Local methods in nonlinear differential equations 0387189262, 9780387189260

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Alexander D. Bruno

Local Methods in Nonlinear Differential Equations Part I The Local Method of Nonlinear Analysis of Differential Equations

Part II The Sets of Analyticity of a Normalizing Transformation Translated from the Russian by William Hovingh and Courtney S. Coleman

With 94 Figures

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Alexander D. Bruno M. V. Keldysh Institute of Applied Mathematics Academy of Sciences of the USSR Miusskaya pl. 4, 125047 Moscow, USSR William Hovingh 721 2nd Avenue Salt Lake City, UT 84103, USA

Courtney S. Coleman Harvey Mudd College Claremont, CA 91711, USA

Supplemented version of the original Russian edition: Lokal’nyj metod nelinejnogo analiza differentsial’nyck uravnenij Publisher Nauka, Moscow 1979

This volume is part of the Springer Series in Soviet Mathematics Advisers: L. D. Faddeev (Leningrad), R. V. Gamkrelidze (Moscow)

Mathematics Subject Classification (1980): 34C, 34D, 14B, 70E, 70K, 52A25

ISBN 3-540-18926-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18926-2 Springer-Verlag New York Berlin Heidelberg

Library of Congress Cataloging-in-Publication Data Briuno, Aleksandr Dmitrievich. Local methods in nonlinear differential equations. (Springer series in Soviet mathematics) Translation of: Lokal’nyi metod nelineinogo analiza differentsial’nykh uravnenii. Bibliography: p. Includes index. 1. Differential equations, Nonlinear. I. Title. II. Series. QA372.B85413 1988 515.3’55 88-25941 ISBN 0-387-18926-2 (U.S.)

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September, 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Typesetting: Asco Trade Typesetting Ltd, Hong Kong. Printing: Saladruck, Berlin. Bookbinding: Liideritz & Bauer-GmbH, Berlin 2141/3020-543210 - Printed on acid-free paper

LIBRARY. UNIVERSITY OF ALABAMA IN HUNTSVILLE HUNTSVILLE, AL 35899

Introduction to the English Edition by Dr. Steven Wiggins When faced with a new problem, an often effective strategy for solving it is to somehow transform it into the form of a problem previously solved. Obviously there are no general recipes for implementing such a procedure; instead one must usually rely on insight and luck. However, in the context of the “local” theory of ordinary differential equa­ tions (“local” means in a neighborhood of some invariant manifold) such a procedure for reducing an ordinary differential equation to the “simplest possible form” (what this phrase means can be precisely defined and depends on the par­ ticular problem) or normal form is available and goes by the name of the method of normal forms. The method of normal forms is not new and is usually attributed to Poincare although some of the basic ideas of the method can be found in earlier works of Jacobi, Briot and Bouquet. Following Poincare, the method was further devel­ oped and utilized in the works of Lyapunov, Dulac, Birkhoff, Cherry, Siegel, Moser, Sternberg, Chen, and Gustavson. In this book, A. D. Bruno gives an account of the work of these mathemati­ cians as well as the results of his own extensive investigations on the subject. The book begins with a thorough development of the analytical techniques necessary for the implementation of the theory as well as an extensive description of the geometry of the Newton polygon. He then proceeds to discuss the normal form of systems of ordinary differential equations giving many specific applications of the theory. An underlying theme of the book is the unifying nature of the method of normal forms regarding techniques for the study of the local properties of ordinary differential equations. Indeed, Bruno shows that many of the techniques used for studying ordinary differential equations having a small parameter (e.g. the method of averaging) may be viewed as a special case of the method of normal forms. In the second part of the book he shows, for a special class of equations, how the method of normal forms yields classical results of Lyapunov concerning families of periodic orbits in the neighborhood of equilibrium points of Hamiltonian systems as well as the more modern results concerning families of quasiperiodic orbits obtained by Kolmogorov, Arnold, and Moser. One of the more beneficial results of the recent explosion of interest in non­

VI

Introduction to the English Edition

linear dynamical systems is the increased interaction between mathematicians and applied scientists. Regarding the method of normal forms, this has led to new developments in the theory as well as a variety of applications of the method to bifurcation theory, fluid mechanics, and structural mechanics. The following references supplement those given by Bruno.

