systems as it will be shown in the hapters 8 and 9. For this reason we extend the de nition of power transformations to the use of any invertible matrix as they have also been studied by L. Brenig and A. Goriely [23℄. As some of those transformations are not inje tive and therefore no dieomorphism we will introdu e some omplementary methods to make them bije tive. Another important role in the geometri study of dierential equations is played by

ones as some systems an have their support within a one. That is why in se tion 2.4 we introdu e some basi notations about ones that an for example be found in A. Goldman and A. Tu ker [4℄. 2.1

The support and the Newton diagram

Power transformations are applied to a system in order to integrate it in the ase of an elementary singular point in se tion 7.1 or in order to simplify it by blowing-ups that are the subje t of hapter 4. To study the geometri al aspe ts of power transformations we will work in the spa e of exponents (whi h is subset of Zn) where the support and the 33

34

2.1. The support and the Newton diagram

Newton diagram for any dynami al system are de ned. To simplify the representation the notations X = (x1 ; x2 ; : : : ; xn ) ;

Q = (q1 ; q2 ; : : : ; qn ) 2 Zn ; X Q = (xq11 ; xq22 ; : : : ; xqnn ) ; A = (aij ) 2 Mn (Z) ;

0 XA = 

xa111 xa212 : : : xan1n ::: a n 1 an2 x1 x2 : : : xannn

1 A;

0 x P Q X Q 1 Q2N A ; Qi 2 R or C F (X ) =  P : : : Q x  X where the sets Ni are de ned as

1

1

n

Q2Nn Qn

1

Ni = fQ 2 N i  N [ f 1g  N n i : 1

will be used. Further the set N de ned as

N=

[n i=1

X

qi  0g

Ni

will frequently appear. Based on the above notations the support and the Newton diagram for F are de ned as follows :

De nition 8 (Support)

The set

supp(F ) :=

[n i=1

fQ 2 Ni : Qi 6= 0g  Zn

is alled the support of the system (2.1).

The de nition of the support allows to onstru t the set

=

[

fQ + P : P 2 Rn g

Q2supp(F )

+

that is used to de ne the Newton diagram.

De nition 9 (Newton diagram) The lower left part of the onvex hull of the set is

alled the Newton diagram of F . It does not ontain horizontal or verti al fa es. It is denoted by (F ).

Chapter 2. The Newton diagram

35

(F ) onsists of a nite number of j -dimensional fa es that are denoted by (ij ) . The (0) fa es (1) i and i are alled edges and verti es of the Newton diagram. The fa es of the Newton diagram will be used to ompute matri es de ning quasihomogeneous blowing-ups in hapter 4. The Newton polygon whi h is almost identi al to the Newton diagram an also be used in a dierent way. It an be applied for the dire t al ulation of solutions of algebrai or dierential equations (see for example J. Della Dora and F. Jung [19℄, F. Beringer and F. Jung [6℄ and J. Cano [37℄). Example 5 Consider the system

_ = X



x

4

+ yx3

13 2 6 y x 9

 2 2

x y

+ xy3

whi h has a nonelementary singularity in the origin. Its support onsists of the points

( ) = f(3; 0); (2; 1); (6; 1); (1; 2)g

supp F

and its Newton diagram is shown in gure 2.1.

2.2

Power transformations

Power transformations are de ned as



n ! kn ; k = R AT : X 7! X k

or k

T denotes the transposed of an invertible matrix

=C

2

(2.2)

n (TZ) with integer oeÆ ients. A ~ We use those transformations as oordinate hange X = X . The ee t of the oordinate hange on the exponents of the system is des ribed by the following theorem. A

A

Gl

T Theorem 3 A hange of oordinates X = X~ A applied to a system (2.1) indu es an ~ = AQ on the points Q 2 supp(F ). aÆne transformation Q

Proof 1 To prove this theorem we will study the system 

log X t

= G(X ) = (

f1 x1

;:::;

f

n) n

x

36

2.3. Power transformations as dieomorphisms

that is equivalent to (2.1). Under the hange of oordinates



0 P = B  P

T

~A log X t

~ log X

(a11 ;a21 ;:::;an1 )

t

:::

~ log X

(a1n ;a2n ;:::;ann ) x ~i

0 B = B 

~ X

~ X

1 C A

x ~i

x ~i

a11 X

(a11 ;:::;an1 )



1

(a1n ;:::;ann )

an

a1n X

~1 1;a21 ;:::;an1 )  x t

~1 1;a2n ;:::;ann )  x

(a1n

t

t

a n x

ann

X

xn

t

t

~  log X

A

log X~ t

nn

=A

A simple al ulation shows that ea h monomial

~

X

belonging to

(~

G X

T

).

+:::

t

t



A

 1 C C A

~ T log X

and the new system

AQ

+ :::

1 C A

~  log X

n

=

we obtain

X

xn

n

T

:::

x

n

A

x ~i

(a11

0 1 ~ ( ~111 ; : : : ; ~ 1 ) B C .. C = B .  A ~ 1 ( ~1 ; : : : ; ~ ) 0 (a11 ; a21 ; : : : ; a 1 ) ::: = B  (a1 ; a2 ; : : : ; a ) a

= X~

t



1

X

T

(~

G X

X

Q

T

A

in

): ( ) is transformed to a monomial

G X

Theorem 3 is used to al ulate appropriate matri es for power transformations. Finding a matrix A that handles the support or the Newton diagram in a suitable way makes sure that exponents of the system transformed by the power transformation X = X~ have the required ara teristi s. T

A

2.3 Power transformations as dieomorphisms The appli ation of power transformations in order to transform a ve tor eld implies that the oordinate hange must be a dieomorphism. However most power transformations are not inje tive in any neighbourhood U of the origin in R or C but we will show that they are in some regions of U . If the power transformations are de ned on those regions instead of U their use is allowed. The methods employed to nd those regions are dierent in C and R . For this reason both ases are treated separately. S fX jx = 0g where For the following onsiderations we work in the set U 0 = U none of the omponents of X is zero. The sets fX jx = 0g will be onsidered in hapter 4. They play the role of ex eptional divisors for a spe ial kind of power transformations that are alled blowing-ups. n

n

n

n

i

i

i

Chapter 2. The Newton diagram

37

(0) 2 (1) 1

R

(0) 1

Figure 2.1: The Newton diagram for the system treated in example 5. 2.3.1

Power transformations in

R

n

f 2

T

In a rst step it will be shown that X is inje tive within the set X U : x1 ; : : : ; x T T 0 . Suppose that X0 = X1 . This an also be written as a system of equations A

g

A

A

x

x

11 xa21 : : : xan1

=

x

12 xa22 : : : xan2

=

x

a

01

02

a

01

0n

02

0n

0

>

n

11 xa21 : : : xan1

a

11

12

1n

12 xa22 : : : xan2

a

11

12

1n

:::

x

1n a2n 01 x02 a

:::x

nn 0n

1n a2n 11 x1n

=

a

a

x

:::x

nn 1n a

:

Exponentiating the k-th lines with a11 and dividing them by the rst line exponentiated by a1 leads to a new equation system k

x

11 xa21 : : : xan1

a

01

02

=

0n

~22 ~n2 1 x02 :::x 0 :

a

11 xa21 : : : xan1

a

x

11

12

1n

~22 ~ n2 = 1 x12 :::x 1

a

:

n

a

a

n

:::

~2n ~nn 1 x02 :::x 0 :

a

= 1 x1~2n : : : x1~nn :

a

:

n

a

n

a

n

6

2 f1

Further exponentiations and divisions (that are allowed as x1 ; x0 = 0 for all i leads to X0 = X1 where R is an upper triangular matrix. Be ause x0 > 0 and x1 > 0 for all i we an on lude that X0 = X1 . This result an be generalized to any of the 2 quadrants i

R

i

g)

;:::;n

R

i

i

2 f1

;:::;n

n

f 2 X

U

0

: x 1; : : : ; x l i

i

>

0;

xi

l+1 ; : : : ; xin


0; z > 0g :

Chapter 2. The Newton diagram

39

that ontains only two quadrants. X A : U^ ! Rn is inje tive. T The list image allows to on lude that the image of the power transformation X A is the set fX : x > 0; y > 0g. That means we also have to use the 3 transformations T

0( X 7!  (

1)j1 X ( 2;0;4) 1)j2 X (2;0; 2) ( 1)j3 X (1;1; 2)

1 A

with

j = (j 1 ; j 2 ; j 3 ) 2 f(1; 0; 0); (0; 1; 0); (1; 1; 0)g that are all de ned on U^ , to onstru t a pie ewise surje tive transformation.

2.3.2

Cn

Power transformations in

Handling the ase of transformations de ned in C n is less ompli ated as U^ does not

onsist of a set of quadrants and any inje tive transformation is surje tive. T Suppose that X A is not inje tive. That means there exist X0 6= X1 su h that T

T

X0A = X1A : Xj an be written in trigonometri form

0 Xj = B 

rj 1 eij1 .. .

rjn eijn

(2.4)

1 C A

with rjk > 0 and jk 2 [0; 2[. With Rj = (rj 1 ; : : : ; rjn ), j = (j 1 ; : : : ; jn ) and k = (k1 ; : : : ; kn ); kj 2 Z the ondition (2.4) holds if the 2n onditions T

R0AT

= R1A

AT 

= AT 1 + k 2

0

(2.5)

are veri ed. That means that the arguments modulo 2 have to be equal. As rjk > 0 the rst n onditions are veri ed only if R0 = R1 so we an fo us on the remaining

onditions on the arguments. The se ond ondition in equation (2.5) an be multiplied by 1 A where A denotes the adjoint matrix to AT . All entries of AT are integer. A T = detA That yields 2  A k: (2.6) 0 1 = Any power transformation X

AT

: U^

detA

! C n is inje tive for U^ = Ui0 with

Ui0 = fX~ 2 U 0 : arg(~xi )
0 (A T )j mod 2 [0; 2[ j = i 2 (A T )i mod 2 [0; detA [

2 2

86

have to be veri ed. (:)j denotes the j-th omponent of a ve tor. The rst two onditions are always true so only the last ondition has to be proved. 2 [. Equation (2.6) Without loss of generality let i = 1. Suppose that (A T )1 [0; detA T 0 A ~ ~ will be used to nd another X U su h that X = X . For any l Z there exists a k0 su h that (A k0 )1 = l and an l Z su h that



(A

De ning

62

2 2

T ) + 2 1

detA



l mod 2

= A T +

2

detA

2

2 2 [0; detA [:

A k0

T we an easily verify that AT = . It has been proved that the transformation X A is bije tive on the set U 0 . T

Example 7 The power transformation from example 6 is bije tive if it is de ned as X A : U10 U 0 with  U10 = X : arg(x) mod 2 [0; [ : 2

!

f

2

g

Remark 2 Power transformations de ned by unimodular matri es are bije tive. For this reason their use for de ning power transformations is advantageous in many ases.

For the power transformations used in the following it an be assumed that they are bije tive on U 0 or that they have been de ned on a subset U^ of U 0 su h that they are bije tive. This se tion has shown that with some additional onstru tions any power transformation is a dieomorphism and an therefore be used to transform ve tor elds.

Chapter 2. The Newton diagram 2.4

41

Cones

In the following a ertain type of ve tor elds will play an important role: ve tor elds having their support within a one V in the spa e of exponents. Those systems are often refered to as lass V -systems. Time- and power transformation that are used to transform a onsidered system (2.1) to an equivalent system an also be interpreted as manipulations of the one V that ontains supp(F ). Further ones an be used to de ne transformations that manipulate a given system in an appropriate way. A set V  Rn is alled a one if along with the point P it ontains any point P with   0;  2 R. Any one an be de ned by a set of ve tors Q0 ; Q1 ; : : : ; Qm as the set m (2.7) V = fQ : Q = Q0 + i Qi ; i  0; i 2 Rg : i=1

X

A one is alled onvex if it is a onvex set. All ones onsidered in the following are

onvex ones. A one V is alled degenerate if it ontains an entire line P with  2 R. Consider a time transformation de ned by dt~ = X Q0 dt that transforms the initial system (2.1) to a system 1 X = F (X ) :  t~ X Q0 It is obvious that ea h point Q 2 supp(F ) is translated to the point Q~ = Q Q0 . With the points of the support of F the time transformation also tranlates the Newton diagram of F or any one V .

The ee t of power transformations on ones is more omplex. Consider a onvex degenerate or non-degenerate one V that is de ned as in equation (2.7) by the ve tor Q0 and the linearly independent ve tors Q1 ; : : : ; Qm 2 Zn where of ourse m  n. A hange T of oordinates X = X~ A an be de ned via the inverse of the matrix A. Using the ve tors Q1 ; : : : ; Qm as the rst row ve tors of A 1 yields the matrix

A

1

= (Q1 j : : : jQm j : : :)

that is ompleted to a n  n matrix su h that A 1 is invertible. T A

ording to theorem 3 the oordinate hange X = X~ A transforms all ve tors Qi ; i = 1; : : : ; m to Q~ i = AQi = detA ei : Therefore the one V has been transformed to a one

V~ = fQ : Q = AQ0 +

X ~iei; ~i  0; ~i 2 Rg

that is de ned by the ve tors e1 ; : : : ; em and AQ0 . However, the matrix A 1 might not be unimodular and A might therefore have fra tional oeÆ ients. For this reason instead of the inverse matrix the adjoint matrix A = detA A 1 = (Q1 j : : : jQm j : : :)

42

2.4. Cones

is used to de ne the power transformation. The matrix A is not uniquely de ned by  and an therefore be hosen su h that it has only integer oeÆ ients. The matrix A  of any matrix B is de ned A is omputed in the following way: The adjoint matrix B k  1 as B = detB B . Now de ne A = detB B with k 2 N su h that 8i; j : aij 2 Z and g d(a11 ; : : : ; ann ) = 1 for the oeÆ ients aij ; i; j = 1; : : : ; n of the matrix A.  is its adjoint as detA = k A is the matrix used for the power transformation and B yields  1 A = kA = det B B 1 = B  : So for A = B  the matrix A is the appropriate matrix for the power transformation.

Chapter 3

Normal Forms The theory of normal forms is due to H. Poin are [47℄ who introdu ed qualitative methods in the study of solutions of ordinary dierential equations. To study the qualitative behaviour of a systems of dierential equations

X_ = F (X )

(3.1)

means to lassify them lo ally into equivalen e lasses of similar behaviour. The lassi ation is performed by formal dieomorphisms. For systems with non-vanishing linear part the orresponding lasses an be represented by a set of elements that are said to be in normal form. These elements are the "simplest" elements of their lass. In general these representative elements are not unique and their hoi e depend on the de nition of what "simplest" means. In the following hapters the Poin are-Dula normal form is used. Here "simplest" means that the matrix hara terizing the linear part of the system is in Jordan form and the nonlinear part of the system ontains as few terms as possible. The omputation of the Jordan form is a diÆ ult problem for higher-dimensional matri es. However it represents the rst step in the omputation of the Poin are-Dula normal form. The redu tion of the non-linear terms is performed step by step for terms of in reasing degree. The omputations yield a normal form and the formal dieomorphism that is used to normalize the onsidered system (3.1). There exist many approa hes for the omputation of normal forms. In this hapter we will fo us on the onstru tion of the Poin are-Dula normal form using Lie transformations and the matrix representation method. Normal forms have been the subje t of many publi ations. The basi results of H. Poin are [47℄ have been extended by H. Dula [21℄ and G. Birkho [7℄. Lie theory has been introdu ed to normal form theory by K.T. Chen [40℄ and W. Groebner [62℄. The

on erning algorithms have been optimized by A. Deprit [3℄. Re ently normal form theory has rapidly developped sin e it is essential in bifur ation theory. See for example the works of A. Bruno [9℄, J. Gu kenheimer and P. Holmes [36℄, S. Wal her [65℄, S. Ushiki [58℄, F. Takens [25℄, Shui-Nee Chow, Chengzhi Li and Duo Wang [15℄, G. Gaeta [27℄, S. Louies and L. Brenig [50℄, L. Vallier [60℄, G. Iooss and M. Adelmeyer [35℄ and F. Zinoun [67℄. There exist other algorithms for normal form omputations that are however not onsidered here. For example the omputation of normal forms by Carleman linearizations (J. 43

44

3.1. The Poin are-Dula normal form

Della Dora and L. Stolovit h [20℄ and G. Chen [13℄) and the omputation of normal forms for systems with nilpotent linear parts (R. Cushman and J. Sanders [16℄). The problem of

omputing the Jordan form an be avoided by using the Frobenius form of a matrix (G. Chen [13℄). The omponents of the map F in equation (3.1) are onsidered to be formal power series. Therefore the only possible singularity is X = 0. As a onsequen e we will suppose that F (0) = 0. All transformations are onsidered to be formal power series.

3.1

The Poin are-Dula normal form

The basi theory of normal forms is due to H. Poin are [47℄. He stated that systems of the form (3.1) are formally equivalent to their linearized system if the eigenvalues of the on erned system are non-resonant. In this ontext resonan e is de ned as a relation between the points Q 2 N that an appear in the support of the ve tor eld F and the eigenvalues of the matrix DF (0). The set N de nes the set of all points that an appear in the support of a ve tor eld F . It has already been de ned in se tion 2.1. De nition 10 (resonan es) Let 1 ; : : : ;  be the eigenvalues of the matrix DF (0). n

They verify a resonan e ondition if

9Q = (q1 ; : : : ; q ) 2 N : hQ; i = n

P

Xq  = 0 n

i

i=1

i

(3.2)

where  = (1 ; : : : ; n ). They verify a resonan e ondition of order k if the ondition (3.2) holds and if jQj = qi = k 1.

Now the Poin are theorem an be formulated as follows: Theorem 4 (Poin are theorem) If the eigenvalues of the matrix DF (0) are non-resonant the system (3.1) is formally equivalent to its linear part. That means that the nonlinear system (3.1) an be redu ed to a linear system X~ = DF (0) X~

by a formal hange of oordinates X = H (X~ ).

The Poin are theorem has been extended by H. Dula [21℄ to systems whose matri es

DF (0) have resonant eigenvalues. He states that any system (3.1) an be redu ed to a system X_ = F~ (X ) where the matrix DF~ (0) is in Jordan form and the nonlinear part

ontains only resonant terms.

Theorem 5 (Poin are-Dula theorem) The dierential equation (3.1) an be redu ed to a system

 X~ = F~ (X~ ) = J X~ + W (X~ ) (3.3) t by a formal hange of oordinates X = H (X~ ). In equation (3.3) the matrix DF~ (0) = J is in Jordan form and F~ ontains only resonant terms. That means that the resonan e

ondition

8Q 2 supp(F~ ) : hQ; i = 0

holds for all exponents of the normal form.

Chapter 3. Normal Forms

45

The transformation X = H (X~ ) is alled normalizing transformation. It an be de omposed into H (X~ ) = P X~ + V (X~ ) where P X~ denotes its linear part and V (X~ ) its nonlinear part. The matrix P is the transition matrix that transforms the matrix DF (0) into Jordan form J =P

1

DF (0)P :

The new system (3.3) is alled Poin are-Dula normal form of the initial system (3.1). The parti ular stru ture of its support is used for further redu tions or for the integration of normal forms. The points of supp(F~ ) lie on the so alled resonant plane. De nition 11 (resonant plane)

The set of points

M = fQ 2 N : hQ; i = 0g is alled the resonant plane for a normalized system (3.3). It is a subset of the spa e of exponents and represents all points that an appear in the support of a normal form.

It is obvious that resonan es o

ur for all Q 2 N if 1 = : : : = n = 0. For this reason the Poin are-Dula normal form only yields a redu tion for systems (3.1) with non-nilpotent linear part. However there exist normal form onstru tions for nilpotent systems with non-vanishing linear part (see R. Cushman and J. Sanders [16℄). EÆ ient algorithms for the omputations of the Poin are-Dula normal form will be introdu ed in the following. These algorithms use Poin are- or Lie-transformations. However the Poin are and the Poin are-Dula theorem are the basis for all redu tions and integrations on erning normal forms that are used in the following hapters. 3.2

The Jordan form

The rst step in al ulating the Poin are-Dula normal form is the al ulation of the Jordan form of the matrix DF (0) that represents the linear part of the system (3.1). Let J =P

1

DF (0) P

be the Jordan form of the matrix DF (0). Then the linear hange of oordinates X = P X~ yields a system  X~ = F (X~ ) t

where the matrix DF~ (0) = J is in Jordan form. However due to the problems of representation of algebrai numbers the omputations of the Jordan form for n  n matri es with n > 2 is diÆ ult. In this ase the omputations an be performed by algorithms that are based on works from I. Gil [29℄, P. Ozello [44℄ and M. Griesbre ht [31℄.

46

3.3

3.3. Computation of normal forms by Poin are Transformations

Computation of normal forms by Poin are Transformations

To understand the basi idea of normal form omputations we will study the ee t of so

alled Poin are transformations to a given system (3.1). Poin are transformations have the form ~ + Hk (X~ ) X = X (3.4)

where Hk 2 Hnk . Hnk denotes the produ t of n opies of the spa e of homogeneous polynomials of degree k with n variables x1 ; : : : ; xn . The ee t of Poin are transformations on the initial system (3.1) is omputed step by step for in reasing degree k  2. To study this ee t the Taylor expansion _ = AX + F2 (X ) + F3 (X ) + : : :

X

(3.5)

of system (3.1) with Fk 2 Hkn is onsidered. Applying a Poin are-transformation (3.4) to the ve tor eld (3.5) yields the so alled homologi al equation for the terms of degree k of the resulting normalized system. Cal ulating normal forms an be redu ed to the problem of nding solutions for the homologi al equation. This problem an be solved for example by using the matrix representation method. 3.3.1

The homologi al equation

The ee t of Poin are transformations on a system (3.1) or (3.5) an be omputed straightforward. The resulting equation that is alled homologi al equation is basi to all normal form theory. Introdu ing the Poin are transformation (3.4) into equation (3.1) yields the new system ~ X = (I d + DHk (X~ )) 1 F (X~ + Hk (X~ )) = F~ (X~ ) (3.6) t

that has been omputed a

ording to H  F = (DH ) 1 F (H ). This expression an be simpli ed by introdu ing 1 (I d + DHk (X~ )) 1 = ( DHk (X~ ))i = I d DHk (X~ ) + O(X~ k ) : i=0

X

Now the system (3.6) an be written as ~

X t

= (I d

( ~ ) + O(X~ k )) 1 F (X~ + Hk (X~ )):

DHk X

Ordering all terms a

ording to their degree yields

~ X t

=

~ + F2 (X~ ) + : : : + Fk 1 (X~ ) +(Fk (X~ ) + AHk (X~ ) DHk (X~ )AX~ ) +O(X~ k+1 ): AX

(3.7)

This equation an be de omposed into three parts. The terms of degree lower than k remain un hanged in equation (3.7). The terms of degree higher than k are hanged but

Chapter 3. Normal Forms

47

they are not onsidered at this step of the algorithm. The term F~k (X~ ) of the normal form is obtained by the relation ~k (X~ ) = Fk (X~ ) + AHk (X~ )

(3.8) k (X~ )AX~ : The task onsists in nding an appropriate Hk su h that F~k is "as simple as possible". For the Poin are-Dula normal form that means to nd Hk (X~ ) su h that a maximum of terms in equation (3.8) vanish. For this purpose we try to solve the equation F

k (X~ ) = DHk (X~ )AX~

F

DH

AH

k (X~ )

(3.9)

that is alled the "homologi al equation". An equivalent formulation ~ ) = LkA (H ) Fk (X

an be given by introdu ing the linear operator  k Hn ! Hnk k LA : ~ ) 7! LkA (Hk ) = DHk (X~ )AX~ Hk (X

AH

k (X~ ):

that is alled the homologi al operator. The subs ript means that the linear operator k only makes use of the informations available from the linear part A of the system L A and k refers to the degree of the polynomials in Hk . Theoreti ally the problem of using Poin are-transformations for al ulating normal forms redu es to al ulating the inverse of the operator LkA(H ). If the eigenvalues of LkA (H ) do not ontain zero the operator LkA (H ) is invertible and equation (3.9) an be solved. If LkA (H ) is not invertible the spa e Hnk is split into Hnk = Rnk (A)  Cnk (A) where Rnk (A) denotes the range of LkA (H ) and Cnk (A) a omplementary spa e. The terms belonging to Cnk (A) an not be removed. In the ase of the Poin are-Dula normal form those terms are the resonant terms. The normal forms are not unique sin e Cnk (A) is not uniqueley determined. Several methods an be used for nding the omplementary subspa es C k for a given matrix A. We will use the matrix representation method. 3.3.2

The matrix representation of the homologi al operator

A possible method for omputing the subspa e Cnk (A) is the matrix representation method. In liteature this method an be found for example in Shui-Nee Chow, Chengzhi Li and Duo Wang [15℄ and L. Vallier [60℄. The linear operator LkA an be represented by a matrix L in a suitable basis. The matrix L has a stru ture that an be derived from the stru ture of the matrix A. The spa e K er(L), that is asso iated to Cnk (A) an easily be omputed for the Poin are-Dula normal form as the matrix A is in Jordan form. The operator LkA an be represented by a matrix with respe t to a basis of Hnk . A basis for Hnk an be given by the basis elements X Q ei with jQj = k and i = 1; : : : ; n. The basis elements are ordered in the lexi ographi order X

Q ej

< X

P ei

,

(i; q1 ; : : : ; qn ) < (j; p1 ; : : : ; pn )

:

(3.10)

48

3.4. Computation of normal forms by Lie transformation

In equation (3.10) the relation (i; q1 ; : : : ; qn ) < (j; p1 ; : : : ; pn ) holds only if i < j or if i = j and the rst omponents qi and pi with qi 6= pi verify qi < pi . The stru ture of the matrix L depends on the stru ture of A. Lemma 1 If A is a diagonal matrix then L is also diagonal. If A is lower (upper) triangular then L is lower (upper) blo k triangular. The element lii is hQ; i j where i means the i-th element in the lexi ographi al ordering and it orresponds to the basis element ej X Q .

The maps Hk and Fk (X~ ) F~k (X~ ) an be represented by the ve tors h and f with respe t to the basis of Hkn . Equation (3.8) an be written as a linear system Lh = f :

If A is in Jordan form the range of L and a omplementary subspa e an be easily read o the matrix L be ause the range of L is spanned by the olumns of the matrix. We an easily see that L has zero eigenvalues if the eigenvalues of A verify resonan e onditions of order k. That means that resonant monomials an not be redu ed by the operator LkA . The terms in the spa e Ckn are the terms of degree k that remain in the normal form. They an not be removed in equation (3.8). However the Poin are transformation (3.4) also ae ts terms of higher degree than k. To perform the next step of the omputation of the normal form (for k + 1) these terms need to be known. They are al ulated using equation (3.7). However these omputations are not very eÆ ient. A better approa h for this problem is the use of Lie transformation methods. 3.4

Computation of normal forms by Lie transformation

A main problem in al ulating normal forms using Poin are transformations is that the

al ulated transformation X = X~ + Hk (X~ ) does not only ae t terms of degree k but it also hanges terms of higher degree. In the previous se tion the appli ation of Poin are transformation to the initial system was omputed a

ording to equation (3.7). However a mu h more eÆ ient way to perform these omputations is to use Lie transformations. The introdu tion of Lie theory to the theory of normal forms is due to W. Groebner [62℄ and K.T. Chen [40℄. Sin e then this subje t has been developped by many others. See for example the works of P. Olver [46℄, G. Chen [13℄ and K. Meyer [43℄. A. Deprit [3℄, S. Chow and J. Hale [14℄ and L. Vallier [60℄ have optimized the organization of the

omputations. 3.4.1 De nitions The elementary operators used in Lie theory are the Lie derivative and the Lie bra ket. De nition 12 (Lie derivative) Let F = (f1 ; : : : ; fn ) be a ve tor eld de ned on an open subset M of Rn or C n and let g : M

! R[[X ℄℄ or C [[X ℄℄ be a fun tion on M . The operator

Chapter 3. Normal Forms

49

X L ( )=

LF (:) with

n

F g

i=1

fi

g xi

is alled the Lie derivative.

LF (g) an be interpreted as the derivative of the fun tion g in dire tion of the ve tor eld F.

De nition 13 (Lie bra ket) Let F and G be two ve tor elds de ned on open subsets. The ve tor eld

G

[F; G℄ = is alled the Lie bra ket of

F

and

G.

X

F (X )

F X

G(X )

(3.11)

F denotes the Ja obian matrix of F . The Lie bra ket In equation (3.11) the expression X is used to de ne the adjoint operator.

De nition 14 (adjoint operator) The operator adF

= [F; :℄

that asso iates a ve tor eld [F; G℄ to any ve tor eld asso iated to a ve tor eld F .

G,

is alled the adjoint operator

The main idea of Lie transformation theory is to introdu e a new parameter  and to

onsider the transformation H (X~ ; ) either as the ow of a ve tor eld G(H ) or as solution of the asso iated dierential equation ~ ; ) H (X 

= G(H (X~ ; ) :

~ ) The normalizing transformation introdu ed in se tion 3.1 is obtained from the ow H (X; for  = 1. It transforms the initial system (3.1) to Poin are-Dula normal form. The ve tor eld H  F is not omputed as the a tion of the transformation H on F but as the a tion of the ve tor eld G on F . The task onsists in transforming F to normal form by the a tion of the ve tor eld G. Having al ulated H  F and G, the transformation H is omputed as the ow of the non-singular ve tor eld G.

3.4.2 A tion on a ve tor eld Consider the ve tor eld F de ned on an open subset M of C n or Rn and the ve tor eld  of Hn G de ned on an open subset M :X2M  G:H 2M F

7! C n orRn 7! Hn :

Introdu ing the parameter  the normalizing transfomation H an be written as ~ ; ) H (X

= X~ + 2 H2 (X~ ) + 3 H3 (X~ ) + : : :

:

50

3.4. Computation of normal forms by Lie transformation

an be onsidered as the ow of the non-singular ve tor eld G(H ) or the solution of the asso iated dierential equation ~ ; ) H (X = G(H (X~ ; )) : (3.12) H



Now the a tion of the ve tor eld G on the ve tor eld F is given by the following theorem: Theorem 6 The a tion of a ve tor eld Taylor serie H

F

F

2

is given by the formal

[G; [G; F ℄℄ + : : : = e adG (F ) 2! denotes the transformed ve tor eld. H

where

on a ve tor eld

G

 F = F + [G; F ℄ +

(3.13)

Proof 2 Equation (3.13) represents the Taylor serie of the ve tor eld H  F . Therefore to prove theorem 6 it is suÆ ient to show that  F j=0

=

H F  =0



= [G; F ℄

2H F 2

= [G; [G; F ℄℄

H



=0

F

(3.14)

:::

The rst equation in (3.14) follows immediately from the de nition of H (X~ ; ). Now we will prove the se ond equation. Applying the transformation X = H (X~ ; ) to the initial equation (3.1) yields ~ X DH = F (H (X~ ; )) (3.15) 

and the new system

~ X 

= (DH )

1

~ ; )) F (H (X

(3.16)

that is also denoted by H  F . Deriving equation (3.15) yields ~ ; ) DH H  F H (X H  F + DH = DF 





that an be simpli ed by the properties DH 

H

F

H 

=

D H 

=

D (G(H ))

=

DG(H ) DH

= (DH ) =

G(H ) :

1 F (H )

(3.17)

Chapter 3. Normal Forms

51

Introdu ing these results into equation (3.17) yields H

DG(H )F (H ) + DH

F



= DF (H ) G(H )

:

With the de nition of the Lie bra ket this an be written as H

F



= (DH ) 1 [G; F ℄

:

For  = 0 this yields the se ond term in the Taylor serie for H  F . The terms of higher degree an be omputed in a similar way (see also G. Chen [13℄). Now H (X~ ; ) an be omputed as the ow of the ve tor eld formation H (X~ ) is obtained for  = 1.

G.

The normalizing trans-

3.4.3 The ow of a non-singular ve tor eld Let G(H ) be a non-singular ve tor eld. The ow H of the ve tor eld

omputed very eÆ iently by omputing the Taylor serie of H . The derivations of H (X~ ; ) are given by ~ ;) H (X  ~ ;)  2 H (X 2

G(H )

an be

~ ; )) G(H (X

= =

~ ; )) D G(H ) G(H (X

:::

~ ) around  = 0 yields Computing the Taylor serie for H (X; ~ ; ) H (X or

= H (X~ ; 0) + 



~ ; ) H (X =0



~ ; ) = H (X~ ; 0) +  G(H (X~ ; )) H (X

+





=0

+

2

2!



~ ) 2  2 H (X;

2!

2

+ :::



=0



~ ; )) D G(H ) G(H (X

=0

+ :::

:

Introdu ing the notation of Lie derivatives this an be written as ~ ; ) H (X

= H0 (X~ ) +  LG (H0 (X~ )) +

2

2!

~ )) + : : : = eLG (H0 (X~ )) L2G (H0 (X

(3.18)

with H0 (X~ ) = H (X~ ; 0). Considering that H is the normalizing transformation de ned in se tion 3.1 and that the linear part has Jordan form we an onsider that H0 = X~ . Equation (3.18) is alled the Lie serie of the ve tor eld G. The property (3.18) is used to ompute the transformation H for the ve tor eld G(H ) that is used to normalize the initial ve tor eld F .

52 3.4.4

3.4. Computation of normal forms by Lie transformation Organization of the omputations

The omputations of the a tion on a ve tor eld (theorem 6) and of the ow of a nonsingular ve tor eld (equation (3.18)) an be performed very eÆ iently when they are organized in so alled Lie triangles. This s heme has been introdu ed by A. Deprit [3℄. Therefore the s alar parameter  2 k is introdu ed. Now the ve tor elds F , G and ~ = H  F and the transformation H are written as F P1 m F (X; ) = Fm (X ) m m =0

where Fk ; theorems

!

~ ; ) G(X

=

P1

~ ; ) F~ (X

=

P1

~m+1 (X~ )  F m!

~ ; ) H (X

=

P1

~) : Hm+1 (X m!

2 Hkn.

~k ; Hk Gk ; F

+1

m

m=0

Gm+1 (X ) m! m

m=0

m

m=0

The al ulations are performed a

ording to the following

Theorem 7 If the sequen e Fi m (X ); i = 1; 2; : : : ; m = 1; 2; : : : ; i 1 is de ned by the re ursive relations Fi = Fi ; i = 1; 2; : : :
P m Fi = Fi m + j i m ji [Gj ; Fi mj ℄ f or i = 1; 2; : : : and m = 1; 2; : : : ; i 1 then i F~i = Fi ; f or i = 1; 2; : : : : This allows to ompute the normal form F~ and the ve tor eld G. It yields an equation that is equivalent to the homologi al equation (3.9). This equation an be solved by using the matrix representation method. The transformation H an be omputed a

ording to the following theorem. (

)

(0) (

)

(

1)

(

1

1

(

1)

+1

1

1)

Theorem 8 If we de ne the sequen e (0)

Gi

= Gi ; i = 1; 2; : : :
P = Gim + j i m ji LGj+1 (Gimj i = 1; 2; : : : and m = 1; 2; : : : ; i 1

(m)

Gi

f or

(

1)

then Hi

(

1

1

= Gii (

1)

;

1

f or i

1)

)

= 1; 2; : : : :

These theorems allow to organize the omputations in Lie triangles : (0)

(0)

F0

H0

(0)

F1

(0)

F2

(0)

F3

F1

F2 F3

(1)

(0)

H1

(0)

H2

(0)

H3

H1

(1)

F2

(2)

(1)

F3

(2)

H2 (3)

F3

H3

.. .. .. .. .. ... . . . . . These Lie triangles an be omputed very eÆ iently.

(1) (1)

H2

(1)

H3

.. .

(2) (2)

.. .

(3)

H3

.. .

...

Chapter 3.

3.5

Normal Forms

53

Example

The following example illustrates all steps of the proposed algorithm.

Example 8 (Pendulum) Consider the dynami al system



x _

=

y _

=

y

g l sin(x)

already introdu ed in the examples 1 and 4. Developping sin(x) into its Taylor serie around x = 0 yields  x _

=

y _

=

y

g l (x

1 3 1 5 7 3 x + 120 x + O (x ))

(3.19)

:

The linear part of the system (3.19) represented by the matrix



A

=

0 g l



1 0

is transformed to its jordan form by a hange of variables X

This yields the new system ~ X t

0 qg l =

1 qg A

0

i

0

i

1 1 ~ + 2y ~ 2x g i i ~ 2 lx 2

q

=

0 i qg 3 48 l~ ~ + q g i x

X

l

48

!

:

~ ly

qg qg i 2~ 2 ~ l 16 l ~~ q q g g i i

3+ l 16 x ~

qg

i 16

x y

l

2

x ~ y ~

qg 3 l~ + q g i i 48

xy

+ 16

l

2+ 48

x ~y ~

This system an be transformed to its normal form ^ X t

0 qg l = i

0

1 0 i qg 2 q g A ^ +  i 16q g l ^ ^ +

0

x y

X

i

by a hange of variables ~ =X ^ + X

l



16

^ O (X

4)

^ 4) ^y ^2 + O (X lx

1 1 3 1 3 ^ + 32 x ^y ^2 + 192 y ^ 96 x 1 3 1 2 1 3 x ^ + x ^ y ^ y ^ 192 32 96

y

~ O (X

4)

3 + O (X ~ 4) l y ~

1 A

(3.20)



The system (3.20) ontains only resonant nonlinear terms. This is the Poin are-Dula normal form of the initial system (3.19). Cal ulating with parameters in the linear part of the system in example 8 is only possible be ause in the resonan e equation the parameters vanish.

1 A

:

Chapter 4 Resolution of singularities by blowing-up Blowing-up is one of the most frequently used methods to redu e ve tor elds. It is mostly used for the redu tion of nilpotent ve tor elds but in hapter 9 it is shown that blowingups an also be used for the redu tion of non-nilpotent systems of dierential equations. The idea is to apply a hange of oordinates that expands or "blows-up" the singularity of a ve tor eld F or of the asso iated system of dierential equations _ = F (X )

X

:

(4.1)

In se tion 4.3 it will be shown that F an no longer be supposed to be given by formal power series. Therefore we will presume that the omponents fi of F are real or omplex analyti power series in the variables x1 ; : : : ; xn . The lo al study of the initial system in the singular point X = 0 is repla ed by a study of the transformed system in the blown-up singularity. The on erned singularity X = 0 is alled the enter and the blown-up singularity is refered to as the ex eptional divisor of the blowing-up. The simplest blowing-up, that illustrates very well all aspe ts of this method, is the introdu tion of polar oordinates. However for systems de ned by power series the use of dire tional and quasi-homogeneous dire tional blowing-ups is more appropriate. Those blowing-ups are represented by power transformations that were already introdu ed in

hapter 2. Dire tional blowing-up has rst been introdu ed for the desingularization of plane algebrai urves by O. Zariski [66℄. Those results were extended to two-dimensional systems of dierential equations by A. Seidenberg [53℄ and A. van den Essen [61℄. Quasihomogeneous dire tional blowing-up, that is ontrolled by the Newton diagram, was introdu ed by A. Bruno [9℄. Like M. Brunella and M. Miari [8℄ he uses only unimodular matri es to

onstru t the power transformations that de ne blowing-ups for two-dimensional systems. In the works of F. Dumortier [32℄, [22℄ and M. Pelletier [45℄ the matri es for the used quasihomogeneous blowing-ups are onstru ted using a dierent approa h. However we will see that the use of the matri es de ning the blowing-ups an be generalized. This is ne essary as unimodular matri es do not allow to treat all appearing problems. Therefore the de nition of blowing-ups is extended to the use of any invertible 55

56

4.1. Two-dimensional polar blowing-up

matrix. The used matri es are omputed via their adjoint matrix. As those matri es do not always de ne inje tive power transformation this generalization is only possible due to the results from se tion 2.3. The de nitions of blowing-ups are given for n-dimensional systems. However blowingups of systems with n > 2 might yield diÆ ulties. For some three-dimensional systems there might not exist a nite hain of su

essive blowing-up that entirely redu es the

on erned system. This has been shown by J. Jouanolou [38℄ and X. Gomez-Mont and I. Luengo [30℄. Three-dimensional ve tor elds an only be desingularized by blowing-ups for some parti ular ases as it was shown for nondi riti al systems by F. Cano and D. Cerveau [10℄. 4.1

Two-dimensional polar blowing-up

The simplest possible blowing-up is the introdu tion of polar oordinates to the dierential equation (4.1). In general polar blowing-up is not used for systems given by power series. However it is onsidered here as it illustrates well all aspe ts of blowing-up. We will

onsider a two-dimensional ve tor eld F = (f1 ; f2 ) where f1 and f2 are given by real analyti power series in the variables x1 and x2 (f1 ; f2 2 Rfx1 ; x2 g). De nition 15 (Polar blowing-up in R2 )

: is alled a polar blowing-up in



The map

R  [0; 2[! R2

(r; ') 7! (r sin '; r os ')

R2 .

A onsidered system is blown-up by applying the hange of oordinates X = (X~ ). This yields a new system F~ =   F . The point X = 0 is "represented" by the set f(r; ') : r = 0; ' 2 [0; 2[g : (4.2) That means that  1 is not de ned for X = 0. Nevertheless the set (4.2) is denoted by  1 (0). It will be alled the ex eptional divisor of the blowing-up. The point X = 0 is refered to as the enter of the blowing-up. In ontrast to dire tional blowing-up, that will be studied in the following se tion, the ex eptional divisor of polar blowing-up is nite as ' 2 [0; 2[. The system resulting from the blowing-up an be omputed a

ording to   F = (D) 1 F () : This yields the system  r_    1 f (r sin '; r os ') sin ' r os ' 1 = os ' r sin ' : '_ f2 (r sin '; r os ') that an also be written as  r_ = sin 'f1 (r sin '; r os ') + os 'f2 (r sin '; r os ') = rk f~1 (r; ') '_ = osr ' f1 (r sin '; r os ') sinr ' f2 (r sin '; r os ') = rk 1 f~2 (r; ')

Chapter 4. Resolution of singularities by blowing-up

57

with f~1 ; f~2 2 R[r; sin '; r os '℄. Applying a time transformation dt~ = rk 1 dt yields the new system ( r = r f~ (r; ') 1  t~ (4.3) ' = f~ (r; ') : 2  t~ The lo al study of the initial system near X = 0 is repla ed by the examination of the system (4.3) near the ex eptional divisor  1 (0). For further studies the new singularities on  1 (0) are of parti ular interest. Those singularities exist as the studied systems are analyti . Two possible ases have to be onsidered for equation (4.3).  f~2 (0; ') 6 0. The singularities on  1 (0) are given by the set S = f(0; ') : f~2 (0; ') = 0g : All other points on  1 (0) are regular points. Here the solution urves are parallel to the ex eptional divisor as r  t~ = 0. For the initial oordinates that means that only the solution urves omputed for the singularities in S might pass through X = 0. Further applying the time transformation dt^ = f~2 (r; ')dt~ yields the new system



(

r = r f~1 (r;')  t^ f~2 (r;') ' = 1 :  t^

that has the solution (r; ') = (0; t^). Therefore the ex eptional divisor is a solution

urve for (4.3). f~2 (0; ')  0. Another time transformation dt~ = rdt^ yields the new system ( r = f~ (r; ') 1  t^ ' 1 ~  t^ = r f2 (r; ') : The singularities of the new system are given by the set 1 ~ ~ S = f(0; ') : f (0; ') = 0 and f (r; ') 1

r

2



r=0

= 0g :

All points (0; ') 62 S are regular points. For any point (0; 0 ) 62 S with f~1 (0; 0 ) = 0 there exits a solution urve that is tangent to the ex eptional divisor. Those solution urves are alled tangen ies. As for all other points (0; 0 ) 62 S on the ex eptional divisor r  t^ 6= 0, there exists a solution urve for equation (4.3) passing through this point. For the initial system (4.1) that means that there is an in nite number of solution urves passing through the singularity in X = 0.

58

4.2. Quasi-homogeneous dire tional blowing up

In both ases the points in the set S an be studied by translating them to the origin and by applying either another blowing-up or by omputing normal forms. Remark 3 The ase f~2 (0; ') 6 0 is alled the non riti al ase. The ase f~2 (0; ')  0 is refered to as the di riti al ase. The name "di riti al" is due to the problems in the di riti al ase for higher dimensional problems.

 is a dieomorphism on the set R  [0; 2[  1 (0). Therefore the initial ve tor eld F and the ve tor eld   F are onjugated.

Remark 4 The transformation

Example 9 Consider the system given by X_ =



x1 4 + x1 3 x2 13 6 2 x1 2 x2 2 + x1 x2 3 9 x1 x2



:

Applying a polar blowing-up and a time transformation with dt~ = r3 dt yields a new system

(

r  t~ '  t~

= =

r sin(') + 2 r sin(') os(')2 + r os(') + : : :

os(') + os(')3 + 2 sin(') os(')2 2 os(')4 sin(') + : : :

(4.4)

that veri es the onditions for the non riti al ase. For r = 0 the system (4.4) has the form ( r  t~ '  t~

= 0 = os(') + os(')3 + 2 sin(') os(')2 2 os(')4 sin(')

that allows to ompute its singularities on the ex eptional divisor. As

os(') + os(')3 + 2 sin(') os(')2 2 os(')4 sin(') = os(') sin(')2 (sin(2 ') 1) the singularities on the ex eptional divisor  1 (0) are given by the set

0); (0; 14 ); (0; 21 ); (0; ); (0; 54 ); (0; 23 )g : The omponents of the systems resulting form polar blowing-up are analyti series in r; sin ' and os '. As all further redu tions are de ned for power series, Taylor series expansions have to be used to allow further omputations. Therefore dire tional blowingup is more appropriate for the use with systems of the form (4.1). S = f(0;

4.2

Quasi-homogeneous dire tional blowing up

Among the most frequently used blowing-up onstru tions for systems (4.1) given by power series are quasihomogeneous dire tional blowing-ups. They are based on the fa t that the qualitative properties of ve tor elds near the singularity in X = 0 are mainly determined by the quasihomogeneous parts of the on erned ve tor eld. Those parts are omputed using the Newton diagram. The degree of quasi-homogenity de nes the oeÆ ients of the matrix used for the blowing-up. That means that quasihomogeneous blowing-up is

ontrolled by the Newton diagram.

Chapter 4. Resolution of singularities by blowing-up

59

4.2.1 Quasi-homogeneous ve tor elds Quasi-homogeneous blowing-ups have been studied by F. Dumortier [32℄ and M. Pelletier [45℄. They are de ned by the type of quasi-homogenity of a quasi-homogeneous ve tor eld.

De nition 16 (quasi-homogeneous fun tions and ve tor elds) A fun tion f de ned on R or C is alled quasi-homogeneous of type  = (1 ; : : : ; n ) 2 N n and degree k with g d(1 ; : : : ; n ) = 1 if f (r1 x1 ; : : : ; rn xn ) = rk f (x1 ; : : : ; xn ) A ve tor eld F = (f1 ; : : : ; fn) is alled quasi-homogeneous of type  and degree k if any fj ; j = 1 : : : n is quasi-homogeneous of type  and degree k + j .

The quasi-homogeneous parts of a ve tor eld with the lowest degree are given by the monomials asso iated to the fa es of the Newton diagram. Consider the ve tor eld

0 x P j a XQ 1 1 Q2 1Q i C : : : Fi(j ) = B  P A Q ( )

xn

Q2

(j )

i

anQ X

ontaining all monomials asso iated to the points of a fa e (ij ) of the Newton diagram. Choose  su h that the ve tor  is orthogonal to the fa e (ij ) and su h that 1 ; : : : ; n  0. Due to the de nition of the Newton diagram this ve tor exists for all fa es (ij ) . Then all Q 2 (ij ) are lying on the hyperplane fQ : h; Qi = kg. Fi(j ) is quasihomogeneous of type  and degree k as

0 Fi(j ) (r x1 ; : : : ; rn xn ) = 

r1 x1

1

rn xn

P rh;Qia X Q 1 1Q Q : : : P rh;Qia X Q A : nQ

Q

The ve tor eld Fi(j ) is alled the quasihomogeneous part of F relative to the fa e (ij ) . Ea h ve tor eld Fi(0) is quasihomogeneous of type  for any ve tor . The ve tor  with 1 ; : : : ; n  0 is used to de ne quasi-homogeneous blowing-ups. Quasihomogenous dire tional blowing-up is de ned by a power transformation m : X 7! X Am T

where the matrix Am is de ned as

0 1 B B B Am = B 1 B B 

...

: : : m : : : ...

(4.5)

1 C C C: n C C C A 1

60

4.2. Quasi-homogeneous dire tional blowing up

The index m refers to the m-th row ve tor in Am that is identi al to the ve tor . A

onsidered system (4.1) is transformed to the onjugated blown-up system m  F by the

hange of oordinates X = m (X~ ). Quasi-homogeneous blowing-ups of the form (4.5) are not ne essarily dieomorphisms. In this ase the transformed system is not onjugated to the initial one. Therefore the additional onstru tions from se tion 2.3 are used to de ne a dieomorphism orresponding to transformation (4.5). Quasi-homogeneous blowing-up is de ned by the ve tor  that is omputed using the Newton diagram. Therefore the ee ts of the blowing-up m on the support and on the Newton diagram of F are of parti ular interest.

4.2.2 The ee t on the Newton diagram All power transformations a t on the exponents of the onsidered system and equivalently on the support of F and its Newton diagram. Consider the n 1 dimensional fa e (in 1) . A

ording to theorem 3 in se tion 2.2 the ee t of the power transfomation (4.5) on the points Q 2 supp(F ) an be omputed as

0 q 1 B .. C B . C B C: ~ B Q = AQ = B h; Qi C C B  ... C A 1

qn

That means that all points on the fa e (in 1) are transformed to points with identi al m- oordinate k = h; Q0 i with Q0 2 (in 1) . As i(n 1) lies on the lower left part of the

onvex hull and as 1 ; : : : ; n  0 it an be stated that all points Q 2 supp(F ) (in 1) are transformed to points AQ whose m- oordinate is greater than k. This result an be interpreted geometri ally. The fa e (in 1) has been straightened up. That means that it has been transformed to a fa e that is parallel to the hyperplane

fQ~ jq~ = 0g : m

(4.6)

This fa e an be translated to the set (4.6) by a time transformation

dt~ = x~km dt : As all points Q 2 supp(F ) (in 1) are transformed to points AQ whose m- oordinate is greater than k the exponents of the system resulting from the blowing-up and the time transformation are positive. Now onsider the n 2 dimensional fa e (in 2)  (in 1) . This fa e an be seen as the interse tion of the fa e i(n 1) and another fa e of dimension n 1. This fa e is either a n 1 dimensional fa e of the Newton diagram or a fa e of the onvex hull of supp(F ). Let be a normal ve tor of this fa e that veri es 2 N n ; g d( 1 ; : : : ; n ) = 1. The ve tors  and are not olinear and all points Q 2 (in 2) lie on a hyperplane hQ; i = k~. Further, for all Q 2 supp(F ) the ondition hQ; i  k~ holds.

Chapter 4. Resolution of singularities by blowing-up

61

For this reason after having applied the transformation (4.5) another quasihomogeneous blowing-up l an be applied to straighten the fa e (in 2) su h that it is transformed to a set of points that is parallel to the set

fQjqm = 0 and ql = 0g : The omposition of those two blowing-ups is given by the power transformation X = X~ AT with 1 0 1 C BB . . . C BB  : : :  : : : : : : : : :  C m n C C BB 1 C: ... A=B C BB : : : : : : : : : : : : C n C l C BB 1 C . . A  . 1 As any fa e (ij ) of the Newton diagram an be seen as an interse tion of n j fa es of dimension n 1 the n j normal ve tors of these fa es are orthogonal to (ij ) . Therefore n j elementary quasihomogeneous blowing-ups of the form (4.5) an be used to straighten the on erned fa e. These elementary blowing-ups an be joined to a single power transformation : X 7! X AT where the normal ve tors of the interse ting fa es de ne the row ve tors of the matrix A. This de nes a blowing-up for ea h fa e (ij ) of the Newton diagram. Ordering the quasihomogeneous blowing-ups (m = 1; l = 2; : : :) yields that the fa e (ij ) is transformed to a fa e that is parallel to the set

fQ~ : q~ = 0; q~ = 0; : : : ; q~n j = 0g : 1

2

(4.7)

For all Q~ 2 supp(  F ) the ondition

q~1  q~01 ; : : : ; q~n j  q~0n j with Q~ 0 = (~q01 ; : : : ; q~0n j ) = AQ0 ; Q0 2 (ij ) holds. Therefore a time transformation

dt~ = x~q1~01 : : : x~nq~0nj j dt

(4.8)

translates the straightened fa e to the set (4.7). The exponents of the system resulting from the blowing-up and the time transformation (4.8) are positive. However quasihomogeneous blowing-ups de ned that way are not uniquely de ned as in a fa e (ij ) with j < n 2, any number of n 1 dimensional fa es of the Newton diagram

an interse t. Ea h of those n 1 dimensional fa es an be used to de ne the matrix A by its normal ve tor. Therefore quasihomogeneous blowing-ups de ned that way do not allow adequate ontrol on the ones of the Newton diagram. That indu es that the se tors that will be introdu ed in se tion 8 an not be ontrolled suÆ iently. A better method to

onstru t matri es that perform well dire ted manipulations on the Newton diagram and its ones is the onstru tion of A via its adjoint matrix.

62 4.2.3

4.2. Quasi-homogeneous dire tional blowing up Constru tion of blowing-ups via adjoint matri es

The onstru tion of the matrix A, that de nes a quasi-homogeneous blowing-up, via its adjoint matrix is based on theorem 3 in se tion 2.2 and on the results in se tion 2.4. It yields a blowing-up that has the same ee t on the Newton diagram as the blowing-ups

onstru ted in se tion 4.2.2. Consider the linearly independent ve tors v1 ; : : : ; vn 2 Zn. They form the olumns of the matrix A = (v1 j : : : jvn ) : A

ording to the results from se tion 2.4 the matrix A an be used to ompute a matrix T A ~ A that de nes a power transformation X = X . A

ording to theorem 3 in se tion 2.2 the ee t of this transformation on the ve tors vk ; k = 1; : : : ; n is given by v~k

= Avk = detA ek :

If there exists a point Q0 su h that any point Q 2 supp(F ) an be written as Q = Q0

n X + k k k=1

v with k

those points are transformed to the points ~ = AQ0 + Q

0

(4.9)

Xn k k k=1

e :

The ondition (4.9) is equivalent to the ondition that supp(F ) has to be ontained in the set Q0 + V where V is the onvex one spanned by the ve tors v1 ; : : : ; vn . A time transformation ~ = X~ AQ0 dt dX translates the point AQ0 to the origin and yields a new system with positive support. These properties an be used to onstru t a quasihomogeneous blowing-up with the same ee ts as the blowing up de ned in se tion 4.2.2. (j ) (j ) Consider the fa e (ij ) and let Q0 = (0) i 2 i be a vertex on the fa e i . There exist j linearly independent ve tors vk

= Q0

Qk ; Qk

j i ; k=n

2

( )

1; : : : ; n

j

on the fa e i(j ) su h that all Q 2 (ij ) an be represented as Q = Q0 +

Xn

k=n j

k vk ; k

2R

+

:

1

With the ve tors v1 ; : : : ; vn j they an be ompleted to a set of ve tors that de ne the

onvex one V

=

Xn

k=n j +1

R v +

n j X R k+ k=1

+

vk

Chapter 4. Resolution of singularities by blowing-up

63

su h that all Q 2 supp(F ) lie within the set Q0 + V . The one V an be omputed using for example the virtual Newton diagram that will be introdu ed in se tion 9.4 and hapter 10. The blowing-up de ned by X = X~ AT with A

 = det AA

1

= (v1 j : : : jvn )

has the same ee ts as the blowing up from se tion 4.2.2. That means that the fa e (j ) i is straightened and an appropriate time transformation yields a system with integer exponents. The hoi e of the ve tors v1 ; : : : ; vn and therefore the onstru tion of V is not unique. Further the onstru tion of the matrix A is not limited to the above hoi es of Q0 and V . Any one V spanned by n ve tors and any point Q0 su h that supp(F )  Q0 + V an be used for the above onstru tion if n j ve tors in the set of ve tors de ning V lie on (j ) i . However the onstru tion of quasihomogeneous blowing-ups by adjoint matri es is in

ertain ases easier to handle than the quasihomogeneous blowing-ups de ned in se tion 4.2.2. The main advantage is that the se tor de nition, that will be introdu ed in hapter 8 and 10, is mu h simpler to ontrol as the inverse matrix of A is known. It is obvious that not all power transformations de ne blowing-ups. Therefore a entral point in the use of power transformations is to determine whether they de ne blowing-ups or not and to onsider this in the onstru tion of the power transformations. 4.2.4

The ex eptional divisor

A power transformation de nes a blowing-up if the enter of the blowing-up an be represented by an obje t of higher dimension in the new oordinates. That means that the ex eptional divisor has to ontain more than a single point. Consider the transformation : X ! X AT where A is an invertible matrix. The ex eptional divisor 1 (0) an be

omputed by onsidering theTinverse of the transformation X = X~ AT where it is de ned. The transformation X = X~ A an be inverted on the set U

0 = Kn

[ k

f 2 X

K

n jx = 0g; k

K

=R

or K

=C

:

If A is de ned by its adjoint matrix A with the olumn ve tors v1 ; : : : ; vn this yields the inverse transformation 0 1 1 ~ = XA T = B 

X

X

det A v1

.. .

vn X det A 1

C A

:

(4.10)

The ex eptional divisor is de ned as the set of values that the expression lim X~

X !0

0 = lim B X! 

X

det A v1

X

det A vn

0

1

.. .

1

1 C A

(4.11)

64

4.2. Quasi-homogeneous dire tional blowing up

an take. To evaluate this expression the point X = 0 is approa hed on so alled urves of lass W. Those urves have been introdu ed by A. Bruno [9℄. They are de ned as 8 x (t) = t ( + O(1=t)) < 1 1 F :: : ::: xn (t) = tn ( n + O(1=t)) with  = (1 ; : : : ; n ) 2 Zn; 1; : : : ; n  0 and 1 ; : : : ; n 6= 0. Evaluating expression (4.11) on the urve F yields 0 ~ t det A hv ;i 1 1 C B ... ~ (4.12) lim j X = tlim A  !1 X !0; X 2 i h v ; i

~n t det A n with ~k = det1 A Pl vkl l for vk = (vk1 ; : : : ; vkn). Ea h line in equation 4.12 an be evaluated separately. For the line k this yields 8 0 if hv ; i < 0 < k

~k thvk ;i ! ~k if hvk ; i = 0 : 1 if hvk ; i > 0 as det A > 0. It is obvious that ~k thvk ;i 6= 0 only if the ve tor vk has at least one negative

oeÆ ient. In this ase there exists an  with h; vk i = 0. That means that x~k an take T A ~ any value if X ! 0. As a onsequen e the transformation X = X is a blowing-up. Any power transformation X = X~ AT is a blowing-up if the matrix A has negative entries. From the onstru tion of the ve tors vn j+1; : : : ; vn it follows that they lie on a hyperplane h:; i = 0 with  2 Zn . As those ve tors are used for the onstru tion of the matrix A the power transformation X = X~ AT with A and A de ned as in se tion (4.2.3) is a blowing-up. The omponents x~n j+1; : : : ; x~n an take any value and the set S = fX~ : x~1 = 0; : : : ; x~n j = 0g (4.13) is part of the ex eptional divisor. We are interested in the singularities of the new system  F on the set (4.13). It is obvious that those singularities are given by the singularities of 0 x~ f~ (0; : : : ; 0; x~ ; : : : ; x~ ) 1 1 1 n j +1 n B C (j ) . .  Fi (0; : : : ; 0; x~n j+1 ; : : : ; x~n) =  A : (4.14) . ~ x~n fn(0; : : : ; 0; x~n j +1 ; : : : ; x~n ) We have to distinguish two ases:  If the ondition 9k 2 fn j + 1; : : : ; ng : f~k (0; : : : ; 0; x~n j+1 ; : : : ; x~n ) 6 0 holds, the singularities are given by the points X~0 = (0; : : : ; 0; ; : : : ; ) that verify f~n j +1(X~0 ) = 0; : : : ; f~n (X~ 0 ) = 0 : 1

1

( )

1

1

Chapter 4. Resolution of singularities by blowing-up

65

All other points ~ are regular points. As for those points the ondition  ( ~ ) = (0 0  ) holds, the solution urves ~ ( ) in the neighbourhood of those points verify ~ ( ) ~n j ( ) = The urves ~ ( ) near ~ are all parallel to the ex eptional divisor. The dimensional system 8 x +1 = ~ ~n j (0 0 ~n j ~n ) < t n j X0

F X0

;:::;

;

;:::;

X t

x

t

n

onst:

x1 t ; : : : ;

X t

X0

j

~n j

:

x

+1 f

;:::;

+1

;x

+1 ; : : : ; x

:::

= ~n ~n(0 0 ~n j ~n ) yields solution urves on the ex eptional divisor. As they are all transformed to the point = 0 they are not interesting for the omputation of solution urves ( ). This ase is alled the non riti al ase.  If the ondition ~n j (0 0 ~n j ~n)   ~n(0 0 ~n j ~n )  0 holds, equation (4.14) is identi al to zero. The entire set is a non-isolated singularity. However there exists an index su h that the time transformation  = ~k ~  x~n t

x f

;:::;

;x

+1 ; : : : ; x

X

f

X t

+1

;:::;

;x

+1 ; : : : ; x

:::

f

;:::;

;x

+1 ; : : : ; x

S

k

dt

x dt

yields a new system with at least one non-vanishing omponent on the set (4.13). The singularities of the resulting system are identi al to the singularities of 0 x1 ~ (0 0 ~n j ~n ) 1 x C B ... C B j C B  i (0 0 ~n j ~n) = B ~ (0 0 ~ C ~ ) n j n k C B ~k C B ... A  x ~ 0 ~n j ~n ) x n (0 In general this system will have negative exponents. If this is the ase a further study with the methods introdu ed here is not possible. This ase is alled the di riti al ase. ~ f ~k 1

F

( )

;:::;

;x

x

+1 ; : : : ; x

f

~n f ~k

;:::;

;:::;

;:::;

;x

;x

;x

+1 ; : : : ; x

+1 ; : : : ; x

:

+1 ; : : : ; x

In this se tion it has been shown how quasi-homogeneous blowing-up an be used to redu e ve tor elds. However in the di riti al ase a further redu tion is not always possible. In general a single blowing-up might not be suÆ ient to redu e a onsidered ve tor eld entirely. This is due to the fa t that the rst blowing-up might still yield nonelementary singularities. In this ase further blowing-ups are needed. The omposition of several blowing-ups is alled su

essive blowing-up.

66

4.3. Su

essive blowing-up q2

(0) 2 (1) 1

V (0) 1

q1

Figure 4.1: This gure shows the one V onstru ted to de ne the blowing-up applied in the ase of an edge (1) in example 10. 1 4.3

Su

essive blowing-up

The singularities of a system resulting from a single blowing-up might still be nonelementary. In this ase a single blowing-up is not suÆ ient to redu e a nonelementary singular point entirey. The on erned singularity is translated to the origin and another blowing-up is applied. This yields a hain of ompositions of a blowing-up, a time transformation and a translation. This hain is alled su

essive blowing-up. An important question arises in this ontext. Is it possible to redu e any nonelementary singular point by a nite su

essive blowing-up ? This is true for two-dimensional ve tor elds if the ve tor eld is analyti in the neighbourhood of an isolated singularity. This has been shown for example by A. van den Essen [61℄, F. Cano [24℄ and F. Dumortier [32℄. For three-dimensional problems this is not true for any ase. This has been shown by J.P. Jouanolou [38℄ and X. Gomez-Mont and I. Luengo [30℄. However some parti ular ases

an be solved. It has been shown by F. Cano and D. Cerveau [10℄ that any non-di riti al system an be redu ed by a nite number of blowing-ups. Non-di riti al means that none of all admissible blowing-ups yield a di riti al ase.

Example 10 (Blowing-up of a two-dimensional ve tor eld) Consider the two-dimensional nilpotent system of dierential equations   3 4 _ = 13x12 +6 x2 x12 2 : (4.15) X 3 x1 x2 + x1 x2 x x 9 2 1


It's Newton diagram is drawn in gure 4.1. It ontains a single edge (1) = 11 . The 1 blowing-up orresponding to that edge an be onstru ted using Q0 = (0) and the matrix 1  A = (v1 jv2 )

Chapter 4. Resolution of singularities by blowing-up with

v2

67

= ( 1; 1). The matrix A an be ompleted by any ve tor v1 = (v11 ; v12 ) with 0

Choosing

v1

v12 v11

= (1; 0) yields the matri es A

The power transformation



=

X



1 0

1 1

>

1;

v11 >

 and A

=

0:



1 1 0 1

 :

= X~ AT yields the system

 ~  x~41 + x~41 x~2 ~ = x~31 x~2 2~x31 x~22 + x31 x32 + 139 x71 x22 : t

X

The points (0) and (0) have been transformed to the point (3; 0) and (3; 2) respe tively. 1 2 Therefore a time transformation

~ = x~31 dt

dt

transforms the straightened edge to the q~2 -axis. The resulting system

 ~  x~1 + x~1 x~2 ~ = x~2 2~x22 + x~32 + 139 x~41 x~22 t

X

(4.16)

veri es the non riti al ase. Therefore the singularities of (4.16) are given by the set S

= fX~ : x~1 = 0; x~2 (1 2~x2 + x~22 ) = 0g = f(0; 0); (0; 1)g

:

In X~ = (0; 0) the system (4.16) is non-nilpotent. The singularity in X~ = (0; 1) however is nilpotent. It an be redu ed by a further blowing-up.

This hapter has shown how power transformations and quasihomogeneous blowing-up

an be used for the desingularization of nilpotent ve tor elds. They are ontrolled by the Newton diagram.

Chapter 5

Classi ation In the previous hapters a number of methods that an eÆ iently be used to redu e ve tor elds were introdu ed. These methods are the basis of the algorithms introdu ed in the following. This hapter will show how those methods an be linked together to integrate any 2-dimensional and a large number of 3- and higher-dimensional system of autonomous dierential equations. Consider a system of dierential equations given by _ X

= F (X )

(5.1)

where the omponents of F are given as real or omplex analyti power series in the variable X . The intention of all algorithms proposed in the following is to redu e this system su h that formal solution urves X (t) an be al ulated in the neighbourhood U

= fX : jjX

X0

jj1  g

of a point X0 . We will presume that the point of interest X0 has been translated to the origin to simplify all onsiderations. (For more details about translations see example 2). The algorithm is split into three main parts a

ording to the lassi ation of the point X0 = 0 that has already been mentioned in se tion 1.2:



The origin is a regular point if F (0) 6= 0. The ow of a ve tor eld in the neighbourhood of a regular point has a very simple stru ture. A

ording to the ow box theorem there exists a hange of oordinates X = H (X~ ) that transforms the system (5.1) into a new system of the form

011 ~ B 0C X B = B .. C t .C A: 0

The integration of this system is obvious. The hange of oordinates an be al ulated as a Taylor serie. The ow box theorem an be found for example in M. Hirs h and S. Smale [42℄ or J. Hale and H. Kor ak [33℄. 69

70  The origin is an elementary singular point if

DF (0)

F (0) = 0 and the Ja obian matrix

is non-nilpotent. In this ase the Poin are-Dula theorem introdu ed in

hapter 3 an be applied. The resulting normal forms an always be integrated in two dimensions. For 3- and higher-dimensional problems, if a dire t integration is not possible, an appropriate power transformation an redu e the normal form to a system of lower dimension. These systems are treated re ursively by applying the entire algorithm again.

 The origin is a nonelementary singular point if

F (0) = 0 and DF (0) is nilpotent.

In this ase there exist two possibilities to redu e the onsidered system. They are

ontrolled by the Newton diagram that was introdu ed in hapter 2. For any of its verti es there exists a time transformation that yields a non-nilpotent system that has its support within a one. Applying a power transformation this system an be redu ed to a form that allows to apply redu tion and integration algorithms for elementary singular points. All other fa es of the Newton diagram an be used to de ne matri es for blowing-ups that where introdu ed in hapter 4. Those blowingups yield new systems that have either regular points, elementary singular points or nonelementary singular points. Those systems an be treated by applying the entire algorithm re urively. Blowing-ups are very well known for 2-dimensional problems (see for example A. van den Essen [61℄, F. Cano [24℄ and F. Dumortier [32℄). For higher dimensional systems there still remain many unsolved problems as it has been shown by J.P. Jouanolou [38℄ and X. Gomez-Mont and I. Luengo [30℄.

An overwiev on the splitting of the algorithms and how the dierent parts are linked is given in gure 5.1. The dierent ases are the subje t of the following hapters. Chapter 6 deals with n-dimensional regular points. Chapter 7 des ribes the integration of systems with 2dimensional elementary singular points. Systems with 2-dimensional nonelementary singular points are treated in hapter 8. In hapter 9 we show how 3 dimensional normal forms an be integrated and where the limits of the proposed algorithms are. The propositions obtained for 3-dimensional systems an partly be extended to higher-dimensional systems. Chapter 8 gives some examples for 3- and higher-dimensional blowing-ups.

Chapter 5. Classi ation

71

X_ = F (X ) regular point

nonelementary singular point

elementary singular point

Redu tion using the ow box theorem and integration. See hapter 6.

Draw the Newton diagram and

onsider its fa es.

Cal ulate the Poin are-Dula normal form.

See hapter 2.

See hapter 3.

for all other fa es

Blowing-ups yield a nite number of less omplex systems. See hapter 4 and 8.

for all verti es

Use a time and a power transformation to obtain a system with an elementary singular point

the normal form is integrable Integrate the normal form See hapter 7

the normal form is not integrable

Use a power transformation to redu e the dimension of the system. See hapter 9.

See hapter 8.

Figure 5.1: An overwiew on the entire algorithm and its omposites.

Chapter 6

Regular Points The simplest examples of dynami al systems are systems in the neighbourhood of a regular point X0 = 0. In this ase the onstant part of the onsidered system _ = F (X )

(6.1)

X

is dierent form zero. The ow or the normal form of a regular system (6.1), where the

omponents of F are given as real or omplex, analyti or formal power series, has a very simple stru ture as it is stated in the ow box theorem.

In a suÆ ient small neighbourhood of a regular point of (6.1) there exists a dierentiable, analyti hange of variables

Theorem 9 (Flow Box Theorem)

X

= H (X~ );

H

= (h1 ; : : : ; hn );

that transforms the initial system to

hi

2 kfX g;

011 ~ B 0C C =B B . . .C A

X t

:

K

=R

or

C

(6.2)

(6.3)

0

in the new oordinates. A proof of this theorem an be found in works from M. Hirs h and S. Smale [42℄ and J. Hale and H. Ko ak [33℄. Theorem 9 guarantees the existen e of a box-like neighbourhood of any regular point su h that the ow of the system enters the box at one end and ows out at the other. The ow of a system in the neighbourhood of a simple point and a

orresponding ow box are sket hed in gure 6.1. The omputation of the hange of variables H an be redu ed to the problem of

al ulating the ow of a regular ve tor eld. An algorithm that solves this problem was introdu ed in se tion 3.4.3. The method is based on Lie theory and is very eÆ ient. It allows to ompute the ow X (t; X0 ) for any initial value X0 = X (0; X0 ) if X0 is a regular point. 73

74 The solution of the redu ed system (6.3) is

0 B X~ (t) = B B 

t + 1

2 .. .

n

1 C C C A

:

Transformed to the initial oordinates the solutions have the form

X (t) = H (X~ (t)) : They must oin ide with the ow X (t; X0 ) al ulated using the algorithm from se tion 3.4.3. Therefore H an be obtained from substituting

t = x~1 ; x01 = 0; x02 = x~2 ; ::: x0n = x~n in X (t; X0 ) where X0 = (x01 ; : : : ; x0n ).

Example 11 The system



x_ = y y_ = gl sin(x)

(6.4)

that issued from the pendulum model in example 1 an be developped into a Taylor serie around the regular point (=2; 0). This yields a system



x_ = y y_ = gl + 21 gl (x

that is transformed to

8 < :

by the transformation

X=



 2

1 2 2 )

 x~ t

=1

 y~ t

=0

1  )4 ) 2

+ O((x

+ y~ 21 gl x2 + : : : gx + : : : l



:

The solution urves X~ = (t; ) for the redu ed system are transformed to the initial oordinates. The obtained urves X (t; ) are solution urves for the initial system (6.4). They are sket hed for g=l = 1=2 in gure 6.1. The al ulation of solutions of dynami al systems (6.1) in the neighbourhood of a regular point is not very interesting for itself. Nevertheless many blowing-ups of nilpotent ve tor elds yield systems with regular points. Therefore solution urves of nilpotent ve tor elds an only be al ulated if the ase of a regular point an be solved.

Chapter 6. Regular Points

75

Figure 6.1: Solution urves in the neighbourhood of a regular point for the system treated in example 11. The lines ow through a so alled " ow box".

Chapter 7

Two dimensional elementary singular points In hapter 3 non-nilpotent singular systems of dierential equations X_ = F (X )

(7.1)

where transformed to their Poin are-Dula normal form. In two dimensions this yields systems of the form ( x P q1 q2 t = x P fQ2N1 :hQ;i=0g aQ x y (7.2) y q1 q2 t = y fQ2N2 :hQ;i=0g bQ x y that have their support on the resonant plane M = fQ 2 N : hQ; i = 0g

with  = (1 ; 2 ). A possible resonant plane is sket hed in gure 7.1. In this hapter it will be shown that 2-dimensional normal forms an always be integrated. The results from se tion 7.1 are based on works from A. Bruno [9℄ and the results from se tion 7.2

an partly be found in works from S. Chow, C. Li and D. Wang [15℄. 7.1

Integration

To integrate the Poin are-Dula normal form (7.2) a entral point of interest are the possible solution of the resonan e equations

hQ; i = 0

(7.3)

Let  := 21 , 2 6= 0 without loss of generality. This allows to lassify the possible solutions of equation (7.3) 1.  2 C Q No resonan e is possible, the normal form is a linear dierential equation



x t y t

77

= 1 x = 2 y:

78

7.1. Integration

Figure 7.1: This gure shows the resonant plane for the normal form in example 12. All points in N1 [ N2 that interse t the line fQ 2 R :< Q;  >= 0g are elements of the resonant plane M and an be in the support of the normal form. Its integration

x = 1 e1 t y = 2 e2 t

is obvious. 2.  2 Q;  > 0 




If  = m; m 2 N;  6= 1 only one ve tor Q = m1 an solve the resonan e equation (7.3). With  = m (1 = m2 ) the normal form is  x t y t

= m2 x + a( = 2 y:

1;m)

ym

This an be integrated to 2 mt x = (a( 1;m) m 2 t + 1 )e  t 2 y = 2 e



using the method of the variation of the onstant. If  = m1 ; m 2 N;  6= 1 we get  x t y t



= 1 x = m1 y + b(m; 1) xm

whi h an be integrated in a same way as the previous ase. If  = 1,  might be dierent from zero but no nonlinear monomials will appear. This ase is similar to the previous one ( = m; m 2 N ) with m=1 and a( 1;1) =  = 1). If  = 0 the resulting system is linear.

Chapter 7. Two dimensional elementary singular points

79

1  If  62 f m ; mjm 2 N g the obtained system is linear.

3.  2 Q ;  = 0 , 
1 = 0. All ve tors Q = 01 with q1 2 N verify the ondition (7.3). The normalized system (7.2) has the form P  =x a( 0) x =: xf (x) P (7.4) = 2 y + y b( 0) x =: yg (x) f and g depend only on x. If f  0 a time transformation q

x t y

k

k;

k

k

t

k;

k

dt~ = yg (x) dt

yields the system

(

=0 =1 ~

x

~ t y t

that an be integrated to with

x(t~) = 1 y (t~) = t~ + 2 t~ = 2 eg( 1 )t :

If f 6 0 the orresponding s alar dierential equation logy g (x) = x xf (x)

is formally integrable as the right hand side does not depend on y. The result is logy =

Z

x

g (s)ds + 1 : sf (s)

This an be parametrized in the following way x = t~ R t~ (s)ds y = exp( gsf + 1 ): (s)

Note that in this ase time transformations t~ = (t) with unknown map are used.

(t) is the solution of the s alar dierential equation dt~ = xf (x) : dt

(7.5)

Therefore to transform the al ulated urves X (t~) to solutions X (t) of the initial system the solution of equation (7.5) has to be known. 4.  2 Q ;  < 0
If  2 Q there exists a ve tor Q = 12 with q1 ; q2 2 N and g d(q1 ; q2 ) = 1 that solves the equation (7.3). This is also true for any ve tor kQ; k 2 N. Figure 7.1 shows the set N = N1 [ N2 of the points that an be in the support of the initial q q

80

7.1. Integration system supp(F ) and the line de ned by the resonan e ondition. The points on this line an be in the support of the resulting normal form



P

x t y t

= 1 x + x k a(kq1 ;kq2) (xq1 yq2 )k P = 2 y + y k b(kq1 ;kq2 ) (xq1 yq2 )k :

(7.6)

This system has two parti ular solutions (x; y) = (t~; 0) with t~ = e1 t (x; y) = (0; t~) with t~ = e2 t :

(7.7)

Further the normal form (7.6) an be brought to a system of the form (7.4), that T is treated by the previous ase, by a suitable hange of oordinates X = X~ A . A

ording to theorem 3 in se tion 2.2 the matrix A is omputed su h that

AQ =



a b q2 q1

  

 

q1 detA 1 = = q2 0 0

(7.8)

and a; b  0. The resulting system an be integrated as in the previous ase ( = 0). T The transformation X = X~ A where A veri es equation (7.8) is not a blowing-up as the inverse of A does not have any negative matrix entries. As a onsequen e a lo al integration of the resulting system of the form (7.4) is suÆ ient to nd solution

urves for system (7.6). However it an be dedu ed from the power transformation that the parti ular solutions (7.7) an not be obtained from the integration of the redu ed system. The solution urve (x; y) = (~xa y~ q2 ; x~ b y~q1 ) = (t~; 0) an only be obtained for y~ = 0 but limy~!0 x = limy~!0 x~a y~ q2 < 1 only if q2 = 0. q2 = 0 is impossible for this ase. For this reason the solutions (7.7) have to be added separately to the set of al ulated solutions.

Example 12 This example omputes solution urves for the pendulum equation introdu ed in example 1 in the neighbourhood of the elementary singularity X0 = (; 0). In a rst step the pendulum equation



x_ = y y_ = gl sin(x)

is translated to move the point of interest to the origin. Developping the translated system into a Taylor serie around 0 we obtain the system

X_ =





y

g (x l

1 6

1 x3 + 120 x5 + O(x7 ))

that we an treat with normal form algorithms. The Poin are-Dula normal form is

0 qg

 X~  = t

l

0

1 1 0 1 qg 2 ~ 4) x ~ y ~ + O ( X A q g A X~ +  16 1 ql g 0

l

16

~y~2 + O(X~ 4 ) lx

:

(7.9)

Chapter 7. Two dimensional elementary singular points

81

Figure 7.2: Solution urves al ulated for the pendulum equation in the neighbourhood of the singular point (; 0) in example 12. The urves obtained from the parti ular solutions (7.7) are dashed. Resonan es o

ur for all Q = k with yields the new system

1
1

 X t

for k 2 N . Applying a power transformation X~

A=

=



0

1 16

2 1 1 1

= X AT

! q

g xy l

q

gy l

This system has been trun ated to order 3. It an be integrated to X (t~) = ( ; t~) a

ording to

ase 3 in se tion 7.1. The orresponding solution urves and the parti ular solution urves X~ = (t~; 0) and X~ = (0; t~) of system (7.9) have been retransformed to the initial oordinates and sket hed for g=l = 1=2 in gure 7.2 without onsidering problems of onvergen e. (The dashed lines represent the solution urves obtained for X~ = (t~; 0) and X~ = (0; t~)).

7.2 Systems with real oeÆ ients

A system (7.1) that has real oeÆ ients an have a linear part with omplex eigenvalues 1 and 2 =  1 . In this ase the previously des ribed method will transform the system into its omplex Jordan anoni al form and therefore introdu e omplex oeÆ ients and yield omplex solutions. In this se tion it will be shown that it is possible to al ulate only the real solutions for real systems even if the omputations use omplex numbers. There are two ases to onsider. The ase where the eigenvalues are purely imaginary 1 = 2 and the ase when they have a non-vanishing real part.  Re(1 ) 6= 0 In this ase we know that the resonan e ondition (7.3) has no solutions for Q 2

82

7.2. Systems with real oeÆ ients N1 [ N2 .

The Poin are-Dula normal form is linear. A

ording to Sternberg [55℄ the system (7.1) an be linearized by a normalizing algorithm even without transforming the linear part of the system into its Jordan anoni al form. To al ulate this so

alled rational or A-normal form we an use the algorithms des ribed in hapter 3. The integration of the resulting linear system X~_ = DF (0)X~ is obvious.  =  In this ase the linear part of F has the form   DF (0)X = 0 0 X; ; > 0; ; 2 R : This linear part is transformed to its Jordan anoni al form   p  0 i ~ ~ p JX = 0 i  X by a linear transformation 1

2

X = P X~ with P

=

i

1 2 q

2

!

1



i

2 q

2

(7.10)



The resulting system has the parti ular stru ture  P Q  ~  X~ x ~ a X Q ~ = P F (P X ) = y~ P bQX~ Q t where the oeÆ ients verify aQ = bQ for all Q with q = q . This stru ture is preserved by the normal form onstru tion. Applying the inverse of transformation (7.10) to the resulting Poin are-Dula normal form ! P x^ P j a^ j;j X^ j;j  X^ = y^ a^ X^ j;j t j j;j yields a new system  0 P q  1  y~ )j Re(^  y~ ~ ~ ~ ( x + a ) x Im(^ a ) j;j j;j X  j  A  q = P   j ~ ~ t a j;j ) x~ + Re(^a j;j )y~ j (x + y ) Im(^ 1

1

2

2

2

2

(

)

(

)

(

)

(

)

(

)

(

)

(

2

)

(

)

that is transformed to 

r_ = 1 r + 3 r3 + : : : _ = d0 + d2 r2 + d4 r4 + : : :

p
by the introdu tion of polar oordinates X~ = rrpsin os . This new system an be integrated using previously introdu ed methods.

Chapter 7. Two dimensional elementary singular points Example 13 (Pendulum)



83

In example 8 the normal form of the system

x_ = y y_ = gl (x

x

1 3 3

+

x

1 5 120

+ O (x7 ))

:

was omputed. This system issued from modelling a planar pendulum in example 1. The normal form

 X^ t

0 qg i l =

1 0 i qg 1 2y ^ 4) x ^ ^ + O ( X A q g A X^ +  i 16q g l

0

i

0

is transformed to the new system

 X~ t

=



0 g l

by a transformation

1 0

l



^ 4) ^y^2 + O (X lx

16

1 2 1 l 3 16 ~ ~ 16 g ~ + 1 3 1 l 2 16 ~ + 16 g ~ ~ +

xy x

X~ +

y xy

O(X~ 4 ) O(X~ 4 )

(7.11)

!

0 q 1 x~ i gl y~ ql A X^ =  x~ + i

g y~

that is the inverse to the transformation that was used to obtain a system with a linear part in Jordan form. The introdu tion of polar oordinates

X~ = yields a new trun ated system

(

r t  t

r

l os()

= 0q =

!

) qr sin( g

g (1 l

r2

16 )

q

1 that an be integrated to (r(t~); (t~)) = ( ; t~) with t~ = gl (1 16 )t + 2 a

ording to ase 3 in se tion 7.1. These urves an be transformed to the initial oordinates. They are sket hed together with the exa t solutions in gure 7.3 without onsidering eventual problems of

onvergen e. 2

This hapter has shown that all 2 dimensional systems an be integrated if they are in Poin are-Dula normal form. However, the obtained results are in general no solutions of the onsidered systems but parametrizations of the solution urves.

84

7.2. Systems with real oeÆ ients

Figure 7.3: This gure shows the exa t solutions and the solution urves al ulated in example 13 for the planar pendulum for g=l = 1=2. We an see that the urves oin ide well in the neighbourhood of (0; 0).

Chapter 8

Two dimensional nonelementary singular points In the ase of a nonelementary singular point the appli ation of the Poin are-Dula normal form to a system X_ = F (X ) (8.1) with F 2 C 2 fX g or R2 fX g fails be ause 1 = 2 = 0 and resonan es o

ur for any exponent. For this reasons other te hniques, that are ontrolled by the Newton diagram of F , are employed. For two-dimensional systems the Newton diagram yields a set of (1) verti es (0) j with j = 1; : : : ; k and a set of edges j with j = 1; : : : ; k 1. For ea h of those fa es there exist algorithms to al ulate solution urves. For ea h vertex a time transformation and a power transformation transform the initial system (8.1) to a non-nilpotent system. This system an be integrated using normal form

onstru tions and the methods introdu ed in hapter 7. The edges of the Newton diagram are used to de ne blowing-ups that yield a nite number of "less ompli ated" new systems. Those systems are treated by applying the entire algorithm re ursively. However the solutions omputed by the algorithms asso iated to the fa es of the Newton diagram are only valid on parts of the initial neighbourhood. Those parts are alled se tors. Those se tors are de ned su h that they over the entire on erned neighbourhood though they might sometimes interse t ea h other. That means that solution urves are omputed for any point of the on erned neighbourhood. Blowing-ups of two-dimensional systems have been subje t of many publi ations. This

hapter is mainly based on works from F. Dumortier [32℄, M. Pelletier [45℄ and A. Bruno [9℄. Though we give a dierent explanation for the appearan e of se tors the se tor notation introdu ed by A. Bruno is largely used in this hapter. 8.1

The Verti es

Ea h verti e Q := (0) j of the Newton diagram an be used to de ne a time transformation dt~ = X Q dt:

85

(8.2)

86

8.1. The Verti es

(1)

j

R

(0)

j

1

(1)

j

1

1

-1

1

-1

-1

1

R

-1

Figure 8.1: The time transformation (8.2) translates the onsidered vertex to the origin. The new system (8.3) has its support within a onvex one V that ontains the rst quadrant. that translates the point Q to the origin in the spa e of exponents (see gure 8.1). This yields a new system 1 X (8.3) ~ = Q F (X ) t

X

that has a non-zero linear part. This system and the initial one in equation (8.1) are equivalent in the sense that the solution urves of both systems are parametrized solutions of the same s alar dierential equation. If the Newton diagram of the system onsists of more than one vertex the support of the new system has negative exponents and it lies within a onvex one +  + V = R R +R R (see gure 8.1). The one V ontains the rst quadrant and it is de ned by the ve tors  and R . For non-extremal verti es those ve tors lie on the edges (1) and (1) . R R j j 1 and R lead away from the origin and they verify s u  g d(s; t) = 1; g d(u; v ) = 1 with R = ; R = : t

v

For the extremal vertex (0) the ve tor R is de ned as R = (1; 0) and for the extremal 1 (0)  vertex k the ve tor R is de ned as R = (0; 1). Systems having their support within a onvex one are alled lass V systems. 8.1.1

The Poin are-Dula Normal Form for Class V Systems

Class V systems have been introdu ed by A. Bruno [9℄. They denote systems that have their support within a onvex one V . In the ase of a vertex this one is de ned by the ve tors R and R . A lass V system (8.3) an either be transformed to normal form dire tly via the de nition of the generalized Poin are-Dula normal form, as it is done by A. Bruno [9℄, or it

an be transformed to a system with positive integer exponents by a power transformation ~ AT : (8.4) X = X

Chapter 8. Two dimensional nonelementary singular points Theorem 10 The hange of oordinates X

A :=



= X~ AT v t

87

des ribed by the matrix 

u s

transforms a lass V system (8.3) to a new system with

integer exponents that has its support within the rst quadrant (R = st and R = uv ; s; v > 0; t; u  0).

Proof 3 All points Q that might appear in the support of the lass written as

V system (8.3) an be

Q = R + R

with ; ; 2 R+ . A

ording to theorem 3 in hapter 2 the ve tors R and R are transformed to the ve tors    

0

detA

and

detA

0

respe tively. For this reason Q is transformed to the point  ~Q =  0



detA



+ 0 ~ are The oordinates of Q detA

that has positive oordinates.

oeÆ ients.



integer as A has only integer

A

ording to the results from se tion 4.2.3 the matrix A an also be omputed via its adjoint matrix. For the matrix A = (R jR ) we obtain the matrix A form theorem 10. The new system is transformed to its Poin are-Dula normal form and integrated with the algorithms introdu ed in hapter 7. The advantage of this method ompared to the omputation of generalized normal forms for lass V systems is that the lassi al normal form algorithms from hapter 3

an be used. They are more eÆ ient than algorithms omputing normal form for lass V systems. Remark 5 If the employed power transformation is not bije tive the onstru tions introdu ed in hapter 2 an be used to de ne inje tive and pie ewise surje tive transformation. 8.1.2

The Se tors

A very important hara teristi of the solution urves al ulated by the algorithms asso iated to a vertex of the Newton diagram is that they are not valid in the entire on erned neighbourhood U = fX : jxj  ; jyj  g : This results from the fa t that the transformation (8.4) is a blowing-up. The origin X = 0 is transformed to the ex eptional divisor of the blowing-up and the neighbourhood U of the origin is transformed to the set U~ that denotes a neighbourhood of the ex eptional

88

8.1. The Verti es y

y~

x~

x

Figure 8.2: The neighbourhood U of the singularity X = 0 is transformed into a neighbourhood U~ of the ex eptional divisor. divisor. This is shown in gure 8.2. The solution urves for the set U~ an no longer be found with lo al methods as U~ is not a neighbourhood of a point. The algorithms from se tion 8.1 yield solution urves that are only lo ally valid in a neighbourhood of X~ = 0. We will presume that this neighbourhood an be denoted as ~j U

(0)

= fX~ : jx~j  Æj ; jy~j  Æj g

with Æj suÆ iently small. Transforming U~j(0) to the initial oordinates yields the de nition sets (0)

Uj

= fX : jX jR  ÆjjdetAj ;

jX jR  ÆjjdetAjg

(0) for the urves omputed for the verti es (0) j . The sets Uj are limited by the urves

8 < jxj = Æj detAs : jyj = ÆjdetAt jxj 8 < jxj = Æj detA u detA : jyj = Æj v jxj

s t u v

=0 if t 6= 0 if v = 0 if v 6= 0

if t

The sets Uj(0) are alled se tors. An example for a se tor obtained in example 14 is sket hed in gure 8.4. An important point in the de nition of the set U~j(0) is the hoi e of the parameter Æj . As it has been shown in hapter 4 the blowing-up might yield new singularities on the ex eptional divisor besides the one in X~ = 0. We have to onsider all those singularities. The singularities dierent from X~ = 0 will be studied in the algorihms asso iated to the edges of the Newton diagram. Therefore the set U~j(0) must be hosen su h that it does not

Chapter 8. Two dimensional nonelementary singular points

89

ontain any further singularities besides X~ = 0. This is guaranteed if the suÆ iently small. To simplify the representation of the se tors we hoose 

n

= min 1; min =1 fÆj k j

detA

j

Æj

are hosen

j go

and work with the neighbourhoods ~j U

(0)

= fX : jx~j   jdetAj ; jy~j   jdetAj g 1

1

in the oordinates X~ and with the the se tors (0)

Uj

= X : jX j

R

 ; jX j   ; jX j1  g R

in the initial oordinates. The parts of U that are not overed by the se tors asso iated to the verti es of the Newton diagram are asso iated to the edges (1) and treated separately. They are denoted by j

(1)

Uj

= fX :   jX j

R

The ve tor R is de ned as lying on the edge



1

j j1  g :

; X

and verifying the properties

(1) j

r1 < 0; r2 > 0; g d(r1 ; r2 )

=1

with R = (r1 ; r2 ). R is identi al to the ve tor R for the vertex (0) . The following example illustrates how solution urves in a se tor asso iated to a vertex of the Newton diagram are al ulated. The used power transformation is not bije tive. This makes additional onstru tions ne essary. j

Example 14 The Newton polygon of the system 

x t y t

= x2 y + xy4 = 2xy2 + x2 y

is sket hed in gure 8.3. It has a vertex

(0) 2

=




1 1

. Applying the time transformation

dt~ = X (1;1) dt

yields a system

(

= x + x 1 y3 = 2y + xy 1 ~

x

(8.5)

~ t y t

that has its support within a one V de ned by the ve tors R =

one is shown in gure 8.1. The transformation

~ T; X =X A

A

=



3 1 1 1



1 3




and R =

1 1




. The

90

8.1.

The Verti es

q

2

(0) 3

R (1) 2

(0) 2 (1) 1

R

q

(0) 1

1

Figure 8.3: The Newton diagram for the system treated in example 14.

is not bije tive. Therefore we use the two transformations ~ AT and X = X =X

de ned on the set

(0)

~ U 2

f

3

x ~ y ~




(8.6)

x ~y ~

j j   21 ; jy~j   21 ; x~  0g

~ : x = X ~

that onsists of two quadrants of a neighbourhood of X~ = 0. The transformations (8.6) transform system (8.5) to the two systems (

x ~

=

~ t y ~

=

~ t

and

(

x ~

~ t y ~ ~ t

The set

(0)

~ U 2

= =

1 3 1 x ~ x ~ 2 2 5 3 2 y ~ + x ~ y ~ 2 2

1 x ~ 2

5 y ~ 2

+

1 x ~y ~2 2 1 3 y ~ 2

1 3 1 x ~ x ~y ~2 2 2 3 2 1 3 x ~ y ~ + 2y ~ 2

+

(8.7)

:

(8.8)

is transformed into the two sets (0)

f j j = fX : jX j

U2;1 = X : X (0) U2;2

(

1;3)

(

1;3)

 ; jX j  ; jX j

(1;

1)

(1;

1)

 ;  ;

g xy < 0g

xy > 0

That form the se tor U2(0) . To nd solution urves valid for U2(0)1 and U2(0)2 the two systems (8.7) and (8.7) are treated like any system with an elementary singular point. Figure 8.4 shows U~2(0) and the se tor U2(0) for  = 21 . ;

;

For  < 1 the onstru tions for the verti es of the Newton diagram yield solution urves that do not entirely over the initial neighbourhood U . Solutions for the remaining se tors are asso iated to the verti es of the Newton diagram. They are found using blowing-ups.

Chapter 8. Two dimensional nonelementary singular points

91

Figure 8.4: The set U~2(0) and the solution urves X~ (t) otained for the system (8.7) are transformed to the initial oordinates. We obtain the set U2(0)1 and solution urves X (t) for the initial system that have been sket hed without onsidering any problems of onvergen e (see se tion 1.5). See also example 14. ;

8.2

The Edges

The se tors

(1)

Uj

() = fX :   jX j

R



1

;

jjX jj1  g

are asso iated to the edges (1) of the Newton diagram. Within these se tors the initial system (8.1) is redu ed by quasihomogenous blowing-ups that were introdu ed in se tion 4.2.3. These blowing-ups yield a nite number of new systems that have no or "less

omplex" singularities . The matrix de ning the orresponding power transformation is de ned by the ve tors R verifying j

r1 < 0; r2 > 0; g d(r1 ; r2 )

and lying on the edges 8.2.1

(1) j

of the Newton diagram.

Redu tion of the Singularity

The se tor

(1)

Uj

() = fX :   jX j

R



1

is limited by the plane algebrai urves 1

jyj =  r2 jxj and

=1

jyj = 

1 r2

jxj

r1 r2

r1 r2

;

jjX jj1  g

92

8.2. The Edges

Figure 8.5: The gure shows that the set set U~j(1) does not entirely over the set (8.11) if any unimodular matrix is used (left). If matri es of the form (8.9) or (8.10) are used, the set U~j(1) , that is drawn with dashed lines, entirely ontains the set (8.11) (right gure). To study the behaviour of the system (8.1) within this se tor an appropriate blowing-up

an separate these urves. This blowing-up is de ned by the hange of oordinates X

with the matrix A

=



= X~ A

T

r2

r1



d

with R = (r1 ; r2 ). The matrix oeÆ ients and d an be hosen su h that detA = 1. This makes sure that the power transformation is a dieomorphism (see se tion 2.3). However, we will also onsider the matri es A

=

if r2 is odd and the matrix =



r2

0



r2

r1



1

r1



(8.9)

(8.10) 1 0 if r2 is even. Those matri es also de ne dieomorphisms. Using those matri es the representation of the se tors Uj(1) is more ompli ated as in the ase of unimodular matri es but the al ulated solution urves are valid in the entire se tor Uj(1) . This is not the ase if unimodular matri es are used (see gure 8.5). T Applying a blowing up X = X~ A to the initial system (8.1) yields a new system A

~

X t

that has the following properties :

= F~

Chapter 8. Two dimensional nonelementary singular points

93

q~2

q2

(0) 2 (1) 1

~ (1) 1

(0) 1

q~1

q1

Figure 8.6: The gure shows the Newton diagram and the support of the system treated in example 15. After having applied a blowing up the edge (1) has been straightened up 1 (1) ~ to the verti al edge 1 .

 For the matri es (8.9) and (8.10) the origin is blown up to the ex eptional divisor f0g  k where k = C or R. In the ase of unimodular matri es dierent from the matri es (8.9) and (8.10) the ex eptional divisor is k  f0g [ f0g  k.  The se tor Uj has been transformed to the set (0)

fX~ :  detA  jy~j   j

1

j

j

1 detAj ; jx ~jr2 jy~ j  ; jx~j r1 jy~d j  g

(8.11)

but we will work with the simpli ed expression ~j U

(1)

1 = fX~ :  detA j

j

 jy~j  

j

detAj ; jx ~j   jdetAj g 1

1

(8.12)

that ontains the set (8.11) for the matri es (8.9) and (8.10). In the ase of unimodular matri es the se tor U~j(1) does in general not entirely ontain the set (8.11). This is sket hed in gure 8.5.

 The edge

has been "straightened up". That means that the ve tor R has been transformed to a verti al ve tor   0 ~ R = AR = : (1)

j

detA

(0) (1) The points (0) j and j +1 lying on the edge j have been transformed to points with identi al q~1 - oordinates. This is illustrated in gure 8.6.

 As the support of the new system lies entirely on the left of the straightened edge the new system has the form



 x~ t  y~ t

= x~s+1 f~(~x; y~) = y~x~s g~(~x; y~)

(8.13)

94

8.2. The Edges y

y~

Im( ~)

S~1 1 ;

S~1 2

S~1 6

S~1 1

;

;

;

S~1 7 ;

S~1 5 ;

y

x~

Re( ~)

S~1 2

S~1 3 ;

S~1 3

;

;

S~1 4 ;

Figure 8.7: The set U~1(1) from example 15 is de omposed in subse tors for the real and the

omplex ase. The only appearing singularity is the one in X~ = (0; 1). where s is the q1 - oordinate of the straightened edge. The initial problem was to nd solution urves for the initial system (8.1) in the neighbourhood U of the origin X = 0. Applying the blowing-up this has been redu ed to a new problem. Now we must nd solution urves of a new system (8.13) in the neighbourhood ~j(1) of the y~-axis. This problem an be solved by splitting the set U~j(1) into so alled U subse tors and by de ning a re ursion. 8.2.2

Subse tors and Re ursion

To solve the problem of nding solution urves within U~j(1) two dierent ases have to be

onsidered. To distinguish those ases the trun ated system

0 P _ = ^j =  P fQ:Q2 x

X

F

y

fQ:Q2

(1) j (1) j

g

aQ X

g bQ X

Q

Q

1 A

(8.14)

is de ned. This system ontains only those monomials in (8.1) that have their support on the edge (1) j . By the blowing up the trun ated system (8.14) is transformed to the system  dx~ s+1 ^ ~ f (~y) dt = x (8.15) dy~ sy = x ~ ~g^(~y): dt that has its support on the straightened edge ~ (0) j . The transformed trun ated system (8.15) is used to distinguish the two ases: 

^(~)  0. Applying a time transformation dt~ = x~s+1 dt to system (8.13) yields ( x~ = f~(~x; y~)  t~  y~ = xy~~ g~(~x; y~):  t~

g y

(8.16)

Chapter 8. Two dimensional nonelementary singular points

S1 1 ;

95

U2(0) S1 3 ;

S1 2 ;

U1(0)

Figure 8.8: The division of the neighbourhood of the origin into se tors and subse tors in the rst and the se ond level of re ursion for the system treated in example 15 (U1(1) = S1 1 [ S1 2 [ S1 3 ). ;

;

;

To al ulate the new singular points on the ex eptional divisor let x~ = 0. The resulting system ( ~ ^ = f (~ y) ~  ~ ~ g ~(~ x; y ~) : ~ = ~ dx dt

dy

y

dt

x

x ~=0

has the following properties: 1. The y-axis is no solution urve. 2. All points in f(0; y~) : f~(0; y~) 6= 0g are regular points and x~(t) is not onstant there. Therefore there exists a solution urve passing through ea h of those points.   3. All points in f(0; y~) : f~(0; y~) = 0; ~ g~(~ x; y ~) 6= 0g are regular points but y

x ~

x ~=0

here the solution urves are parallel to the y~ axis as x~(t) is onstant. Those points are alled tangen ies.   = 0g are singular points. 4. All points in f(0; y~) : f~(0; y~) = 0; ~~ g~(~x; y~) ~=0 Those points are translated to the origin and the entire algorithm is applied re ursively to the resulting systems. y

x

x

This ase is alled the di riti al ase as tangen ies and singularities an o

ur.



^(~) 6 0. Applying the time transformation dt~ = x~ g y

(

s

. (8.13) yields the system

dt

= x~f~(~x; y~) = y~g~(~x; y~): ~

x ~

~ t y ~ t

(8.17)

96

8.2. The Edges On the ex eptional divisor, for x~ = 0 the system is (

=0 ~g^(~y): ~ = y

x ~

(8.18)

~ t y ~ t

Some properties for the system (8.17) an now be given: 1. The y-axis (~x(t) = 0; y~(t) = t) is a solution urve of system (8.17). 2. All points fX~ 2 U~ (0) : g~(0; y~) 6= 0g are regular points. From equation (8.18) shows that here the solution urves will be parallel to the y-axis. 3. All points in fX~ 2 U~ (0) : g~(0; y~) = 0g are singular points. They might be elementary or not. Those points are translated to the origin and the entire algorithm is applied re ursively to the resulting systems. j

j

This ase is alled the non riti al ase as tangen ies do not o

ur. The above lassi ation has shown that in any ase the resulting system an be solved in any point. A entral point is the re ursive appli ation of the entire algorithm. Those re ursions have to be de ned su h that they yield solution urves for the whole set U~ (1) . Therefore the set U~ (1) is split into so alled subse tors S~ 1 ; : : : ; S~ . Any of those subse tors ontains either one singular point or only regular points. The subse tors are onsidered as neighbourhoods of the points X~ 1 ; : : : ; X~ . The point X~ with i = 1; : : : ; k is either a singular points or any regular point. A possible splitting of a set U~ (1) into subse tors in the ase of real and omplex variables is sket hed in gure 8.7. With the systems (8.16) or (8.17), the subse tor S~ and the point of interest X~ the entire algorithm an be alled again. The rst step in the next level of re ursion will be a translation of the point X~ to the origin. If the singularity has not been redu ed by a rst appli ation of a blowing-up another one is applied. A. van den Essen [61℄ has proved that any isolated singularity an be redu ed entirely by a nite number of blowing-ups. So this algorithm will ome to an end after a nite number of steps. j

j;

j

j;

j;k

j;i

j;k

j

j;i

j;i

j;i

Example 15 The Newton diagram of the onsidered system

_ = X has only one edge

(1) 1

with (1)

U1

R



=

= fX :






+ yx3

13 2 6 y x 9 1 1

1 2

x4

x2 y 2

(8.19)

+ xy3

(see gure 8.6). For  =

1 2

the se tor

 jxj jyj  2; jxj  12 ; jyj  21 g 1

is asso iated to this edge. The blowing-up A

=

X



= X~ 1 1 0 1

T

A



with

Chapter 8. Two dimensional nonelementary singular points X~ 1 = 0

97

y~1

U~1(0) U~1(1)

y~2 x~2

X~ 2 = 0

y~3 x~3

X~ 3 = 0

x~4

X~ 4 = 0

Figure 8.9: This gure shows the stru ture of the ex eptional divisors of the blowing-ups used for the fa es of the Newton diagram. transforms

U

(1) 1

into

~1(1) = fX~ : 1 2

U

and yields the new system

~_ = X



 j ~j  2 j ~j  12 g y

;

x

~4 + x~4 y~ x ~3 y~ 2~x3 y~2 + x3 y3 + 139 x7 y2 x

 :

That means a non riti al ase. A new time transformation with s = 3. yields a resulting system that has a nonelementary singular point in (0; 1). For the real ase the se tor U~1(1) is divided into 3 subse tors as shown in gure 8.7. The subse tors S~1 1 and S~1 3 ontain only simple points but the remaining subse tor S~1 2 ontains a nonelementary singular points. Therefore in the se ond level of re ursion another blowing-up has to be performed. In the se ond level of re ursion the solution urves for all se tors and subse tors of S1 2 an be al ulated dire tly as the blown-up system has only simple points. Figure 8.8 shows the se tors and subse tors resulting for this example in the rst and se ond level of re ursion. ;

;

;

;

This hapter has shown how blowing-ups and time transformations an be used to nd solution urves for two-dimensional systems of dierential equations in the neighbourhood of a nonelementary singular point. The omputed solutions are valid only on parts of the neighbourhood that are alled se tors.

98

8.2. The Edges

For both ases, for the ase of an edge and for the ase of a vertex, blowing-ups are used to redu e the onsidered system. This fa t imposes to study the stru ture of the blowing-ups and their ex eptional divisors more losely. Consider the blowing-ups for the (1) (0 verti es (0 j and j +1 and the edge j . They are denoted by X

AT

= X~ j j ;

X

AT

j +1 = X~ j +1

and X

BT

= X^j j

respe tively. As those power transformations are invertible anywhere ex ept in their ex eptional divisors we an onstru t the transformations j +1 Aj T ~j = (X~ jA+1 ) T

(8.20)

X

and

~ j = (X^jBj )Aj T T

(8.21) Those power transformations an be ompleted in the ex eptional divisors of the initial power transformations. Computing the transformations (8.20) and (8.21) expli itely yields X

1 ~j = k ; x ~j +1

y

k >

0;

:

k

2Q

and

1 ~j = k ; k > 0; k 2 Q y ^j if the matrix (8.9) is used to de ne the blowing-up for the edge and y

~j = y^jk ;

y

k >

0;

k

2Q

if the matrix (8.10) is used. That means that a part of the ex eptional divisor for all three

onsidered blowing-ups is identi al. However as x +1 !0 y~j =

lim ~j

1

the point X~ j = (0; 1) is identi to the point X~ j +1 = (0; 0). To visualize this relation the lines x~j = 0 and y~j = 0 of the ex eptional divisors an be displayed as ir les as it has been done in gure 8.9. This gure shows the stru ture of the ex eptional divisors omputed for a Newton diagram that onsists of four verti es and three edges. The neighbourhoods of the ex eptional divisors are split into the sets U~1(0) ; : : : ; U~3(1) a

ording to this gure. Now, it is obvious that the power transformations applied for the redu tions of lass V systems ould also be used for the ase of an edge to de ne a re ursive pro ess. Nevertheless the ase of an edge and the ase of a vertex of the Newton diagram are treated separately for two reasons. The se tor de nition is simpler and the use of transformations that are no dieomorphisms to de ne re ursions would in rease the ost of the algorithm. This is due to the additional onstru tions introdu ed in se tion 2.3 that have to be used in the

ase of non-inje tive power transformations.

Chapter 9 Three- and higher-dimensional elementary singular points The ase of a two-dimensional elementary singular point has been subje t of hapter 7. There it was shown that any two-dimensional system an be integrated in the neighbourhood of an elementary singular point. In this hapter we are interested to know whether these results an be extended to higher dimensional systems. Like in the ase of two-dimensional systems the rst step is the transformation of the

onsidered system to Poin are-Dula normal form. For higher-dimensional systems the

omputation of the Jordan form might ause some problems that were already mentioned in se tion 3.2. However, here we will presume that the Jordan form an be omputed and that the onsidered system X_ = F (X ) (9.1) is given by a formal power serie and that it is in Poin are-Dula normal form. The resonant plane M = fQ 2 N : hQ; i = 0g of the normal form (9.1) has been de ned in se tion 3.1. It plays a entral role in the redu tion and integration of the system (9.1). The set M is used to de ne the ve tor spa e M = fP : P =

X  Q ;  2 R; Q 2 M g : i

i

i

i

The methods used for the redu tion of the normal form depend on m that denotes the maximum number of linearly independent ve tors in M . It is obvious that m is smaller than the dimension of the system. For m = 0 the system (9.1) an be integrated dire tly as it is linear and the matrix DF (0) is in Jordan form. For m = 1 the onsidered normal form an always be integrated. The integration methods are similar to those already used in se tion 7.1. If M ontains two linearly independent ve tors one or several power transformations an be used to redu e the normal form. The redu tion yields systems that an be integrated via the integration of a m-dimensional nilpotent system. However, the employed power transformations have to verify very stri t onditions. The methods for redu ing systems with elementary singular points are extensively studied for three-dimensional systems. A generalization to n-dimensional systems is pos99

100

9.1. Integration of n-dimensional normal forms for m = 1

sible but some further diÆ ulties arise from the fa t that a three- or higher-dimensional

one an be spanned by any number of ve tors. The general ideas for the algorithms des ribed in this hapter have been studied by A.Bruno [9℄ and L. Brenig and A. Goriely [23℄. In this hapter many aspe ts of those algorithms are studied more intensely and some further onstru tions are introdu ed. The se tions 9.2 and 9.3.3 deal with real normal forms for systems with two omplex onjugated, purely imaginary eigenvalues. They are based on works from S. Chow, C. Li and D. Wang [15℄. 9.1

n-dimensional normal forms for m = 1

Integration of

If the resonant plane M of a normalized system (9.1) ontains only one linearly independent ve tor (m = 1) the results from se tion 7.1 ( ase 3 and 4) an be generalized. Two ases have to be studied separately:



= f0g Without loss of generality it an be presumed that M

\N

n

Q0

0 B =B B 

1 C C C A

1 q02

.. .

q 0n

with q0 2 N is the only ve tor dierent from Q = 0 that lies in M . The normalized system (9.1) has the form i

X t

= DF (0)X

0 B +B B 

a1;Q0 X

(0;q02 ;:::;q0n )

0 .. . 0

1 C C C A

:

It an be integrated using the method of the variation of the onstant.



M

\ N 6= f0g n

There exists a ve tor Q0 2 M su h that all other ve tors Q 2 M an be written as kQ0 with k 2 N. The normalized system (9.1) has the form _ = DF (0)X

X

0 P + P x1

xn

a1k X

k

kQ0

:::

k

ank X

kQ0

Like in se tion 7.1 there exists a power transformation this system to a system ~ X t

0~P ~ (0) ~ +  P ~

= DF

x1

X

k

~ ~

k

a1k x1

:::

xn

k

~ ~

k

ank x1

1 0 A=

1 A

X

~

(9.2)

:

= X~

A

T

(~ )

x1 g1 x1

~

:::

(~ )

x n gn x 1

that transforms

1 A

(9.3)

Chapter 9. Three- and higher-dimensional elementary singular points

101

that an be integrated. If g1 6 0 the resulting equation (9.3) an be transformed to a new system of dierential equations

(

 x~1 = 1  t~ logx~i = gi (~x1 ) x~1 g1 (~x1 ) ;  t~

by a time transformation

i

(9.4)

= 2; : : : ; n

~ = x~1 g1 (~x1 ) dt

dt

In equation (9.4) ea h line is integrable. The lines 2; : : : ; n an be integrated to ~i =

Zt

~

log x

g

i (s )

( )

sg1 s

ds

+ i

:

 0 yields x~1(t) = 1 for the rst variable. As a onsequen e equation (9.3) has the form 0 0 1 B x~2 k2 C X C =B B t  ... C A x ~ n kn with the onstant terms ki = gi ( 1 ); i = 2; : : : ; n. This an formally be integrated to

g1

~ ( ) = ( 1 ; 2 ek2 t ; : : : ; n ekn t ) :

X t

The matrix A for the power transformation is best al ulated via its adjoint matrix  1 A = detA A . Any matrix A with  A = ( Q0 j : : : )

su h that A has no negative entries an be used. The fa t that A has no negative entries guarantees that the power transformation is no blowing-up. Like in se tion 7.1 the parti ular solutions ~) = (t~; 1 ; : : : ; n 1 ) with t~ = e1 t ; 1 ; : : : ; n 1 2 R or C X (t ~) = ( 1 ; : : : ; n X (t

:::

~ ~ n t ; 1 ; t) with t = e

1 ; : : : ;

n

1

2R

or

C

of equation (9.2) have to be onsidered separately if they an not be obtained from solution urves al ulated for the redu ed system (9.3). Example 16

The normal form of the onsidered system _

X

has the form ~

X t

01 = 1

0 1 + p2 =

2 1 0 0

1

0 0 2

1 A

X

2

x1 x2 x3

0

2

1 A

x1 x2 x3

p

2

0 +

2

1 0 A ~ +B  X

p2 2 16 p2x~1 x~2 x~3 x ~ x~2 x~ 16 1 2 3 0

1 C A

:

102

9.2. Integration of real n-dimensional normal forms for m = 1

The resonant plane

8 0 < M = k :

9 11 = 1 A : k 2 N; 1

q~1 -axis by the hange of oordinates X~ = X AT

is transformed to the

1 1 01 A =  1 0 1 A 1 0 0

with

0

0 0 1 1 and A =  1 0 1 A 0 1 1

0

:

This yields the new system

 X t

0

=

0

p

1+ 2

0 0p A X + B  162px2 x1 1

1 p2

2 16

x3 x1

1 C A

that an be integrated to



p2+1+ p2

X (t) = 1 ; 2 e( with

9.2

1 ; 2 ; 3 2 R

or

t ; 3 e(

16 1 )

p2+1 p2 )t  16 1

C.

Integration of real

n-dimensional normal forms for m = 1

A parti ular normal form with m = 1 an issue for a system with real oeÆ ients and two

omplex onjugated, purely imaginary eigenvalues. The two-dimensional ase has been the subje t of se tion 7.2. The system X_ = F (X ) that has real oeÆ ients and the purely imaginary eigenvalues  and  =  an be transformed to a system  X~ ~ ~ = F (X ) t 1

2

1

with real oeÆ ients where the linear part is in real Jordan form 0 B1 B ~ DF (0) = 

The blo k

B1 =



... 0



Bk

1 C : A



0 represents the eigenvalues  and  ( = Im( )). The other blo ks are either Jordan blo ks for real eigenvalues or real Jordan blo ks for omplex onjugated eigenvalues. The used linear transformation X = T X~ is also real. 1

2



1

Chapter 9. Three- and higher-dimensional elementary singular points

103

The next step is the appli ation of a linear transformation

0 X~ = P X with P = B 

1 2

1 2

i

i

2

2

..

.

1 C A

that yields a new system with a linear part in Jordan form

0 i DF (0)X = B 0

0

i

..

.

1 C A X :

This system is transformed to its Poin are-Dula normal form

0 i ^ X B 0 = t 

0

i

1 0 x^1 P a1k x^k1 x^k2 1 P ^k x^k C B x ^2 a2k x C 1 2 C ^ B AX +  A : : : .. P . k k x^ a x^ x^ n

2n 1 2

^ = that has the parti ular form a1k = a2k . Applying the linear transformation X the normalized system yields a new system

 X~~ t

0 B =

0



 0

0 P ~ 2 ~2 k 1 ~ ~ ( x ~ + x ~ ) (Re( a ) x ~ Im( a ) x ~ ) 1 B P(x~~21 + x~~22 )k (Im(a1k )x~~1 + Re(a1k )x~~2) C B 1k 1 1k 2 C 1 2 B C C P ~ 2 2 C ~ B k ~~3 a ~~3k (x ~~1 + x ~~2 ) X +B A x C .. B C . ::: A  P 2 2 k ~ ~ ~ ~ x~ a~ (x~ + x~ ) n

nk

1

2

0 r sin  1 B r os  C C B B ~ X~ = B x~~3 C C B .. A  . C

Introdu ing polar oordinates

x~~n

~~1 and x ~~2 yields the integrable system for the oordinates x

8 > < > :

Example 17

r_ _

~~3 x t

:::

= 1 r + 3 r 3 + : : : = d0 + d2 r 2 + d4 r 4 + : : : ~~3 (e0 + e2 r 2 + e4 r 4 + : : :) = x

Consider the system

0 X_ = 

0 4 0 4 0 0 0 0 2

1 0 3x 2 x AX +  0 1 2

:

x3 3

x1 2 x2 x3 x1 x2

1 A:

P X~~

to

104

9.3. Redu tion of three-dimensional normal forms for m=2

Its linear part is in real Jordan form. Transforming it to Jordan form using the transformation X P X yields the new system

= ~

 X~ t

0 4i 0 0 1 0 =  0 4i 0 A X~ +  0 0 2

3 8 3 8 1 4

I x~31 I x~31 I x~21

3 8 3 8 1 4

I x~21 x~2 + 38 I x~1 x~22 + 83 I x~32 I x~21 x~2 + 38 I x~1 x~22 + 83 I x~32 I x~22 + : : :

x~33 x~33

1 A:

This system is transformed to Poin are-Dula normal form

0 4I 0 0 1 0 4 I x 3 I x2 x + 27 I x3 x2 + : : : 1 1 1 2 1 2  8 1024 X  27 I x21 x32 + : : : A = 0 4 I 0 A X +  4 I x2 + 83 I x1 x22 1024 t 75159 x3 2211825664 0 0 2 x41 x3 x42 + : : :  = P 1 X^ . This yields the new system and retransformed by the transformation X 0 1 0 3 2 3 3 ::: 1  X^  0 4 0 A ^  8 x^3 1 x^32 +38 x^2 + = 4 0 0 X + 8 x^1 8 x^1 x^22 + : : : A : t 2245477 ^41x^42x^3 + : : : 0 0 2 1105912834 x 0 r sin  1 X^ =  r os  A

Introdu ing polar oordinates

x^3

yields a new integrable system

8 < :

9.3

r t  t  x^3 t

843792 9 = 5399965 r + ::: = 4 6 r2 + 274 r4 + : : : = 2 x^3 + : : : :

Redu tion of three-dimensional normal forms for m=2

The redu tion and integration of three dimensional normal forms is of parti ular interest as the dimension of the resonant plane does not ex eed 2 and two-dimensional systems

an be integrated by methods introdu ed previously in the hapters 5 to 8. Three-dimensional normal forms have a resonant plane with 0,1 or 2 linearly independent ve tors. For m = 0 the integration of the normal form is obvious. For m = 1 the algorithms from se tion 9.1 an be usedT to integrate the onsidered normal form. For m = 2 a power transformation X = X~ A an be used to redu ed the normal form to a two-dimensional system. The hoi e of the matrix A is the main problem. Systems that are treated in a parti ular way in se tion 9.3.3 are systems with real

oeÆ ients and two omplex onjugated, purely imaginary eigenvalues. It will be shown that they an be redu ed to systems that are also real. A If M ontains 2 linearly independent ve tors a power transformation an be used to simplify the normal form (9.1). 9.3.1

The Choi e of the Matrix

Chapter 9. Three- and higher-dimensional elementary singular points

105

Suppose that the ve tors Q1 ; Q2 2 M form a basis for M . Then the normal form (9.1)

an also be written as 0 P P 1

x1 Pk0 PjRj=k a1R X R X_ = B A  x2 Pk0 PjRj=k a2R X R C x3 k0 jRj=k a3R X R

with

R = 1 Q1 + 2 Q2

and

jRj = jr1 j + jr2 j + jr3 j where R = (r1 ; r2 ; r3 ). 1 2 Z and 2 2 Z are hosen su h that R 2 N . A

ording to T theorem 3 in se tion 2.2 applying a power transformation X = X~ A with

0 1 0 1 1 0  A  AQ1 = det A 0 and AQ2 = det A 1 A 0

yields a new system

0 P P 1 x ~1 k0 jR~ j=k a~1R~ X~ R~  X~ B P P =  x~2 k0 jR~ j=k a~2R~ X~ R~ C A t P P ~ R~ x~3

with

0

k0

a X

(9.5)

~ j=k ~3R ~ jR

R~ = 1 AQ1 + 2 AQ2 ; jR~ j = jr~1 j + jr~2 j) :

The system (9.5) an also be written in the form

0 1 x~1 g1 (~x1 ; x~2 ) =  x~2 g2 (~x1 ; x~2 ) A : t x~3 g3 (~x1 ; x~2 )

 X~

(9.6)

The rst two lines only depend on the variables x~1 and x~2 . The 2-dimensional system



 x~1 = x ~1 g(~x1 ; x~2 ) t  x~2 = x ~ x1 ; x~2 ) 2 g2 (~ t

(9.7)

is formally integrated by previously introdu ed methods. If the system (9.7) is integrable the solution urves X~ (t) an be obtained from the solution urves (~x1 (t); x~2 (t)) by formally integrating the s alar dierential equations log x~3 = g3 (~x1 (t); x~2 (t))

t

that is equivalent to the remaining equation in system (9.6). The problem of nding solution urves for the initial normalized system (9.1) has been redu ed to the problem of nding an appropriate power transformation and an appropriate matrix A. A is best

onstru ted via its adjoint matrix

A = (Q1 jQ2 j : : :) where the ve tors Q1 and Q2 de ne the rst and the se ond row ve tors. However some essential properties are required for the system (9.6). Those properties ae t the hoi e of the ve tors Q1 and Q2 and therefore the omputaion of the matrix A.

106

9.3. Redu tion of three-dimensional normal forms for m=2

 

as it has been shown the ve tors Q1 and Q2 have to form basis of M . the oeÆ ients of the system (9.6) have to be integer and positive. Otherwise the resulting system an not be integrated by the previously introdu ed algorithms.



T the power transformation X = X~ A has to be a dieomorphism. This is veri ed if the matrix A is unimodular. If A is an non-unimodular invertible matrix the

onstru tions introdu ed in se tion 2.3 an be used to de ne a orresponding bije tive dieomorphism.



T if the power transformation X = X~ A is not a blowing-up, the solutions of the resulting systems are valid in the whole neighbourhood of X = 0. T If the power transformation X = X~ A is a blowing-up, the redu tion and integration yield solutions that are only valid in se tors As a single transformation is not suÆ ient to over a neighbourhood of X = 0 with se tors, a serie of blowing-ups

ontrolled by a Newton diagram are needed.

The following lassi ation allows to onstru t appropriate matri es A for any threedimensional normal form.

9.3.2 The Classi ation In this se tion a lassi ation of three-dimensional Poin are-Dula normal forms is given. This lassi ation uses the eigenvalues of the linear part of the system and an easily be implemented. It allows an exa t de nition of the resonant plane M . Further it makes sure that all ases are onsidered and it allows the redu tion of any three-dimensional normal form. In the study of three-dimensional normal forms we will fo us on systems with rational eigenvalues and show later that all other ases an be derived from these systems. 1. 1 > 0; 2 < 0; 3 = 0 Let

k1 := minfk 2 Q :

0 the set M an be written as M = f k  1

1 2 Z and k2 2 Zg k 1 001 2 1 A +  0 A :  2 N ; 2 N ;  +  0

1

0g The support lies on a plane (m = 2). All points in supp(F ) are in N 3 . Let Q1 ; Q2 2 M \ N 3 be the two ve tors that span the one V that in ludes entirely T the support of the onsidered system. Then the power transformation X = X~ A de ned by the matrix 0 0 11

a A = Q1 jQ2 j  b

AA

with a; b; 2 N yields a system that veri es all onditions for the further integration of the orresponding two-dimensional system.

Chapter 9. Three- and higher-dimensional elementary singular points q~3

q3

q~2

q2 Q2

107

Q3

Q~ 2

Q1

q~1

Q~ 1

q1

Figure 9.1: This pi ture shows the plane < ; Q >= 0, supp(F ) and the ve tors that are

hoosen for the onstru tion of the matrix A in example 18. The support of the new system lies in the q~1 q~2 -plane. Proof 4 Any point Q 2 supp(F ) an be written as

Q = 1 Q1 + 2 Q2 with 1 ; 2  0. Q is transformed to the point

011 001 Q~ = 1 det A  0 A + 2 det A  1 A 0

0

that has only positive oordinates. As A has only integer oeÆ ients the exponents Q~ = AQ of the new system are integer and positive. The rst and the se ond line in the resulting system only depend on the variables x~1 and x~2 . As the matrix A = det A A 1 has no negative oeÆ ients the transformation X = X~ AT is not a blowing-up.

Example 18 Consider the normalized system

01 X_ = 

3

3

1 0 x4 x x2 1 2 A X +  x221 x3 3 A : 0

As we an see in gure 9.1 the one V that ontains the set M is spanned by the ve tors Q1 = (3; 0; 1) and Q2 = (0; 1; 1). They are used to build the matrix

03 A =  0

0 2 1 0 1 1 1

1 A:

108

9.3.

The power transformation

Redu tion of three-dimensional normal forms for m=2

X = X~ AT

00 ~ X  =

yields the new system

0

t

1

with positive, integer support.

1 0 3~x2 x~ 1 A X~ +  x~22 1 2 A : xxx

2~1 ~2 ~3

Using a one that does not ontain

example the one spanned by the ve tors

Q1

and

Q3 = (3; 1; 2),

supp(F ),

for

yields a system with

negative support.

2.

1 > 0; 2 > 0; 3 > 0 or 1 < 0; 2 < 0; 3 < 0

 9k; l 2 0N : 1 =1 k2 ;02 = l13 0 1 0 M = f k A +   1 A ;  l

1 001 1 A ;  0 A :   k;  2 N g

0

l

0

0 The resulting system an be written as

0 1 X_ = 

where

1 2 2 3

1 0 p (x ; x ) 1 A X +  p21(x32 ) 3 A 0

p1 is a power serie in x2 and x3 , p2 is a power serie in x3 and i 2 f0; 1g.

The system an be integrated dire tly using the separation of the onstant.

 9k 2 N0: 1 =1k20 , and there does not exist a l 2 N 1 M

1

=

2 = l3

f k A ;  0 Ag 0



su h that

0 0

Like above the system is dire tly integrable.

0 1 M = f 0 Ag

There does not exist a 0

k; l 2 N

su h that

1 = l2 ; 2 = k3

0

The system is diagonal and therefore integrable. 3.

1 6= 0; 02 = 01; 3 =00

1 M = f  0 A +  1 A : 0 1

0

(

0

; ) 2 N 2 [ f( 1; k); (k;

The system has the form

0 1 X_ = 

1) :

k 2 N; k

 1gg

1 0 x f (x ; x )) 1 1 2 3 A X +  g(x2 ; x3 ) A

0

h(x2 ; x3 )

0 A two-dimensional system in the variables

x2

and

x3

an be split dire tly. After

having solved the two-dimensional system the third equation an be integrated. 4.

1 > 0; 2 > 0; 3 = 0 or 1 < 0; 2 < 0; 3 = 0

Chapter 9. Three- and higher-dimensional elementary singular points

109

 1 = 2 0 1 0 1 0 1 0 1 1   A  A 1 + 1 A :  2 N ; ; 2 f0; 1gg M = f 0 + 1

0

1

As and basis of M ontains at least one ve tor with negative oordinates the

onstru tion of a power transformation that is not a blowing-up is not possible. For this reason blowing-ups ontrolled by the Newton diagram of the on erned system will be used. The support of the onsidered system lies in the one V de ned by two ve tors Q1 = ( 1; 1; 1 ) and Q2 = (1; 1; 1 + 2 ). Then the Newton diagram onsists of the three verti es (0) 1 = ( 1; 1; 1 ); (0) 2 = (0; 0; 0); (0) 3 = (1; 1; 1 + 2 ) (1) 1

and the edges

and

(1) 2

de ned by the ve tors

R1 = ( 1; 1; 1 ); R2 = (1; 1; 1 + 2 ) respe tively. Like in the ase of a two-dimensional nonelementary singular point the edges and verti es are onsidered separately. The verti es (0) are transformed to the point Q = 0 by a time transformation j

dt~ = X

(0) j

dt :

Now, the power transformation de ned by the matrix

0 a 11 0 A = R jR j  b AA

where a; b;  0 is used to redu e the system. The ve tors R and R are de ned as the ve tors lying on the edges adjoinig the vertex and leading away from the vertex. For the extremal verti es the remaining ve tor is de ned as Q3 = (0; 0; 1) 2 M . As the ve tors R and R form a one that in ludes the support of F , the resulting system veri es all onditions required for a further redu tion and lo al integration. The solution urves omputed for the resulting system are valid in a neighbourhood U~ (0) = fX~ : jX~ j1  g of X~ = 0. Therefore the urves X (t) are valid in a se tor j

U (0) = fX : jX j j

R

 ; jX j   ; jX j( R

)

a;b;

 g

in the initial oordinates. The remaining se tors are asso iated to the edges.

110

9.3. Redu tion of three-dimensional normal forms for m=2 The blowing-up for the edges

is de ned by the matrix

(1)

0 a 11 0  A = R jQ j  b AA j

3

j

with a; b; 2 N . In the new oordinates the se tor = fX :   jX j

(0)

Rj

Uj



1

; :::

g

is transformed to a neighbourhood of the ex eptional divisor fkg00. Solution

urves on those sets an be omputed like in se tion 8.2 by the de nition of subse tors and re ursions. The system redu ed by the power transformation veri es all onditions required for the redu tion to a two-dimensional system. Example 19 Consider the system

01 _ = X

1 1

1 00 AX +  x

1 x23

0

x3

1 A

with the three verti es (0) = ( 1; 1; 0), (0) = (0; 0; 0) und (0) = (1; 1 2 3 The support and the resonant plane M are shown in gure 9.3. For the rst vertex (0) the time transformation 1 dt~ = X

yields the system X  t~

0x = x

1

(

2

1

1;1;0)

x2

1

dt

+ x1

+ x1 2 x3 x2 1

x3 2 x1 x2

1

1 A

A power transformation de ned by the matri es A

=A

1

0 =

yields the new system

1 0 1 1 0 0 0 1 0

1; 1).

1 00 A and A =  0 1

0

x ~2 x~31 + x~1 ~ X =  x~22 x~1  t~ x ~1 x~3 + x~3

1 0 0 1 1 0

1 A

1 A (9.8)

that an be integrated by the algorithms introdu ed in se tion 9.3.1. Its solution

urves are valid in the se tor (0)

U1

= fX : jX j(1

;

1;0)

 ; jx j  ; jx j  g : 3

1

Blowing-ups for the other verti es de ned by the matri es

0  A = 2

1 1 1

1 1 1 0 0 0

1 A;

0  A = 3

1 0 1 1 0 0 1 1 0

1 A

Chapter 9. Three- and higher-dimensional elementary singular points

111

Figure 9.2: The gure shows the three se tors omputed by applying blowing-ups to the system in example 19. yield solution urves for the se tors

U2(0) = fX : jX j( U3(0) = fX : jX j(

1;1;0) 1;1;

 ; jX j  ; jx j  g  ; jx j  ; jx j  g :

1)

(1;

3

1;1)

1

1

As no further singularities ex ept the one in X~ = 0 appear on the ex eptional divisors of the power transformations we an hoose  = 1. The sets U1(0) , U1(0) and U1(0) over an entire neighbourhood of X = 0. The se tors omputed for this example are sket hed in gure 9.2

 9k 2 N : 0

1

M = f 

= k 12 ) 0 0 1 k A+  0 0 1

1 A:

; 2 N g.

112

9.3. Redu tion of three-dimensional normal forms for m=2 q3

q2 (0) 3

(0) 1 (0)

q1

2

Figure 9.3: The support and some points of the set M for the system treated in example 19. The Newton diagram has the two verti es = ( 1; k; 1 ); 1 2 N ; = (0; 0; 0) : A pro edure similar to the one used in the previous ase yields systems that verify all onditions for a further lo al integration. As the used power transformations are blowing-ups, the omputed solution urves are only valid in se tors. 6 9k 2 N :01 =1k2 ) 0  M = f 0 A :  2 N g 1 All points in M are on a line. That means that the algorithms for m = 1 proposed in se tion 9.1 an be applied. (0) 1 (0) 2



5. 1 > 0; 2 > 0; 3 < 0 The notations

with Q1

0 := 

 1 2 Z ^ 2 2 Zg k k 2 3 k2 := minfk 2 Q : 2 Z ^ 2 Zg k 0 0 1k 3 1 k1 0 A and Q2 :=  k23 A are used to simplify the representation

1 k1

k1 := minfk 2 Q :

of the set M in this ase.

2 k2

Chapter 9. Three- and higher-dimensional elementary singular points

 9k 2 N : 1 = k2 and

0 M = f

113

9a; b > 0 : a1 2 + b3 = 0

1 0 a 1 A + Q1 + Q2 ;  b

1 1 k1 A + Q1 + Q2 : 0

(9.9)

; 2 N g

We will show that in this ase k1 = k2 and that the onsidered normal form an T be simpli ed using a power transformation X = X~ A de ned by the matrix A with 0 1 0 0 1 0 1 0 0 1 1 0 A: 1 0 A; A 1 =  0 A= 0 1 k1

A and A

1

2 k1

1 3

3 k1

verify all onditions required and det(A) =

2 3 3 . k1

k1 3

= k2 ) This needs to be shown only for 1 ; 2 ; 3 2 N otherwise k1 = k2 = 1 or equation (9.9) is not veri ed. suppose k1 6= k2 . 1 = k2 yields k1 = g d(k2 ; 3 ) = n g d(2 ; 3 ) = n k2 ; n 2 N; n > 1. (9.9) an be written as k   n k2 (a 2 + b 3 ) = k2 2 : n k2 n k2 k2 Dividing by nk2 yields a nkk22 + b nk3 2 = nk2 2 where the right hand side is in N but the left hand side is not (if n > 1). This is a ontradi tion so k1 = k2 . Proof 6 The points with negative entries in supp(F ) an be written in the form Proof 5 (k1

0 Q=

0 Q=

or

1

k1 0

1 A + Q2 = k1 Q1 3

1 k 1 A + Q1 = 1 Q2

a

3

b

2 k1 Q + Q2 1 3 2 ak1 Q + Q1 3 1

T after having applied the transformation X = X~ A the points in the support of the new system are given by

Q~ = AQ =

k1 AQ + ( 3 1

and

Q~ = AQ =

k1 AQ + ( 3 2

0 2 k1 )AQ2 = 

1 3

0 a k1 )AQ1 =  3

1 0 0 0 1 0

1 A + ( 3

001 k)  1 A

1 A + ( 3

001 a)  1 A :

k1

k1

0

0

That means that all new points are in N . The matrix A is hoosen su h that the points in N1 N 3 only appear in the rst equation and all points in N2 N 3 appear only in the se ond equation of the new system.

114

9.3. Redu tion of three-dimensional normal forms for m=2

 9k 2 N0: 1 =1k2 and 6 9a; b > 0 : a1 2 + b3 = 0 M = f

1

k

0 The matrix

A + Q1 + Q2 :

; 2 N g

0 A=

1 0

0 1

0 0

2 k2

1 k2

3 k2

1 A

T and the power transformation X = X~ A an be used to simplify the onsidered system. This an be proved like above.

Example 20 Consider the system

01 X_ = 

2

1

1 0 x2 x 3 A X +  1x21 + x31 x3 0

1 x22 x23 A

T that is in normal form. The power transformation X = X~ A de ned by the matrix 01 0 01 A =  0 1 0 A 1 2 1

with the ve tors Q1 = (1; 0; 1) and Q2 = (0; 1; 2) yields a system

 X~ t

00 =

0

1

1 0 x~2 A X~ +  1x~21 + x~31 0

1 x~22 A

with positive exponents that an be integrated. The resonant plane of the initial system and of the resulting system are sket hed in gure 9.4.

 6 9k 2 N0: 1 =1k2 and 9a; b > 0 : 1 + a2 + b3 = 0 1

M = f a b

A + Q1 + Q2 :

; 2 N g

T The power transformation X = X~ A de ned by the matrix

0 A=



1 0

1

0 1

2

0 0

3

1 A ; = g d(k1 ; k2 )

simpli es the system. This an be proved like above. 6 9k 2 N : 1 = k2 and 6 9a; b > 0 : 1 + a2 + b3 = 0 M = fQ1 + Q2 : ; 2 N g In this ase the matrix A for the power transformation an be found like in example 18 as the ve tors Q1 and Q2 span a one V that ontains supp(F ).

Chapter 9. Three- and higher-dimensional elementary singular points q3

q~3

q2 Q~ 2

115

q~2

Q~ 1

Q~ 2 q1

Q~ 1

q~1

Figure 9.4: This gure shows the resonant planes of the initial system and the redu ed system from example 20. 6. 1 > 0; 2 < 0; 3 < 0 In this ase the redu tion to a two dimensional system is similar to the previous point. All ases where non-rational eigenvalues appear an be derived from the ases treated above. If there exists an eigenvalue  62 Q resonan es appear if there exist ve tors Q 2 N su h that h; Qi = q =0: In this ase m = 1 or m = 2. Further there exists a ve tor ~ = (~ 1 ; ~ 2 ; ~ 3 ) su h that ~ >= 0g. Finding ~ is equivalent to nding a ve tor in Q 3 that M  fQ 2 N :< Q;  is orthogonal to one or two ve tors in N . As for the resonan e equation it makes no dieren e if  or ~ is used. All ases with 1 ; 2 ; 3 2 C have a orresponding ase with 1 ; 3 ; 3 2 Q . i

9.3.3

X

i i

Real three-dimensional systems

This se tion deals with systems that have real oeÆ ients but a pair of omplex onjugated eigenvalues. In ertain ases the redu tion to a two-dimensional system an be performed su h that it yields a system that has real oeÆ ients. Without loss of generality it an be supposed that 1 = 2 . Three ases have to be

onsidered. 

Re(1 ) = 0 and 3 6= 0. In this ase the resonant plane M ontains only one linearly independent ve tor. This ase has alredy been treated for n-dimensional systems in se tion 9.2.



Re(1 ) = 0 and 3 = 0. The redu tions in this ase are similar to those applied in se tion 9.2. However here

116

9.3. Redu tion of three-dimensional normal forms for m=2 the resonant plane ontains two linearly independent ve tors. As a onsequen e the redu ed system is not integrable dire tly but we an integrate a orresponding two-dimensional system. Consider without loss of generality that the linear part of the onsidered system is in real Jordan form. The transformation to Jordan form by the transformation X = P X~ and the transformation of the resulting system to Poin are-Dula normal form yields the system

 X^ t

0 i =

i

with the parti ularity that

1 0 x^ P P a (^x x^ )k x^l 1 1 1 2 A X^ +  x^2PPkkPPll a12klkl(^x1 x^2 )k x^3l3 A

0

 X~~ t

0 =

0



 0

1

3

and 8k; l : Im(a3kl ) = 0. For this ~ ~ X yields the new real system

8k; l : a1kl = a2kl

^ =P reason the transformation X

k xl

l a3kl (x1 x2 )

k

1 0 P P (x~~2 + x~~2)k x~~l (Re(a1kl )x~~1 Im(a1kl )x~~2) 1 3 1 2 A X~~ + B Pkk Pll (x~~21 + x~~22)k x~~l3 (Im(a1kl )x~~1 + Re(a1kl )x~~2) C A: P P (x~~2 + x~~2)k x~~l a 0 k

Introdu ing polar oordinates

1

2

3 3kl

x~~1 = r sin  x~~2 = r os 

8 > < > :

yields the real system

l

 = f (r; x ~~3 ) 1 t r = f (r; x ~ ~3 ) 2 t  x~~3 = f (r; x ~ ~3 ) : 3 t

(9.10)

The system (9.10) is either dire tly integrable or a two-dimensional system in the ~~3 an be split from it. variables r and x

Example 21 Consider the system

0 X_ = 

0 4 4 0

0

1 0 3x 2x 1 2 AX +  0

x3 3

x1 2 x2 x3 x1 x2

1 A

that is transformed to Jordan form by the linear transformation Poin are-Dula normal form

 X~ t

0 4I =

4I

~ =P The transformation X

 X^ t

0 =

0 1

0 4 4 0

1 0 A X~ + 

X^ 0

X

=

27 I x I x~21 x~2 1024 ~31 x ~22 + : : : 27 3Ix 2 ~2 + 1024 I x ~21 x ~32 + : : : 8 ~1 x 1x 4 ~1 x ~2 + : : : 8 ~3 x 3 8

yields

1 0 A X^ + 

x^32 + 83 x^21 x^2 + : : : 3 ^1 x ^22 38 x ^31 + : : : 8x 1 2 4 ^2 x ^3 + 81 x ^43 x ^21 + : : : 8x

3 8

1 A:

P X~

1 A:

and to

Chapter 9. Three- and higher-dimensional elementary singular points

117

Introdu ing polar oordinates yields the system

8 < :



9.4

 t r t  x^3 t

= 4 323 x^53 + 83 r2 3 3 3 = ^3 + : : : 32 r x 1 2 4 = 8 r x^3 + : : : :

27 4 1024 r

+ :::

Re(1 ) 6= 0. In this ase a onstru tion like in the previous ases is not possible. This is due to the fa t that the transformation matrix P is more omplex. However if no resonan es o

ur (m = 0) the normal form is linear. Therefore the omputation of a real normal form is possible. (Compare se tion 7.2). If M 6= 0 the omputations have to be performed in C 3 . Redu tion of

n-dimensional normal forms for m > 2

The lassi ation in the previous se tion has shown that any three-dimensional Poin areDula normal form an be redu ed to a system of the dimension of its resonant plane. In this se tion these results are generalized to n-dimensional normal forms. This generalization yields some additional problems as the one ontaining supp(F ) an be spanned by any number of ve tors for m > 2. The algorithms are illustrated by examples for fourdimensional normal forms with m = 3. The basi pro edure is similar to the pro edure used for three-dimensional systems in se tion 9.3.1. Suppose that the ve tors Q1 ; : : : ; Qm 2 M form a basis for M. Then the normal form (9.1) an be redu ed to a system

0

 X~  x~1 g1 (~x1 ; : : : ; x~m ) = ::: t x~ngn (~x1 ; : : : ; x~m )

1 A:

(9.11)

by a power transformation de ned by the matrix A = (Q1 j : : : jQm j : : :) where the ve tors Qi ; i = 1; : : : ; m de ne the row ve tors of A . The rst m lines of the system in equation (9.11) only depend on the variables x~1 ; : : : ; x~m . That means that it is formally integrable if the m-dimensional system

8 < :

 x~1 t

= x~1 g(~x1 ; : : : ; x~m ) :::  x~m = g (~ x1 ; : : : ; x~m ) m t

(9.12)

~ for equation (9.11) an be obtained

an be integrated formally. The solution urves X(t) from the solution urves (~x1 (t); : : : ; x~m (t)) for equation (9.12) by an integration of the n m s alar dierential equations log x~i = gi (~x1 (t); : : : ; x~m (t)); i = m + 1; : : : ; n : t

118

9.4. Redu tion of n-dimensional normal forms for m > 2

That means that the problem of nding solution urves for the initial normalized system (9.1) has been redu ed to the problem of nding an appropriate power transformation or an appropriate matrix A. Further, the system (9.12) has to be integrated. In general the system (9.12) will have a nonelementary singular point in X~ = 0. Therefore, in pra ti e we will only be able to integrate systems (9.12) of dimension m = 2. Nevertheless, the used methods will be developped using approa hes that an be generalized to m > 2. In those

ases the problems that the integration of system (9.12) might ause are not onsidered. However, some essential properties are required for the system (9.11) in order to integrate it. Those properties ae t the hoi e of the ve tors Q1 ; : : : ; Qm and therefore the

omputaion of the matrix A. 9.4.1

Conditions for the hoi e of the matrix

~ AT

A

The power transformation X = X that transforms the initial normal form to a system of the form (9.11) is de ned by the matrix A. This matrix must be hosen su h that the power transformation and the new system verify the following properties:



the ve tors Q1 ; : : : ; Qm have to form a basis of does not have the form (9.11).



the oeÆ ients of the system (9.11) have to be integer and positive. Otherwise the resulting system an not be integrated by the previously introdu ed algorithms. T the power transformation X = X~ A has to be a dieomorphism. This is veri ed if the matrix A is unimodular. If A is a non-unimodular invertible matrix the

onstru tions introdu ed in se tion 2.3 an be used to de ne a orresponding bije tive dieomorphism. T if the power transformation X = X~ A is not a blowing-up, the solutions of the resulting systems are valid in the whole neighbourhood of X = 0. T If the power transformation X = X~ A is a blowing-up, the redu tion and integration yield solutions that are only valid in a se tor. As a single transformation is not suÆ ient to over a neighbourhood of X = 0 with se tors, a serie of blowing-ups

ontrolled by a Newton diagram are needed.

 

M

. Otherwise the redu ed system

To verify whether an appropriate matrix A exists or not it is useful to onsider 3 possible

ases for the position of the resonant plane M in the spa e of exponents:  M  Nn In this ase all ve tors in M lie within the rst quadrant. An appropriate matrix A exits if the support of F is in luded in a one V spanned by m ve tors Q1 ; : : : ; Qm 2 M . Otherwise, the onsidered system an be redu ed by de ning a so alled virtual Newton diagram and by using blowing-ups.  M 6 N n and the set M \ N n ontains m linearly independent ve tors. To ompute a power transformation that is not a blowing-up yields the same problem as the previous ase. Further the matrix must be hosen su h that the ve tors in M N n are transformed to points that lie within N N n . This matrix does not always exist.

Chapter 9. Three- and higher-dimensional elementary singular points

119

The use of blowing-ups ontrolled by the Newton diagram of the onsidered system is more appropriate here.  M 6 N n and the set M \ N n ontains less than m linearly independent ve tors. Any basis of M ontains at least one ve tor with negative entries. Therefore the

on erned systems an only be redu ed by blowing-ups. Those three possibilities for the position of the resonant plane M will be onsidered more

losely in the following se tions. 9.4.2

The resonant plane lies entirely within

Nn

Consider all ones V that ontain the support of F and that are spanned by ve tors Qi 2 M . The problem arises from fa t that all ones V might be spanned by more than m ve tors. Let V be a one that ontains supp(F ) and that is de ned by the minimum number of ve tors Q1 ; : : : Qk 2 M with k  m. We have to distinguish two ases :  If

k = m the rst rows of the matrix A = (Q1 j : : : jQm j : : :)

are de ned by the m ve tors Q1 ; : : : ; Qm . In this ase all onditions for the hoi e of A are veri ed.

X iQi; i

Proof 7 Any ve tor Q 2 supp(F ) an be written as

Q=

X i

2 R;

i  0 :

A

ording to theorem 3 in se tion 2.2 Q is transformed to the point

Q~ =

det Aei :

Therefore its oordinates are positive and the rst m lines in the new system only depend on the variables x~1 ; : : : ; x~m . As A has only integer oeÆ ients the oordinates ~ are integer. of Q T As all Qi have only positive entries A an be hosen su h that X = X~ A is not a blowing-up.

k > m the ve tors Q1 ; : : : ; Qk are linearly dependent. As the matrix A has to be invertible only m ve tors R1 ; : : : ; Rm 2 Zn \ M an be used for the onstru tion of

 If

the matrix

A = (R1 j : : : jRm j : : :) :

The one W spanned by those ve tors has to ontain all Q 2 supp(F ) and therefore ~ = AQ the one V . Otherwise some points Q 2 supp(F ) are transformed to points Q with negative oordinates and the orresponding system has negative exponents. As W has to in lude all Q 2 supp(F ) at least one ve tors Ri ; i = 1; : : : ; m has negative T

oordinates. As a onsequen e the power transformation X = X~ A is a blowing-up.

120

9.4. Redu tion of n-dimensional normal forms for m > 2 For this reason the on erned systems an only be redu ed by blowing-ups. If blowing-ups are used they have to be ontrolled by a Newton diagram. However the Newton diagram for the on erned systems onsists of a single vertex (0) = 0. 1 For this reason an additional onstru tion, alled the virtual Newton diagram is used.

The virtual Newton diagram Consider a normalized system that has its support on the resonant plane M with m linearly independent ve tors. Any one W that ontains supp(F ) and that is de ned by the ve tors R1 ; : : : ; Rm has at least one ve tor with negative oordinates. Therefore the matri es A = (R1 j : : : jRm j : : :) de ne blowing-ups. For this reason the solution urves omputed for the redu ed systems are only valid in a se tor of the initial neighbourhood U . Several se tors and several blowing-ups are needed to over the entire neighbourhood U . For this reason several

ones W are required too. Those ones are omputed by the virtual Newton diagram. The name "virtual Newton diagram" is used as the employed te hniques yield a stru ture that is similar to the Newton diagram. The idea of the virtual Newton diagram is simple. The only vertex (0) = 0 of the 1 Newton diagram of the onsidered system (9.1) is onsidered as a set of identi al verti es ~ (0) , edges ~ (1) of length 0 and higher dimensional fa es ~ (in) without spa ial expansion. i i By omputing the virtual Newton diagram those fa es are visualized. Any virtual vertex ~ (0) is joined by m fa es. Those fa es are not ne essarily fa es of i the virtual Newton diagram but fa es of the onvex hull of the onsidered set of points. The interse tion of those m fa es de ne m ve tors Ri1 ; : : : ; Rim that de ne the one Wi and that are used for the onstru tion of the matrix 

A

= (Ri1 j : : : jRim j : : :) :

For any virtual vertex ~ (0) the virtual Newton diagram yields a one Wi that ontains i supp(F ) and that de nes the matri es A and A for a blowing-up. This blowing-up is applied to the initial normal form (9.1) without using a time transformation as the only real vertex of the onsidered system is (0) = 0. 1 The virtual Newton diagram is not unique. The de ned blowing-ups yield no further solutions. For this reason we an let  = 1 in the de nition of the orresponding se tors. The onstru tion of the virtual Newton diagram is simple. It is based on the omputation of the omplex hull of a set of points. Possible algorithms for the omputation of the onvex hull are for example the gift wraping method [48℄. Computing the onvex hull of all points Q 2 supp(F ) yield k fa es of dimension m 1 that are joining in (0) =0 1 and that limit the one V . The one V is de ned by the k ve tors Q1 ; : : : ; Qk . Now take m of the fa es joining in (0) and hoose m 1 of them at a time. The interse tions of those m 1 fa es are lines 1 that are either hara terized by a ve tor Qi ; i 2 f1; : : : ; kg or a ve tor Pi 62 fQ1 ; : : : ; Qk g. For all hoi e of m 1 fa es this yields a set of ve tors Pi 2 M; i = 1; : : : ; l. Now we

Chapter 9. Three- and higher-dimensional elementary singular points

121

ompute the new set of points fQ : Q 2 supp(F )g [ fQ : 9j; Q0 2 supp(F ) : Q = Pj + Q0g ; (9.13) and its onvex hull. The Newton diagram of the set of points (9.13) ontains the verti es ^ (0) =0: = Pi ; i = 1; : : : ; l and ^ (0) i l+1 All of these verti es ^ (0) , that are the virtual verti es of the initial system, are onsidered i separately. If the onvex hull of (9.13) de nes m fa es joining the vertex ^ (0) they de ne i m ve tors R1 ; : : : ; Rm that an be used for the onstru tion of the matrix A . If more than m fa es are joining the vertex ^ (0) they de ne k~ > m ve tors those fa es an be used i for a re ursive all of the entire algorithm. The virtual Newton diagram allows the omputation of blowing-ups for systems with k > m. This is illustrated by the following example. Example 22

Consider the 4 dimensional system

0 B X_ = B 

1

1

1

1

1 0xx C B xx C X + A B xx 0

2 1 2 2 2 3

2

4

4

+ x23 x1

1 C C A

that is in normal form. The set

supp(F ) = fv1 ; v2 ; v3 ; v4 g  M

with

v1

011 B 1 CC ; v =B 0A

2

0

001 B0C =B 1C A; v

3

1 is re tangular to the ve tor ( 1; 1; 1; 1).

011 B0C =B 1C A; v

4

0

1 None of the points in supp(F ) an be expressed

by a positive linear ombination of the other three points.

supp(F ) form a onvex the ve tors v1 ; : : : ; v4 .

one

V

supp(F ) is needed.

onstru t W by using the

~ W

ontains all points in in

supp(F ).

Therefore the 4 ve tors in

that is not in luded in any one spanned by only three of

To onstru t a power transformation a one We an try to

001 B1C =B 0C A

planes

des ribed by only three ve tors that

pi ; i = 1 : : : 4

spanned by two ve tors

This one is limited by three planes and spanned by the 3 ve tors on the

interse tion of the planes. (Two arbitrarily hosen planes always interse t in the origin and therefore in a line passing through the origin.) It is obvious that in

supp(F ).

The planes

pi

are given by

p1 = v1 + v3 p2 = v3 + v2 p3 = v2 + v4 p4 = v4 + v1

W

ontains all points

9.4. Redu tion of n-dimensional normal forms for m > 2

122 with ;

2 R.

We obtain 4 possible ones W

W1 = v2 + v3 + v5 W2 = v1 + v4 v5 W3 = v2 + v4 + v6 W4 = v1 + v3 v6 with ; :  0 where v5 = (0; 1; 1; 0) and v6 = (1; 0; 0; 1). v5 and v6 are the ve tors lying on the interse tion of the planes p1 and p3 and the interse tion of the planes p2 and p4 respe tively. That means all ones ontaining supp(F ) ontain at least one ve tor with negative T entries. Therefore any power transformation X = X~ de ned by A = (v 1 jv 2 jv 3 j : : :) (9.14) is a blowing up. In equation (9.14) the ve tors v 1 ; : : : ; v 3 are the three ve tors spanning the one W ; i = 1; : : : ; 3. A

i

i

i

i

i

i

To ontrol those blowing-ups a virtual Newton diagram is onstru ted. It is omputed using the three planes p1 ; p2 and p3 . Their interse tions yield the lines

p1 \ p2 = v3 p1 \ p3 = v5 p2 \ p3 = v2 wit  2 R. The Newton diagram of the set fQ1 = (0; 0; 0; 0); Q2 = (0; 1; 1; 0)g yields the virtual verti es ~ (0) = Q1 and ~ (0) = Q2 .For those verti es the ones in luding the 1 2 supp(F ) are the ones W1 and W2 . Those ones de ne the matri es 00 1 0 01 01 0 0 01 B0 0 1 0C B1 1 1 0C A1 = B 1 1 1 0C A and A2 = B 0 0 1 0C A: 1 0 0 1 0 1 0 1 The blowing-ups yield the new systems

0 x~  X~ B x~ =B  x~ t

and

0 x~  X~ B x~ =B  x~ t

2 1 2 2 2 3

+ x~2 x~1 x ~ ~22 + x~3 x~22 2 1 +x 2 ~1 x~1 x~3 x~2 x~3 3x x~4 2 1

+ x~2 x~1

x~1 + x~2 x~23 x~2 x~3 x~4

1 C C A 1 C C A:

Solution urves omputed for those systems are valid in the se tors

U1(0) = fX : jX j(0 1 1 0)  ; : : :g U2(0) = fX : jX j(0 1 1 0)  ; : : :g ; ;

;

;

; ;

with  = 1. The virtual Newton diagram is sket hed in 3 dimensions in gure 9.6 ane gure 9.5 shows the ones V; W1 and W2 .

Chapter 9. Three- and higher-dimensional elementary singular points

123

V

W1

W2

Figure 9.5: The suport of the normal forms treated in example 22 lies within a one V that is spanned by 4 ve tors. As only 3 of these ve tors an be used to de ne the power T transformation X = X~ A , the ones W1 and W2 are used instead.

(0) 1

~ (0) 1

~ (0) 2

Figure 9.6: To obtain an appropriate ontrol stru ture for the de nition of the blowing-ups used in example 22 the virtual Newton diagram that onsists of two verti es and one edge is omputed..

124

9.4. Redu tion of n-dimensional normal forms for m > 2

9.4.3

M \N

The set

n

ontains

m

linearly independent ve tors

In this ase there exits a basis Q1 ; : : : ; Qn 2 M \ N n for M . For this reason for some parti ular normal forms the redu tion by a power transformation that is not a blowing-up is possible. However the matrix A that de nes this transformation has to verify some very stri t onditions. In general it does not exist. Therefore the use of blowing-ups is the more appropriate method for this ase. The Newton diagram of the on erned systems onsists of more than one vertex. For this reason it an be used to ontrol the blowing-ups. However the Newton diagram of the system (9.1) might ontain higher-dimensional fa es than edges and verti es. Blowing-ups for those fa es have not been de ned. The time transformation ~ = X (0) dt dt that tranforms the vertex (0) to the point Q = 0 yields a new system i i

~ ~ =F

X

:

t

The

fa es of the onvex hull of supp(F ) that are joining in Q = 0 de ne k ve tors . Those ve tors de ne the one V that ontains the support of F~ . As the 1 number of ve tors spanning V might ex eed m the onstru tion of a virtual Newton diagram in for the on erned vertex might be ne essary. The following examples ilustrate this ase. In the rst example a power transformation that is not a blowing-up an be used for the redu tion of the on erned system. This is not possible for the system treated in the se ond example were blowing-ups are used. k

Q ; : : : ; Qk

Example 23 Consider the normalized system

_ = F (X

X

0 B )=B 

1

1

1

1

1 C C A

X

0 B +B 

2 3 x4 2 2 x2 x4 + x3 x1 2 2 x3 x4 + x3 x1 x

0

1 C C A

that also has the points ( 1; 0; 1; 2) and (1; 1; 2; 0) in its support. The set M \ N 4 lies within a one V spanned by the ve tors (1; 0; 1; 0), (0; 1; 0; 1) and (0; 0; 1; 1). Therefore the power transformation de ned by the matri es

01 B0  =B 1

A

0 1 0 0 1

0 0 1 1

0 0 0 1

1 C C A

and A

0 B =B 

1 0 1 1

0 1 0 1

0 0 1 1

0 0 0 1

is used to simplify the onsidered system. This yields the new system

0 ~2 + ~ ~ + ~ 2 1 3 1 3 1 ~ B ~3 ~1 + ~22 C C =B  ~ ~ + ~2 A x

x

x

X

x

x

x

t

x

3 x1 x ~4

x

x

3

:

1 C C A

Chapter 9. Three- and higher-dimensional elementary singular points

Example 24 Consider the normalized system _ = F (X

X

0 B )=B 

1

1

1

1

1 C C A

X

0 B +B 

x3 x4 x2 x3

0

2x 2x

2

+ x3 2 x1 2 2 4 + x3 x1 + x2 x4 4

125

1 C C A

that also has the points ( 1; 0; 1; 2), (0; 2; 1; 1) and (1; 1; 2; 0) in its support. Redu ing this system by a power transformation de ned by the matrix A used in the previous example is not possible. 9.4.4

The set

M \ Nn

ontains less than

m

linearly independent ve tors

In this ase any basis Q1 ; : : : ; Qm of M has at least one ve tror with negative oordinates. As a onsequen e matrix A de ned via its adjoint matrix 

A

= (Q1 j : : : jQm j : : :)

that does not de ne a blowing-up does not exist. The only possibilty to redu e the

on erned system to a m-dimensional system is to use blowing-ups ontrolled by the Newton diagram of the system (9.1). Eventually the virtual Newton diagram an be used if the use of the Newton diagram is not suÆ ient. 9.5

Con lusion

In this hapter it has been shown that three-and higher-dimensional normal forms an be redu ed to m-dimensional systems. m denotes the maximum number of linearly independent ve tors in the resonant plane M of the on erned normalized system. The redu tion is performed by a power transformation. If the power transformation is a blowing-up it has to be ontrolled either by the Newton diagram or the virtual Newton diagram. However in ertain ases blowing-ups asso iated to higher-dimensional fa es of the Newton diagram are needed. Those transformations have not yet been de ned. They are studied in some examples in the following hapter. The redu tion of n-dimensional normal forms yields m-dimensional systems with a nonelementary singular point. The integration of those systems is not always possible for m > 2 (see [30℄). For m = 2 however any normal form an be integrated by the proposed algorithms.

Chapter 10 Three-dimensional nonelementary singular points In hapter 9 blowing-ups have already been used for the redu tion of three-and higherdimensional non-nilpotent systems. In parti ular the virtual Newton diagram has shown to be an eÆ ient tool to de ne blowing-ups. In this hapter blowing-ups will be used to redu e three-dimensional systems in the neighbourhood of a nonelementary singular point. However the use of blowing-ups for three-dimensional systems yields problems. This has been shown by J. Jouanolou [38℄, X. Gomez-Mont and I. Luengo [30℄ and mentioned in se tion 4.3. Therefore it will be presumed that the onsidered dierential equation _ X

= F (X )

(10.1)

is non-di riti al. That means that at ea h step of the desingularization the non riti al

ase o

urs. It has been shown by F. Cano and D. Cerveau [10℄ that those systems an be redu ed to a nite number of systems with regular or elementary singular points by a nite number of su

essive blowing-ups. The redu tion of three-dimensional systems (10.1) by blowing-up is based on the onstru tion of quasi-homogeneous blowing-ups in se tion 4.2. Further the Newton diagram and the virtual Newton diagram will be used. As in the ase of a two-dimensional nonelementary singular point the elementary operations as translations and time and power transformations. The used power transformatins are dieomorphisms (see se tion 2.3). In the study of a three-dimensional nonelementary singular point we will fo us on the algorithmi point of view and on the onstru tion of the se tors. The de nition of the se tors is ae ted by the hoi e of the blowing-ups and it is not unique. It will be shown that the se tors an be onstru ted su h that they over the entire onsidered neighbourhood U

= fX : jX j1  g

of the singularity in X = 0. The se tors and the algorithms are onstru ted a

ording to the fa es of the Newton diagram. 127

128

10.1. The verti es

10.1

The verti es

Consider the vertex (0) i of the Newton diagram of the initial system (10.1). As in the two-dimensional ase a time transformation (0) dt~ = X i dt translates (0) i to the point Q = 0. The resulting system _ = F (X(0)) = F~ (X ) X X

i

is a lass V system. It has its support in a one that is de ned by m ve tors v1 ; : : : ; vm . Three ases have to be distinguished.  m=3 The one V ontaining supp(F~ ) if de ned by the 3 ve tors v1 ; v2 and v3 . They de ne the olumns of the matrix  A = (v1 jv2 jv3 ) T that de nes the power transformation X = X A . The power transformation transforms all points in supp(F~ ) to points with positive integer oordinates. For this reason the resulting system has positive integer exponents. As it has a non-nipotent linear part it an be integrated using the methods introdu ed in hapter 9. All further redu tions and integrations are valid lo ally in a neighbourhood U =  : jX j1  g. For this reason in the initial oordinates the solution urves are fX valid in the se tor (0)

Ui

= fX : jX jv1

 det A ; jX jv2   det A ; jX jv3  det A g : 1

1

1

 m3

The one V ontainig supp(F ) is spanned by more than three ve tors. However for the onstru tion of A via A a maximum of 3 linearly independent ve tors is needed. Therefore a ve tor W thet ontains V is onstru ted su h that W is de ned by the three ve tors w1 ; : : : ; w3 . The power transformation X = X AT with 

A

= (w1 jw2 jw3 )

yields a new system with positive integer exponents. The resulting system an be integrated with previously introdu ed methods. Its solution urves are valid in the se tor 1 1 1 (0) w w Ui (W ) = fX : jX jw1  det A ; jX j 2  det A ; jX j 3  det A g

Chapter 10. Three-dimensional nonelementary singular points

129

q3

R

q2

(1) i

v2 v1

P

q1

Figure 10.1: This gure shows the one V onstru ted to de ne the blowing-up applied in the ase of an edge (1) . i that depends on the hoi e of W . This hoi e and therefore the onstru tion of A is not unique. The proposed algorithm is similar to the methods used for the redu tion of two-dimensional systems. However in se tion 10.4 it will be shown that for the ase m > 3 some additional

onstru tions are needed to guarantee an entire overing of the initial neighbourhood U by se tors. 10.2

The edges

Consider the edge (1) of the Newton diagram and the ve tor R = (r1 ; r2 ; r3 ) with i g d(r1 ; r2 ; r3 ) = 1 that lies on that edge. A

ording to the results from se tion 4.2 we will onstru t a quasihomogeneous blowing-up that straightens that edge. Further the blowing-up yields a system with positive integer exponents after having applied a time transformation that translates the edge (1) to the set i

fQ : q = 0; q = 0g : 1

2

(10.2)

with 2 (0) Any blowing-up de ned in the previous se tion for the ase of a vertex (0) i i w3 = R or v3 = R an be used to perform the blowing-up for an edge. However there are dierent possbilities to onstru t the matrix A that de nes the blowing-up. If r3 6= 0 we an presume without loss of generality that the ve tor R is hosen su h that r3 > 0. Otherwise R an be hosen instead of R. The line Q0 + R with Q0 2 (1) , i  2 R uts tha q1 q2 -plane in the point P . The ve tor R leads away from that point. Consider that the (2) and (2) are the fa es of the Newton diagram that join in the i j (1) edge i . They interse t the q1 q2 plane in the lines P + v1 and P + v2 with  2 R

130

10.2. The edges

respe tively. Without loss of generality it an be presumed that the ve tors v1 and v2 verify

v1

0v 1 =  v A with v 11 12

0

11 > 0 and v12  0; v2

0v 1 =  v A with v 11 12

0

11

> 0 and v12  0 :

The ve tors v1 ; v2 and R form a one V su h that Q0 + V ontains supp(F ). Therefore T the blowing-up X = X~ A de ned by the matrix

A = (v1 jv2 jR) straightens the edge (1) i . The straightened edge is parallel to the set (10.2). A time transformation dt~ = x~q1~01 x~q2~02 dt with Q~ 0 = (~q01 ; q~02 ; q~03 ) = A Q0 : Q0 2 (1) i translates the straightened edge to the set (10.2). The exponents of the resulting system  X~ ~ ~ = F (X ) (10.3)  t~ are positive and integer. An example for su h a one V is shown in gure 10.1. (2) (2) If the edge (1) i is an extremal edge the fa es i or j might not exist. Then the (2) ve tors v1 2 (2) i and v2 2 j an be repla ed by the ve tors e1 and e2 respe tively. If r3 = 0 the ve tor R an be hosen su h that r2 > 0. The line Q0 + R uts the q1 q3 (2) plane in the point P . The ve tors v1 2 (2) i and v2 2 j are de ned su h that

v1

0v 1 =  0 A with v 11

v13

11 > 0 and v13  0; v2

0v 1 =  0 A with v 11

v13

11

> 0 and v13  0 :

The matrix A and the time transformation an be de ned like above. They yield equivalent results. In fa t any one V de ned by the ve tors v1 ; v2 and R su h that

8Q 2 supp(F ) : Q 2 P + V with P = Q0 + R, Q0 2 (1) i an be used to de ne a blowing-up that straightenes the (1) edge i . For any blowing-up for the edge (1) i the resulting system (10.3) is studied on the set S = fX~ : x~ = 0; y~ = 0g (10.4) that is a part of the ex eptional divisor. The singularities of the system (10.3) on the set (10.4) are identi to the singularities of the quasihomogeneous part 0 ~ ~ 1  X~ ~ (1)  x~1 f~1 (X~ ) A = Fi = x~2 f2 (X ) (10.5)  t~ ~ ~ x~3 f3 (X )

Chapter 10. Three-dimensional nonelementary singular points

131

on the set (10.4). The quasihomogeneous par of a system (10.3) has been de ned in se tion 4.2. It ontains only the monomials asso iated to the points on the straightened edge. It is used to distinguish two ases.  f~3 6 0 This is the non riti al ase. The points X~ 0 2 S with f~3 (X~0 ) = 0 are the singularities of system (10.5) and system (10.3). All other points are regular points. In the neighbourhood of these regular points the solution urves are parallel to the ex eptional divisor. Further the ex eptional divisor itself is a solution urve for the blown-up system (10.3).  f~3  0

This is the di riti al ase. For the resulting system there might not exist a serie of blowing-ups that entirely redu e the nilpotent system to a nite number of nonnilpotent or regular systems (see J. Jouanolou [38℄, X. Gomez-Mont and I. Luengo [30℄). However the di riti al ase also yields problems in it's algorithmi aspe ts. The entire set S is a non-isolated singularity of the system (10.5). However a time transformation  = x~1 dt~ dt or

 = x~2 dt~

dt

might transform the system (10.5) into a system with isolated singularities. The resulting system has the form

0 ~ B  =

X t

or

0 ~ B  =

X t

~ ( ~) x~2 f~ (X ~) x~1 2 x~3 ~ ~ x~1 f3 (X ) f1 X

1 C A

1 ~2 ( ~ ) C A

(10.6)

x~1 f~ (X ~) x~2 1 f

X

x~3 ~ ~ x~2 f3 (X )

:

(10.7)

These systems might have negative exponents. Therefore a further redu tion and integration using the methods proposed here is not always possible. The singularities of the system (10.6) or (10.7) are identi to the singularities of the system ~ 1 X = F~ (X~ )  t x ~1 or ~ X 1 ~ ~  = x~2 F (X ) : t They are given by the points X~ 0 2 S that yield a vanishing right hand side in the equations (10.6) or (10.7). All othe points are regular points. They are either tangen ies or there exists a solution urve passing through X~ 0 . Therefore the di riti al

ase yields an in nity of solution urves passing through X = 0 fr the initial system (10.1).

132

10.3. The fa es

The system (10.3) is studied in a neighbourhood of a nite part of the set S . This neighbourhood is denoted by ~ U

= fX~ : jx~1 j  ~; jx~2 j  ~; ~  jx~3 j  ~

1

g:

This study yields results that are valid in a se tor (1)

Ui

= fX : jX jv1

 ~det1 A ; jX jv2  ~ det1 A ; ~det1 A  jX jR  ~det1A g

of the initial neighbourhood U . 10.3

The fa es

AT Consider the fa e (2) i . It an be straightened by a blowing-up X = X~ de ned by the matrix  A = (v1 jv2 jv3 ) v1 and v2 are linearly independent. with the olumn ve tors v2 ; v3 2 (2) i . The ve tors  3 The matrix A is ompleted by a ve tor v3 2 Z . v3 is hosen su h that the support of F lies within the set R + V where R is a point that lies on the plane passing through (2) i . V is the one spanned by the three ve tors v1 ; v2 and v3 . The fa e (2) i is transformed to a fa e that is parallel to the set

fQ : q = 0g :

(10.8)

1

The time transformation

dt~ = x ~q~01 dt

with Q~ 0 = (~q01 ; q~02 ; q~03 ) = AQ0 , Q0 2 (2) i yields the new system ~ X  t~

= F~ (X~ )

(10.9)

that has positive integer exponents. The straightened fa e has been translated to the set (10.8). The new system (10.9) is studdied in a neighbourhood of a nite part of the set S

= fX~ : x~1 = 0g :

The singularities of the new system (10.9) on quasihomogeneous system

S

(10.10)

are identi to the singularities of the

0 x~ f~ (X~ ) 1 1 ~ X ~i(2) (X~ ) =  x~2 f~2 (X~ ) = F  t~ x ~3 f~3 (X~ )

1 A:

The system (10.11) is used to distinguish the non riti al and the di riti al ase:

(10.11)

Chapter 10. Three-dimensional nonelementary singular points

133

 f~2 6 0

or f~3 6 0 The points X~ 0 2 S are singularities of the system (10.9) if f~2 (X~0 ) = 0 and f~3 (X~ 0 ) = 0. All other points are regular points. The solution urves in the neighbourhood of all regular points are parallel to S .  f~2  0 and f~3  0 In this ase the nonelementary singularity of the initial system (10.1) might not be redu eable by nite su

essive blowing-up. As S represents a non-isolated singularity for the system (10.11) a time transformation dt = xdt~ is used to transform the system (10.11) to the new system 0 ~ ~ 1 ~ B x~2 f~1 (X ) C X =  x~1 f2 (X~ ) A  t x~3 ~ ~ x~1 f3 (X ) that has no negative exponents. Its singularities are given by the points X~0 with ~ 0 ) = f~2 (X~ 0 ) = f~3 (X~0 ) = 0. f~1 (X All other points X~ 0 are regular points. If f~1 (X~ 0 ) = 0 they are tangen ies. Otherwise there exists a solution urves passing through X0 . The system (10.9) is studdied in a neighbourhood of a nite part of the set (10.10). This neighbourhood is given by ~ = fX~ : jx1 j  ~; ~  jx2 j  ~ 1 ; ~  jx3 j  ~ 1 g : U Therefore the se tors asso iated to the fa es are given by = fX : ~det A  jX jv1  ~det A ; ~det A  jX jv2  ~det A ; jX jv3 where v3 represents the olumn ve tor used to omplete the matrix A . 10.4

1

1

(2)

Ui

1

1

1

 ~det A g

The se tors

As it has been shown previously the de nition of blowing-ups for verti es, edges and fa es of the Newton diagram also ae ts the shape of the se tors. In this se tion it will be shown that there exist blowing-up onstru tions su h that a on erned neighbourhood U = fX : jX j1  Æ g with Æ suÆ iently small an be overed. First we will presume that in any vertex (0) i of (0) the Newton diagram the one V is de ned by 3 ve tors. For the vertex i , the edge (1) i (2) and the fa e i the orresponding se tors are de ned as = fX : jX jv1  ; jX jv2  ; jX jv2  g (1) (1) (1) Ui = fX : jX jv1  ; jX jv2  ;   jX jR   1 g (2) (2) (2) (2) Ui = fX :   jX jv2   1 ;   jX jv3   1 ; jX jv1 (0)

Ui

(0)

(0)

(0)

(10.12)  g

134

10.4. The se tors

The onstants ~ that de ne the size of the sets U~ in se tion 10.1, 10.2 and 10.3 are hosen su h that ~ = det A . To show that the entire neighbourhood U is overed by the se tors (10.12) the singularity is approa hed for t ! 1 on lass W urves

8 x (t) = t ( + O(1=t)) < F :: :::  x (t) = t ( + O(1=t)) 1

1

3

3

1

:

3

with  = (1 ; 2 ; 3 ) 2 Z3 and 1 ; 2 ; 3  0. The neighbourhood U is entirely overed if for any  and any = ( 1 ; 2 ; 3 ) and t ! 1 the urve F lies within a se tor (10.12). Consider any ve tor v. The point X = 0 is approa hed on a urve X = F (t). The

ondition jX jv   holds if h; vi < 0 or if h; vi = 0 and j jv  . The ondition

  jX jv  

1

holds if h; vi = 0 and   j jv   1 . That means that asso iated to the vertex (0) i if  veri es the onditions

F lies within the se tor Ui

h; v i  0 h; v i  0 h; v i  0 : (0) 1 (0) 2 (0) 3

Further the onditions

(0)

(10.13)

j jv  ; j jv  ; j jv   (0) 3

(0) 2

(0) 1

(10.14) have to be veri ed if the orresponding equalities in equation (10.13) are veri ed. The

ondition (10.13) is equivalent to the ondition that  has to lie in the dual one asso iated to the one de ned by the ve tors v1(0) , v2(0) and v3(0) . This dual one is asso iated to the vertex (0) i . The urve F lies within the se tor Ui(1) if the onditions

h; v i  0 h; v i  0 h; Ri = 0 (1) 1 (1) 2

and hold. Further the onditions

  j jR  

(10.15)

1

j jv  ; j jv   (1) 1

(1) 2

(10.16)

(10.17) have to be veri ed if the orresponding equalities in equation (10.15) are veri ed. The

ondition (10.15) is equivalent to the ondition that  has to lie in the dual one of the degenerate one V = v1(1) + v2(2) + R; ; ; 2 R; ;  0

Chapter 10. Three-dimensional nonelementary singular points

135

that is de ned by the ve tors v1(1) , v2(1) and R. This dual one is asso iated to the edge (1) . i The urve F lies within the se tor Ui(2) if the onditions

h; v i = 0 h; v i = 0 (2) 2 (2) 3

and

  j jv1 (2)   j jv2

(2)

hold. The inequality

 

(10.18) 1 1

(10.19)

h; v i < 0 (2) 3

is always veri ed for t suÆ ient large if (10.18) holdes. The ondition (10.18) is equivalent to the ondition that  has to lie in the dual one for the degenerate one

V = v2(2) + v3(2) + v1(2) ; ; ; 2 R;  0 de ned by the ve tors v1(2) , v2(2) and v3(2) . This dual one is asso iated to the fa e (2) . i However the dual ones asso iated to the ones de ned for the fa es of the Newton diagram entirely over fQ 2 R3 : q1 ; q2 ; q3  0g. Therefore every lass W urve F lies within a se tor if the following onditions are veri ed :

 All ve tors 2 R or C lie within the set denoted by (10.14) or the set denoted 3

3

by (10.16) if  lies in the dual one asso iated to an edge and not in a dual one asso iated to a neighbouring fa e. This an easily be shown as there always exist 2 verti es (0) and (0) 2 (1) . The i j i

ones asso iated to those verti es ontain the ve tor R or R that de nes the edge. Therefor either the ondition (10.14) for the edge (1) or the ondition (10.16) for i (0) one of the edges (0) ok holds. i j

 All ve tors 2 R or C lie within the set denoted by (10.14) or (10.17) and (10.16) 3

3

or (10.19) if  lies within the dual one asso iated to the fa e (2) . i This is not always true as the omputations for simple examples show. To solve this problem a number of methods an be taken into onsideration. The parameters  and ~ an be varied for ertain fa es. Another possibility is to extend the solutions

omputed in the neighbourhood of the sets that are not overed by the se tors. This is not possible if those sets ontain singularities. Further the sets U~ in the se tions 10.1, 10.2 and 10.3 an be hoosen in a dierent way. This in ludes a smaller ~ or a dierent shape of the se tors.

It has been shown that the se tors de ned by the blowing-ups mainly over U if m = 3 for all on erned verti es. If m > 3 it is obvious that this is no longer true. Therfore we will introdu e an additional onstru tion to the Newton diagram that adds virtual verti es and edges su h that m = 3 is veri ed for any real and virtual vertex.

136

10.5. The virtual Newton diagram

10.5

The virtual Newton diagram

The previous se tion has shown that with the proposed methods an entire study of a three-dimensional system is possible if the Newton diagram has a regular stru ture. The is regular. That stru ture of a Newton diagramm will be alled regular if in any vertex (0) i (0) means that the one V that veri es i + V  supp(F ) is de ned by 3 ve tors. Those ve tors lie on adjoining edges if the vertex is not an extremal vertex. Otherwise the three ve tors are de ned by the edges adjoining (0) and by ve tors from the set fe1 ; e2 ; e3 g. i Therefore for any regular vertex the orresponding blowing-up is uniquely de ned up to a permutation of the olumn ve tors in the matrix A . In general the Newton diagram of a given system does not yield su h a regular stru ture. The blowing-ups asso iated to any non-regular vertex are not uniquely de ned. The dual ones of the ones omputed to de ne the blowing-ups orresponding to non-regular verti es do not allow to entirely over the set fQ 2 R3 : q1 ; q2 ; q3  0g. Therefore the se tors resulting form the blowing-up onstru tion do not over the entire neighbourhood U. To solve this problem we will introdu e an additional onstru tion that allows to

ompute a Newton diagram that extends the onventinal diagram su h that it has a regular stru ture. This additional onstru tion is alled the virtual Newton diagram. In a rst step the Newton diagram is omputed. Its non-regular verti es are onsidered as a nite number of identi verti es onne ted by edges of zero length. Any of those virtual verti es owns a one V that is de ned by 3 ve tors. The blowing-ups asso iated to all real and virtual fa es yield se tors that over the neighbourhood U as mentioned in se tion 10.4. It is obvious that the blowing-ups asso iated to virtual edges yield no singularities on the set S . 10.5.1

The onstru tion of the virtual Newton diagram

The onstru tion of the virtual Newton diagram is based on the omputation of the onvex hull and the Newton diagram of a set of points. An algorithm for the omputation of the

onvex hull is for example the gift wraping method (see for example F. Preparata and M. Shamos [48℄). Consider the vertex (0) and presume that it lies on the interse tion of k > 3 fa es i of the onvex hull of F . For non-extremal verti es those fa es also belong to the Newton diagram of F . The interse tions of two of those fa es de ne the edges joining in (0) and i the ve tors de ning the asso iated one. Now hoose 3 of those fa es and ompute the interse tion of ea h two of them. This yields 3 lines that interse t in (0) . They an be used to de ne three ve tors v1 ; v2 ; v3 su h i that these ve tors de ne a onvex one V . The one veri es supp(F )  (0) + V . Two i possible ases have to be onsidered for ea h of the ve tors vk ; k = 1; : : : ; 3 :  

vk

lies on the onvex hull of F . In this ase vk does not de ne a virtual edge.

does not lie on the onvex hull of F . That means that the ve tor virtual edge. vi

vk

de nes a

If none of the omputed ve tors vk ; k = 1; : : : ; 3 de nes a virtual edge the introdu tion of virtual verti es is not ne essary. Otherwise the algorithm de nes the virtual verti es

Chapter 10. Three-dimensional nonelementary singular points v5

v4

v1

v4

v5

v3

v1

v3

v2

137

v2 v7

v6

~

(0)

(0) 1

3

~

(1) 1

~ (0) 3

~ (1)

~ (0)

2

2

Figure 10.2: The virtual Newton diagram allows to repla e the one ~ (0) three ones V1 , V2 and V3 for the virtual verti es ~ (0) and ~ (0) 1 , 2 3 .

W

for

(0) 3

by the

+ vk where the vk are the ve tors de ning virtual edges. Now the Newton diagram of the set of points (0) i

( ) [ fQ + vk : Q 2 supp(F );

supp F

vk

de nes a virtual edge g

is omputed. The virtual edges and verti es for the initial dierential equation are real edges and verti es of the omputed Newton diagram. If the resulting ones asso iated to the verti es ~ (0) are still de ned by more that three ve tors the whole algorithm is i repeated for the on erned verti es. The virtual Newton diagram yields verti es, edges and fa es and their asso iated ones. These ones an be blown-up with the onstru tions introdu ed previously. The resulting se tors over the initial neighbourhood U with the restri tions mentioned in se tion 10.5. 10.6

Examples for the redu tion of three-dimensional nilpotent systems by blowing-ups

Consider the three-dimensional system of dierential equations

0 _ =

X

+ x1 3 x2 2 + x1 2 x2 x3 6 3 2 x2 + x2 x3 x1

x3

6

6

1 A

:

(10.20)

Its Newton diagram ontains 5 fa es, 10 edges and 6 verti es. All onstru tions introdu ed previously an be illustrated by this example. (1; 1; 1).) As shown in gure 10.2 5 edges are joinig in the = (1; 1; 1). Therefore (0) is repla ed by the 3 virtual verti es ~ (0) = ~ (0) = ~ (0) 3 1 2 3

Example 25 (The vertex

vertex

(0) 3

138 10.6. Examples for the redu tion of three-dimensional nilpotent systems by blowing-ups that oin ide with (0) and the virtual edges ~ (1) and ~ (1) de ned by the ve tors v6 and v7 3 1 2 respe tively. The virtual verti es and edges have been sket hed in gure 10.2. The ones (0) Vk ; k = 1; : : : 3 for the virtual verti es are de ned by the ve tors v1 ; v5 ; v6 for ~ 1 , by (0) (0) v2 ; v6 ; v7 for ~ and by v3 ; v4 ; v7 for ~ . The ve tors v1 ; : : : ; v7 are given by 2 3

= ( 1; 4; 1) = (1; 1; 1) v3 = (4; 1; 1) v4 = ( 1; 1; 4) v5 = ( 1; 1; 1) v6 = ( 11; 19; 1) v7 = (3; 7; 3) v1

v2

They hara terize the matri es for the blowing-ups. For the vertex ~ (0) that yields the 1 T A ~ blowing-up X = X by the matri es A

0  =

1 1 1

1 4 1

11 19 1

1 A

; A

0 15 =  20

10 25 12 8 5 2 3

1 A

:

The blowing-up and an apropriate time transformation yields the system

0

1 x 20 1 1 x 20 2

~ X  ~ = t

11 x 20 3

1 A

+ 201 x1 x2 20 + 201 x1 21 + : : : + 15 x2 21 + 51 x1 20 x2 + : : : 19 19 20 20 x x x3 x3 + : : : 20 2 20 1

:

de ned by the ve tor R = ( 4; 1; 1) (0) and the verti es (0) = (5 ; 0; 0), = (1 ; 1; 1) 2 . The simplest way to ompute a 1 3 (1) blowing-up for the edge 1 is to use the blowing-up asso iated to the virtual vertex ~ (0) 3 T It is given by X = X~ A with Example 26 (The edge

A

(1) 1 .)

0  =

Consider the edge

3 7 3

1 1 4

4 1 1

1 A

; A

(1) 1 (1) 1

01 = 2

3 1 3 5 5 3 2

1 A

To straighten the edge (1) the ve tor R = v4 has to appear in the third olumn of the 1 matrix A . After an appropriate time transformation this yields the system

0

~  ~ = t

X

~

1 x 5 1 1 x 15 2

~ ~ +

4 x 15 3

~ ~ + ~ ~ + ~ ~ ~ ~ ~ + ~ ~ ~ ~ ~ +

1 4 6 7 4 6 x x x x ::: 5 1 3 15 1 2 1 1 6 3 3 7 ::: x1 x2 x3 x1 x2 ; 15 15 1 4 3 7 3 6 ::: x1 x3 x1 x2 x3 15 15

1 A

:

The verti es (0) and (0) have been transformed to the points (5; 10; 15) and (5; 10; 10) 1 3 respe tively. As the rst and se ond oordinates of those points are identi the edge (1) 1 has been straightened up.

Chapter 10. Three-dimensional nonelementary singular points (2) 1 ) (0) 1 ;

(2) 1

de ned by the normal ve tor ( 2; 3; 5) and the verti es = (2; 2; 0) 2 (2) 1 . The one (0) V de ned by the three ve tors v2 ; v3 and v8 = ( 13; 2; 7) veri es supp(F )  + V. 3 Therefore it an be used to de ne the matri es

Example 27 (Blowing-up of the fa e

A

0  =

13 2 7

1 1 1

4 1 1

1 A

Consider the fa e

139

; A

(0) 3 ;

(0) 2

02 = 5

3 5 15 5 9 6 15

1 A

for the blowing-up. The ve tors v2 and v3 represent the se ond and third olumn ve tors in A . After having applied an appropriate time hange the system resulting from the power T transformation x = X~ A has the form

0

~ X  ~ = t (0) The verti es (0) and 1 ; 2 (10; 25; 30) respe tively.

~

~ ~ ~ ~ + ~ + ~ + ~ + + ~ + ~ + ~

13 13 13 15 15 x x x x x ::: 15 1 15 1 3 15 1 2 1 1 1 16 15 x x x x ::: 15 2 15 2 15 2 3 4 4 4 15 16 x2 x3 x3 x3 ::: 15 15 15

(0) 3

1 A

are transformed to the points (10; 15; 45); (10; 40; 30) and

The previous se tions have shown how blowing-up an be redu ed for the redu tion of three-dimensional nilpotent ve tor elds. Further it has been shown how the study of the blown-up system in a part of the ex eptional divisor yields se tors for the initial

oordinates. Therefore solution urves within the se tors an be omputed by al ulating solution urves in a neighbourhood of the on erned parts of the ex eptional divisors. Those neighbourhoods are denoted by U~ in the se tions 10.1, 10.2 and 10.3. To ompute solution urves in U~ the sets U~ are divided into subse tors asso iated to regular and simple points of the redu ed system on the ex eptional divisor. The solution urves for those subse tors are omputed by a re ursive all of the entire algorithm. However it an not be guaranteed for all three-dimensional systems that they an be entirely redu ed by a nite number of blowing-ups. Further algorithmi problems may o

ur if the di riti al

ase is veri ed for an edge of the Newton diagram. 10.7

Higher-dimensional nonelementary singular points

In the previous se tions algorithms for the redu tion of three-dimensional dynami al systems have been introdu ed. Espe ially the virtual Newton diagram has shown to be a powerful tool. Those results an be extended to higher-dimensional problems. The main diÆ ulties for higher dimensional blowing-ups are the same as for three-dimensional systems. The virtual Newton diagram an be used to solve large number of those problems. It an be used to enlarge the onventional Newton diagram by introdu ing virtual fa es of dimension n 2 and lower. This new stru ture allows well dire ted manipulation on the

ones and a se tor de nition that overs the studied neighbourhood U .

Chapter 11

The FRIDAY pa kage In the previous hapters several algorithms for the redu tion and the integration of two-, three- and higher-dimensional systems of autonomous dierential equations

X_ = F (X )

(11.1)

Maple

were introdu ed. Those algorithms are implemented in the FRIDAY 1 pa kage. In parti ular the FRIDAY pa kage ontains pro edures for the omputation of n-dimensional normal forms for systems of the form (11.1) in the neighbourhood of regular and elementary singular points. Further, it allows to integrate any real and omplex two-dimensional system and real three-dimensional systems in the neighbourhood of elementary singular points. The intention in implementing the FRIDAY pa kage was to design a program that is easy to use and that an handle a large number of systems (11.1). Therefore the data stru ture is on eived obje t-like be ause obje t-oriented programming an't be realized in . The organisation in modules simpli es the addition of new pro edures and fun tions. Primitives a t on the obje t-like data stru ture and perform elementary operations. These primitives are used by the ontrol stru ture to exe ute the dierent steps of the algorithm. The ontrol stru ture of the program is split into 4 main parts. A

ording to the lassi ation of dynami al systems three modules deal with the ase of regular points, elementary singular point and nonelementary singular points. One module performs the

lassi ation of the onsidered system (11.1). Only a few pro edures are visible and an be manipulated by the user. The main part of the fun tions are apsuled. The pro edure FRIDAY redu es a given system (11.1) as far as possible and performs eventual integrations. Besides this main pro edure in parti ular the pro edures for normal form omputations are a

essible. They allow omputations of Poin are-Dula normal forms, normal forms for systems with real oeÆ ients and omplex eigenvalues and normal forms for systems in the neighbourhood of a regular point. Further some pro edures that handle the obtained solutions, the transformations and the se tors are available. A large number of examples in se tion 11.3 show how these pro edures an be used.

Maple

1

FRIDAY stands for

F

ormal

R

edu tion and

I

ntegration of

141

D

ynami al

A

y

utonomous S

stems.

142

11.1. Organisation

There is a restri ed number of possible tests for the omputed solutions. These methods were already mentioned in se tion 1.4. They were performed for a large number of arbitrarilly hosen systems. 11.1

Organisation

The FRIDAY pa kage is split into 4 main modules. A

ording to the lassi ation of dynami al systems the modules SP n, ESP n and NESP n deal with the ases of regular points, elementary singular points and nonelementary singular points respe tively. The module separate ases performs the lassi ation of the onsidered system. The pro edure separate ases distinguishes the dierent possible ases as it has been introdu ed in hapter 5. Further it performs eventual translations and de nes neighbourhoods. Therefore it uses the primitives newtrans n and initial se tor n. The module SP n redu es and integrates n-dimensional ve tor elds in the neighbourhood of regular points. The appli ation of the ow-box theorem redu es the given system to a normal form that an easily be integrated. The program is based on the algorithms introdu ed in hapter 6. The module ESP n is based on the algorithms de ned in the hapters 7 and 9. It integrates two- and three-dimensional systems in the neighbourhood of elementary singular points. Therefore it omputes normal forms using the primitive fun tions jordan sys, PDNF n, GNF n and PI NF n. All two- and a large number three-dimensional normal forms are integrable. The integrations are performed by a large number of elementary integration pro edures. The redu tion of the remaining three-dimensional normal forms is ontrolled by the pro edure two d solutions. If the redu tion is performed without blowing-ups an T appropriate matrix A for the power transformation X = X~ A is omputed by the fun tion find matrix A. The power transformation is applied to the initial system in the pro edure power trans n and the re ursion is de ned within the module two d solutions. If blowing-ups have to be used their appli ation is ontrolled by the module two d solutions that uses the Newton diagram omputed using the fun tion ND. On e solution urves for two-dimensional systems have been omputed, those solution urves, the se tors and the transformations are extended to the three dimensions by the fun tion solution 23. Therefore the proje tion of the se tors on the x1 x2 -plane has to be reversed. If problems of bije tivity o

ur the de nition of the se tors is adapted. Further the equation remaining from the splitting of the two-dimensional system is integrated to omplete the solution

urves. The module NESP n handels the integration of two-dimensional systems in the neighbourhood of nonelementary singular points. It is based on the algorithms introdu ed in

hapter 8. It uses the primitive ND for the ompuatation of the Newton diagram. Any of the omputed fa es are used to de ne blowing-ups that redu e the system and enables the omputation of solution urves. The main transformation used in this ontext are the elementary operations as time- and power transformations. They are de ned in the fun tions ntt n and power trans n. The used methods lead to a re ursive all if the entire algorithm. The de nition of the re ursion is based on the omputations of the new singularities on the ex eptional divisor and on the omputation of subse tors. Those operations are performed for real and omplex problems by the elementary pro edure subse tors.

Chapter 11. The FRIDAY pa kage

143

The primitive fun tions transform se tor n, vi se tor 2 and vi se tor 3 are not used by the main pro edure but they represent helpful tools for handling se tors and for displaying them. Their use will be illustrated by examples in se tion 11.3. Only a part of the implemented modules are visible to the user. All integrations and redu tions for two- and three-dimensional systems an be performed using the main pro edure FRIDAY. The use of the visible pro edures is illustrated by examples in se tion 11.3. 11.2

Using the pa kage

Maple

Maple

The FRIDAY pa kage is written in V, release 5. It is available as a pa kage or as sour e ode les. If the pa kage form is used, the pa kage has to be installed using the ommand

> with(FRIDAY);

This makes all visible pro edures and fun tions available. The with ommand an only be exe uted if the variable libname

:= libname; `user=pa kages=F RI DAY ` :

in the le .mapleinit has been set to the dire tory that ontains the pa kage. The pa kage version ontains help topi s that explain the use of the most important fun tions that are visible to the user. Those help pages an be onsulted using the ommand

> ?fun tion

If the sour e ode of the FRIDAY pa kage is available the program an be installed using the ommand

> read(FRIDAY);

or an equivalent ommand if the on erned les are not in the working dire tory. Informations on the urrent state of the omputations an be displayed during runtime. Therefore the onstant infolevel[FRIDAY℄ has to be set to a value between 1 and 3. For infolevel[FRIDAY℄ = 1 only basi informations are displayed. For higher values those informations are more and more pre ise. is set The FRIDAY pa kage and its sour e ode are available on the internet. More informations are available on the following adress

Maple http

: ==www

lm :imag:f r=C F =logi iel:html :

The pa kage version adresses to users that are only interested in omputation results. The sour e ode allows a user to perform only elementary operations and to modify or extend the FRIDAY pa kage. 11.3

Introdu ing examples

The way the FRIDAY pa kage works is best illustrated by some examples. In this se tion examples for the use of pro edures ontained in the FRIDAY pa kage are given for a number of representative problems. The pro edures are apable of treating more omplex problems and are not restri ted to the given simple examples.

144

11.3. Introdu ing examples

FRIDAY

Initial system X_ = F (X ) separate ases

dierentiate 3 possible ases

SP n

ESP n

simple point

elementary singular point

normal form and integration

Poin are-Dula normal form

PDNF n

m

NESP n

nonelementary singular point ND

Series of blowing-ups ontrolled by the Newton diagram

Computation of , the dimension Re ursive appli ation of the entire algorithm for the redu ed systems of the resonant plane

m F := [3+x[1℄+x[2℄,3*x[1℄*x[2℄-2*x[1℄^2℄; > X:=[x[1℄,x[2℄℄; S:= [0,0℄;epsilon:=1/2;index:=4; F

:= [3 + x1 + x2 ; 3 x1 x2 X

:= [x1 ;

2 x1 2 ℄

x2 ℄

:= [0; 0℄

S 

:= 1=2

index

:= 4

With these de nitions the main pro edure an be exe uted. The varibles and t are used for the omputed solution urves.

> sol:=FRIDAY(F,X,S, ,t,epsilon,index):

The number of omputed solution urves is 1.

> nops(sol);

1 The omputed solution is represented by the list sol[1℄ with 5 elements.

> nops(sol[1℄);

5 The rst element of the list sol[1℄ represents the redu ed system.

> print(sol[1℄[1℄);

[1; 0℄ This system an be integrated. The integration results are given in the se ond element of the list.

> print(sol[1℄[2℄);

[t1 ;

1 ℄

The third element of the list represents the transformation X = H (X~ ) that was used to redu e the initial system.

146

11.3. Introdu ing examples

> print(sol[1℄[3℄); 1 3 5 3 11 4 3 2 1 2 x1 + x1 x2 + x1 + x1 x2 x ; 2 2 2 3 8 1 9 2 3 2 2 9 4 5 3 x2 + x1 x2 + x1 x2 x1 x2 x ℄ 6 x1 3 2 2 2 2 1

[3 x1 + x1 x2 +

The fourth element sol[1℄ ontains the neighbourhood in the new oordinates that denotes the set where the omputed solution is valid.

> print(sol[1℄[4℄); [[

1 1 1 1 1 1 1 1 1 1 1 1 ; t1 ; [ ; ℄℄; [ ; t1 ; [ ; ℄℄; [t1 ; ; [ ; ℄℄; [t1 ; ; [ ; ℄℄℄ 2 2 2 2 2 2 2 2 2 2 2 2

For a simple point this se tor is the entire neighbourhood U~ = fX~ : jx~1 j  ; jx~2 j  g. ~ U is given by a list of 4 urves that denote the borders of U . Ea h urve is given in a parametrized form. The rst urve for example is given by X~ (t1 ) = (1=2; t1 ) with t1 = 1=2 : : : 1=2.

> print(sol[1℄[4℄[1℄);

[

1 1 1 ; t1 ; [ ; ℄ 2 2 2

In the ase of an elementary singular point the se tors in the initial oordinates are obtained by transforming the neighbourhood U~ . to the initial oordinates. The transformation used for this purpose is given as last element of the list.

> print(sol[1℄[5℄);

[x1 ; x2℄ In the ase of a simple point this transformation is identity. The integration results

omputed for the redu ed system an be transformed to the initial oordinates using the pro edure newtrans n and the transformation ontained in the third element of the list.

> sol2:=newtrans_n(sol[1℄[3℄,X,sol[1℄[2℄); 

3 2 1 2 1 3 5 3 11 4 t1 + t1 1 + t1 + t1 1 t ; 2 2 2 3 8 1  3 2 2 5 3 9 4 9 2 3 6 t1 t t

1 + t1 1 + t1 1 2 2 2 1 1 2 1

3 t 1 + t 1 1 +

As for the omputation of solution urves for simple points no time transformations are used the omputed urves are the aproximated exa t solutions of the initial system. The pre ision of the algorithm an be tested by substituting X =sol2 in the equation _ X

F (X )

= ina

uar y

(11.2)

Chapter 11. The FRIDAY pa kage

147

The lowest degree in t1 and 1 of the terms remaining in ina

uar y denotes the degree of approximation. As the omputations were exe uted with the index 4 the lowest degree of the remainig terms is 4 in both lines of equation (11.2). > ina

uar y := diff(sol2[1℄,t[1℄)-subs(x[1℄=sol2[1℄,x[2℄=sol2[2℄,F[1℄): > ldegree(expand(ina

uar y),[t[1℄, [1℄℄); 4

> ina

uar y := diff(sol2[2℄,t[1℄)-subs(x[1℄=sol2[1℄,x[2℄=sol2[2℄,F[2℄): > ldegree(expand(ina

uar y),[t[1℄, [1℄℄); 4

11.3.2 Computation of n-dimensional normal forms for non-nilpotent singular ve tor elds Non-nilpotent singular ve tor elds an be transformed to normal form. In the FRIDAY pa kage there exist 3 pro edures PDNF n, PI NF n and GNF n that ompute normal forms for dierent purposes. Computation of the Poin are-Dula normal form

The pro edure PDNF n omputes the Poin are-Dula normal form for a given ve tor eld. It is based on the algorithms introdu ed in se tion 3.4. The pro edure PDNF n

an be used separately from the main pro edure and allows the omputation of ndimensional normal forms for ve tor elds F if the matrix A = DF(0) is in Jordan form. If this is not the ase the pro edure jordan sys that is based on the Maple fun tion jordan an be used to ompute a new system JF su h that its linear part B = D JF(0) is in Jordan form. The linear transformation that transforms F to JF is assigned to the optional variable tr1. > F :=[x[1℄+2*x[1℄-x[2℄+2*x[1℄*x[2℄-x[3℄*x[2℄,x[2℄-3*x[4℄, > x[3℄+2*x[4℄,-x[3℄+x[4℄℄; F

:= [3 x1

x2

+ 2 x2 x1

x3 x2 ; x2

3 x4 ; x3 + 2 x4 ;

x3

+ x4 ℄

> A:=lin_part_n(F,X):print(A);

2 3 66 0 40

0

1 1 0 0

0 0 1 1

3

0 37 7 25 1

> JF:=jordan_sys(F,X,'tr1'):B:=lin_part_n(JF,X):print(B);

148

11.3. Introdu ing examples 2

1 6 0 6 4 0 0

0 0 3 0p 0 1+I 2 0 0 1

3

0 7 0 7 0p 5 I 2

For the given example the pro edure PDNF n omputes the Poin are-Dula normal form of the ve tor eld JF up to order 3. Further the used transformation is assigned to the optional variable tr. The transformation trans that transforma the initial ve tor eld F into its Poin are-Dula normal form NF an be omputed from tr1 and tr using the pro edure newtrans n.

> NF:=PDNF_n(JF,X,3,'tr');

NF := [x1 ; 3 x2 +

p 34 3 x1 x3 x4 + 12 x1 ; (1 + I 2) x3 ; (1 3

I

p

2) x4 ℄

> trans:=newtrans_n(tr1,X,tr):print(trans); 

1 1 x2 + x 6 2 1

1 2 x x +::: 3 2 1



Normal forms for real systems In example 13 in se tion 7.2 the Poin are-Dula normal form for the pendulum equation in the neighbourhood of the singularity X = 0 was omputed. The onsidered sysem has only real oeÆ ients but two omplex onjugated, purely imaginary eigenvalues. The parti ularity of those omputations are that parameters are allowed in the linear part of the system as the resonan e equation an be solved expli itely. The omputation of the normal form for systems with 2 omplex onjugated, purely imaginary eigenvalues an be performed using the pro edure PI NF n. In the onsidered example the omputations are performed up to order 6. The last input variable in the fun tion all indi ates weather polar oordinates = x~1 sin x~2 x2 = x ~1 os x~2 x3 = x3 x1

:::

should be introdu ed of not.

> F :=ve tor([x[2℄, onvert(taylor(-g/l*sin(x[1℄),x[1℄,4),polynom)℄); F

:=

 x2 ;

> sol:=PI_NF_n(F,X,6,false);

g x1 l

1 + 6

g x1 l

3

Chapter 11. The FRIDAY pa kage

sol

:= [ x1

g x2 l

1 16

+

1 16

g x1 x 2

g x2 x1

2

l

l

2

+

1 3 x 16 1

1 16

g

149

2x 3 2 2 l

17 3072

g

+

17 3072

17 g2 x1 2 x2 3 17 + 2 l 1536 l 3072 17 17 g x2 2 x1 3 5 x1 ℄ 3072 1536 l

x1

2x x 4 1 2 2 l

4gx

2

+

g

3x 5 2 ; 3 l

> sol:=PI_NF_n(F,X,6,true);

sol

2

3

6

1 g ( 3072 l2 + 192 g x1 2 l + 17 x1 4 g2 )7 7 r 5 3072 g 3

:= 6 40;

l

l

The generalized normal form Another kind of normal form for n-dimensional systems an be omputed using the pro edure GNF n. The implemented algorithm is based on the matrix representation method and the Maple optimisation pro edure leastsqrs. It redu es a maximum of nonlinear terms, even if the matrix DF (0) if not in Jordan form. If the matrix DF (0) is in Jordan form the algorithm yields the Poin are-Dula normal form. However the used algorithm is less eÆ ient than the algorithms implemented in PDNF n and for the example of the pendulum it does not yield the optimal normal form ( ompare se tion 7.2). > g:=1:l:=2:sol:=GNF_n(F,X,6); sol

3 22156 11078 2 2 3 4 3 2 x2 x1 x2 x2 x1 x x := [x2 161 161 129475395 129475395 2 1 1 12 3 2 22156 44312 2 3 5 2 x1 + x1 + x1 x2 + x1 x2 + x 2 161 161 129475395 25895079 1 11078 4 + x x ℄ 129475395 2 1

5539 5 x ; 25895079 2

In the main pro edure FRIDAY the pro edure GNF n is used if it is known that the normal form of a given ve tor eld has no nonlinear resonant terms (see se tion 7.2). 11.3.3

Integration of two-dimensional elementary singular points

The integration of two-dimensional ve tor elds F in the neighbourhood of an elementary singular point is handeled by the main pro edure FRIDAY. The on erned ve tor elds

an have real or omplex oeÆ ient and parameters if they do not ae t the resonan e equation. The implemented algorithms have been introdu ed in hapter 7. > F := ve tor([x[2℄, onvert(taylor(-g/l*sin(x[1℄),x[1℄,4),polynom)℄); F

:=

 x2 ;

g x1 l

1 + 6

g x1 l

3

150

11.3. Introdu ing examples

> sol:=FRIDAY(F,X,S, ,t,1/2,4):

WARNING : parameters in the system ! WARNING : parameters in linear part of the system ! Like in the example of a regular point in se tion 11.3.1 the returned solution list sol

ontains the redu ed system sol[1℄[1℄, the integration result sol[1℄[2℄ and the used transformation sol[1℄[3℄.

> sol[1℄[1℄;

3

2

6 1 g ( 16 l + x1 2 g)7 7 60; r 5 4 16 g 2 l

l

> sol[1℄[2℄; [ 1 ;

t1

℄

> sol[1℄[3℄; "

r 1 192

x1

l

sin(x2 ) (192 l + 5 x1 2 g + 4 g x1 2 os(x2 )2 ) ;

l

1 64

11.3.4

g

g x1

os(x2 ) (64 l

5 x1 2 g + 4 g x1 2 os(x2 )2 ) l

#

2

Integration of two-dimensional nonelementary singular points

The integration of two-dimensional ve tor elds F in the neighbourhood of a nonlelementary singular point is based on the method introdu ed in hapter 8. The ve tor eld an have real or omplex oeÆ ients. The integration is handeled by the main pro edure FRIDAY. The variable infolevel[FRIDAY℄ has been set to 1 to give some informations on the omputations during runtime. Those informations indi ate the

urrent state of the omputations.

> F:=[-x[1℄^4+x[1℄^3*x[2℄,13/9*x[1℄^6*x[2℄^2-x[1℄^2*x[2℄^2+x[1℄*x[2℄^3℄; F

:= [

x1

4

+ x1 3 x2 ;

13 6 2 x x 9 1 2

> sol:=FRIDAY(F,X,S, ,t,1/2,4):

x1

2

x2

2

+ x1 x2 3 ℄

Chapter 11. The FRIDAY pa kage

151

separate_ ases_23: NONELEMENTARY SINGULAR POINT separate_ ases_23: NONELEMENTARY SINGULAR POINT separate_ ases_23: SIMPLE POINT NF_SP_n: simple point al ulated separate_ ases_23: SIMPLE POINT NF_SP_n: simple point al ulated

For the onsidered example, that has already been studied in example 15 in se tion 8.2, 5 dierent solutions are omputed. The se tors returned in the variable sol[i℄[4℄ for i = 1; : : : ; 5 an be transformed to the initial oordinates by the pro edure transform se tor n and the transformation in sol[i℄[5℄ for i = 1; : : : 5. > nops(sol); 5

> sol[1℄[4℄; [[

1 1 1 1 1 1 1 1 1 1 1 1 ; t1 ; [ ; ℄℄; [ ; t1 ; [ ; ℄℄; [t1 ; ; [ ; ℄℄; [t1 ; ; [ ; ℄℄℄ 2 2 2 2 2 2 2 2 2 2 2 2

> print(sol[1℄[5℄); [x1 x2 ; x2℄

> se t:=transform_se tor_n(sol[1℄[4℄,X,sol[1℄[5℄,1/2): The resulting se tors an be visualized by the pro edure vi se tors 2. All 5 se tors are sket hed in gure 8.8 in se tion 8.2. > vi_se tors_2(se t,0.6);

If the optional variable omplex is set in the all of the pro edure FRIDAY, omplex solutions are omputed. This yields 8 solutions as the de nition of subse tors in C 2 yield more subse tors than in R2 ( ompare se tion 8.2). The se tors are de ned by the urves (x1 (t1 ; t2 ; t3 ); x2 (t1 ; t2 ; t3 )); t1 ; t2 ; t3 2 R.

152

11.3. Introdu ing examples

> read(bruno_2):sol:=FRIDAY(F,X,S, ,t,1/2,4, omplex): > nops(sol); 8

> sol[1℄[4℄;  

1 2

1 2 I

11.3.5

t3 ;

 

 

 t 1

+ I t2 ; 21 + I t3 ; t1 + I t2 ; t1 + I t2 ; 12 + I t3 1 1 t1 + I t2 ; 2 I + t3 ; t1 + I t2 ; t1 + I t2 ; 2 I + t3 ;

I t3 ; t1

+ I t2 ; t1 + I t2 ;

1 2 1 2

I t3 I

 ; 

t3

Integration and Redu tion of three-dimensional elementary singular points

The pro edures for the integration of three-dimensional ve tor elds with elementary singular points is based on methods introdu ed in hapter 9. They an treat systems with real oeÆ ients. The integration of those ve tor elds is handeled by the main pro edure FRIDAY. The variable infolevel[FRIDAY℄ is set to 1 to give some informations on the

omputations during runtime. The support of the onsidered ve tor eld F lies within a two-dimensional one and within the rst quadrant in the spa e of exponents. Therfore it an be integrated.

> F := [-2*x[1℄+x[1℄^2*x[3℄^2,x[2℄,x[3℄+x[1℄*x[2℄^2*x[3℄℄; F

:= [ 2 x1 + x1 2 x3 2 ;

x2 ; x3

+ x1 x2 2 x3 ℄

> sol:=FRIDAY(F,X,[0,0,0℄, ,t,1/2,4): separate_ ases_23: two_d_solutions: separate_ ases_23: separate_ ases_23: separate_ ases_23: two_d_solutions: two_d_solutions: separate_ ases_23: separate_ ases_23: separate_ ases_23: two_d_solutions:

ELEMENTARY SINGULAR POINT 2d pro edure NONELEMENTARY SINGULAR POINT SIMPLE POINT SIMPLE POINT 2d solution al ulated ! 2d pro edure NONELEMENTARY SINGULAR POINT SIMPLE POINT SIMPLE POINT 2d solution al ulated !

The ve tor eld F is redu ed to a two-dimensional ve tor eld with a nonelementary singular point. This ve tor eld an be integrated but the obtained solution urves are only valid in se tors. Those se tors are extended to three-dimensional se tors. They an be transformed to the initial oordinates using the pro edure transform se tor n and visualized in three dimensionas by the pro edure vi se tors 3.

> > > >

se t := transform_se tor_n(sol[i℄[4℄,X,sol[i℄[5℄,1/2); vi_se tor_3(se t,t,2): a:=1/2: for i from 1 to nops(sol) do display(p[i℄,view=[-a..a,-a..a,-a..a℄); od;

Chapter 11. The FRIDAY pa kage

153

The 12 resulting se tors are shown in the gures 11.2 and 11.3. It an be observed that in every gure three identi se tors appear. This is due to the fa t that in addition to the general solutions omputed for a se tor two further solutions, the so alled parti ular solutions, have to be onsidered. (See also ase 4 in se tion 7.1). The neighbourhood of X = 0 is de omposed into 6 se tors for x1 > 0 and 6 se tors for x1 < 0. That shows that the power transformation, that is used to redu e the initial three-dimensional system, is not inje tive. Therefore additional onstru tions as they were introdu ed in se tion 2.3 are used. For the onsidered system F the stru ture of the two-dimensional se tors is well preserved by the extention to three oordinates and the retransformation to the initial oordinates. This is not always true as the inverse of the power transformation, that is used to simplify the ve tor eld F, is a blowing-up. 11.4

Tests

A large number of tests have been run to ensure the viability of the FRIDAY pa kage. The possible tests for the omputed solutions are the following :  Introdu ing the solution urves into the initial dierential equation and omputing

the a

uar y of the results yields an evaluation for the pre ision of the omputations. This test an only be used if the omputed results are approsimations of the exa t solutions. It has for example be used to proove the viability of the omputation of normal forms for systems with simple points. If time transformations are used at any step of the omputations, this test fails.

H for Hamiltonian systems or into the s alar dierential equation asso iated the twodimensional systems. This yields another indi ator for the a

uar y of the omputed results. This method also works if time transformations were applied to the

onsidered system. However it fails if power transformation were used. Power transformations ause problems as negative exponents may appear in the retransformed solution urves.

 The omputed solution urves an be introdu ed into the energy fun tion

Due to the limited possiblities it is in general not possible to test the omputed results. However the primitives an be tested individually and the tests proposed above an be used for some parti ular exampes. As far as possible every module was tested for arbitrarilly

hosen ve tor elds.

154

11.4. Tests

Figure 11.2: This gure shows 6 se tors omputed for the example in se tion 11.3.5. They form a de omposition of a neighbourhood of X = 0 for x1 < 0.

Chapter 11. The FRIDAY pa kage

155

Figure 11.3: This gure shows 6 se tors omputed for the example in se tion 11.3.5. They form a de omposition of a neighbourhood of X = 0 for x1 > 0.

Con lusion The obje tive of this thesis is the study of the theoreti al and pra ti al aspe ts of the redu tion and formal integration of two- and three-dimensional systems of autonomous dierential equations. The ase of two-dimensional systems has been solved ompletely. Any onsidered system an be redu ed and integrated by the proposed algorithms. The three-dimensional ase yields mu h more problems. Redu tions are only possible for some parti ular ases. We have introdu ed an algorithm that allows the formal integration of any three-dimensional system in the neighbourhood an elementary singular point. These results an be obtained due to a generalization of power transformations and blowing-ups. The used transformations an be interpreted geometri ally. This interpretation allows a very eÆ ient handling of all redu tions. All transformations are interpreted geometri ally in the spa e of exponents as manipulations on the support of the system. The use of those geometri methods also allows to over the on erned neighbourhood entirely by se tors and to ompute all solutions. The generalization of the proposed algorithms to higher-dimensional systems with elementary singular points is possible. For this purpose the virtual Newton diagram has been introdu ed. The virtual Newton diagram ompletes the information obtained for the

onsidered system by the Newton diagram. Therefore it allows ontrolled blowing-ups and an entire overing of the on erned neighbourhood by se tors. However the integration of the redu ed systems is only possible if the redu tion of higher-dimensional nilpotent systems an be ontrolled. The redu tion of nilpotent three- and higher-dimensional systems by blowing-ups is only possible for some parti ular ases. In these ases the virtual Newton diagram an be used to onstru t blowing-ups that yield se tors that entirely over the on erned neighbourhood. The proposed algorithms have been implemented in the FRIDAY pa kage. This program formally integrates any two-dimensional and a large number of threedimensional systems. The problem of onvergen e is always arising in the ontext of the redu tion of dynami al systems by symboli omputations. Although many results on the resummation of formal power series are known, many problems of onvergen e for normalizing transformations remain unsolved. Further problems are en ountered in generalizing the obtained results to higher dimensional problems. These problems oer many possibilities for further work in this domain. Three and higher dimensional blowing-ups ertainly ause the main theoreti al problems as it has been shown that nite su

esive blowing-up an not entirely redu e any

Maple

157

158

11.4. Tests

nonelementary singularity. For an entire study of those ases the use of some further methods might be ne essary. Three dimensional blowing-ups also ause many algorithmi problems as it has been shown for the di riti al ase of an edge. This problem will have to be onsidered more

losely to entirely over the ase of a nonelementary singular point. In this work an algorithm for the omputation of the virtual Newton diagram has been proposed. However this algorithm performs several omputations of the onvex hull of a set of points. As this is not very eÆ ient this algorithm ould be repla ed by a more appropriate method. The additional onstru tion introdu ed for power transformations makes the use of several similar transformations ne essary. As the solutions omputed for the dierent transformations are similar too, some simpli ations in the algorithms might yield a mu h more eÆ ient program. The pa kage FRIDAY works very eÆ iently for two dimensional and some three dimensional problems. However generalizations to higher dimensional problems will need some basi modi ations of the se tor notation. Handling se tors in several re ursions and dierent dimensions auses implementation problems. Handling the se tors is mu h more ompli ated than handling the solutions and the su

essive transformations. The possible appli ations of the proposed methods are the studies of bifur ations as normal forms play a very important role in this domain. As the use of parameters is possible up to a ertain degree this represents an important advantage. The use of parameters

ould also be used to perform algebrai optimization on simple physi al models.

Maple

Con lusion L'obje tif de ette these etait d'etudier les aspe ts theoriques et pratiques de la redu tion et de l'integration des systemes d'equations dierentielles ordinaires en deux et trois dimensions. Le as des systemes en deux dimensions a ete resolu entierement. Tout systeme en deux dimension peut ^etre integre par les algorithmes que nous avons proposes. Dans le as des systemes en trois dimensions nous ren ontrons des problemes qui rendent l'etude de tels systemes beau oup plus omplexe. Nous proposons un algorithme permettant l'integration de tout systeme au voisinage d'un point singulier elementaire. Ces resultats sont bases sur la generalisation de l'utilisation des transformations quasimonomiales et des e latements. La generalisation des algorithmes proposes aux systemes en dimension superieure est possible. Les problemes ren ontres peuvent ^etre resolus gr^a e au diagramme de Newton virtuel que nous avons introduit auparavant. Le diagramme de Newton virtuel omplete les informations obtenues par le diagramme de Newton. Il permet de ontr^oler les e latements utilises et de nit un ensemble de se teurs ouvrant entierement le voisinage on erne. L'integration des systemes que nous obtenons gr^a e a es methodes n'est possible que si les systemes nilpotents obtenus sont integrables. La redu tion des systemes nilpotents en trois dimensions en utilisant des e latements n'est possible que pour ertains as parti uliers. Dans es as nous pouvons egalement utiliser le diagramme de Newton virtuel qui permet de de nir des e latements ouvrant entierement le voisinage on erne par un ensemble de se teurs. Les algorithmes que nous avons proposes dans ette these ont ete implantes en Maple dans le pa kage FRIDAY. Ce logi iel permet l'integration formelle de tout systeme en deux dimensions et d'une large partie des systemes en trois dimensions. Im me semble que les points suivants peuvent ompleter e travail. Du point de vue theorique, de nombreux problemes restent a resoudre pour des systemes de dimension trois et superieure. L'aspe t de la resommation et de la onvergen e devient essentiel si nous voulons utiliser les methodes proposees pour resoudre des problemes reels. Mais les travaux onnus sur la resommation ne permettent pas en ore de formuler des algorithmes ou d'implanter un logi iel. Du point de vue du ode de al ul, la partie des formes normales et le al ul de l'enveloppe onvexe ne essite une ree riture dans un langage ompile. L'algorithme introduit pour le al ul du diagramme de Newton virtuel n'est pas tres rapide. Il pourrait ^etre rempla e par un algorithme plus sophistique. Le logi iel FRIDAY est tres eÆ a e pour resoudre des problemes en deux et trois dimensions. En generalisant les pro edures existantes a des dimensions superieures, les se teurs utilises vont devenir de plus en plus ompliques. Un probleme fondamental est 159

160

11.4. Tests

don elui de la gestion des se teurs en plusieurs etapes re ursives et en dierentes dimensions. Les appli ations possibles des methodes proposees sont surtout les problemes des bifur ations, ar les formes normales jouent un r^ole essentiel dans e domaine. Le fait que des parametres soient partiellement permis peut representer un avantage non negligeable. Il permet egalement d'optimiser les parametres d'une equation issue d'un probleme de modelisation.

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