Supplementary References Arneodo, A., Coullet, P.H., Spiegel, E.A., Tresser, C.: 1985, Asymptotic chaos. Physica 14D (1985), 327-347 Bibikov. Y.N.: 1979, Local Theory of Nonlinear Analytic Ordinary Differential Equations. Leet. Notes Math. 702 (1979) Birkhoff, G.D.: 1927, Dynamical Systems, A.M.S. Coll. Publications IX, New York Broer, H.W.: 1979, Bifurcations of singularities in volume preserving vector fields. Thesis, University of Groningen (1979) Broer, H.W.: 1980, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case. In: Dynamical Systems and Turbulence, Warwick 1980 ed. D.A. Rand, L.-S. Young, Leet Notes Math. 898 (1980) Buchanan, D.: 1939 A transformation to the normal form. Rend. Circ. Matem. Palermo 62 (1939), 385-387 Burgoyne, N. and Cushman, R.: 1974, Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues. Cel. Meeh. 8 (1974), 435 443 Burgoyne, N. and Cushman, R.: 1977, Normal forms for real linear Hamiltonian systems. In: The 1976 Ames Research Center (NASA) Conference on Geometric Control Theory, ed. C. Martin, Math. Sci. Press, Brookline, Mass., 1977 Chenciner, A., looss, G.: 1979 Bifurcation de tores invariants. Arch. Rat. Meeh, and Anal. 69 (1979), 109-198 Chenciner, A., looss, G.: 1979, Persistance et bifurcation de tores invariants. Arch. Rat. Meeh, and Anal. 71 (1979), 301-306 Cherry, T.M.: 1927, On the solution of Hamiltonian systems of differential equations in the neighborhood of a singular point. Proc. London Math. Soc. series 2, 27 (1979), 151-170 Chow, S.N., Hale, J.K.: 1982, Methods of Bifurcation Theory. New York, Heidelberg, Berlin: Springer (1982) Churchill, R.C., Kummer, M. and Rod, D.L.: 1983, On averaging, reduction and symmetry in Hamiltonian systems. J. Diff. Eq. 49 (1983), 359-414 Coullet, P.H. and Spiegel, E.A.: 1983, Amplitude equations for systems with competing instabilities. SIAM J. Appl. Math. 43 (1983), 774 819 Cushman, R.: 1984, Normal form for Hamiltonian vector fields with periodic flows. In: Differential Geometric Methods in Mathematical Physics, ed. S. Sternberg, Reidel, Dordrecht, Holland (1984), 125-144 Cushman, R., Deprit, A. and Mosak, R.: 1983, Normal form and representation theory. J. Math. Phys. 24 (1983), 2103-2116 Cushman, R. and Rod, D.L.: 1982, Reduction of the semisimple 1:1 resonance. Physica 6D (1982), 105-112 Deprit, A. and Henrard, J.: 1968, A manifold of periodic orbits. Adv. Astron. Astroph. 6 (1968), 1-124 Deprit, A.: 1969, Canonical transformations depending on a small parameter. Cel. Meeh. 1 (1969), 12-30 Deprit, A.: 1982, Delaunay normalisations. Cel. Meeh. 26 (1982), 9 21 Deprit, A., Henrard, J., Price, F. and Rom, A.: 1969, Birkhoffs normalization. Cel. Meeh. 1 (1969), 225-251

Introduction to the English Edition

Vil

Duistermaat, J.J.: 1981, Periodic solutions near equilibrium points of Hamiltonian systems. In: Analytical and Numerical Approaches to Asymptotic Problems in Analysis (Proc. Conf. Univ. Nijmwegen, Nijmwegen, 1980), ed. O. Axelsson et al., North Holland Math. Studies 47 (1981), 27-33 Duistermaat, J.J.: 1983, Non-integrability of the 1:1:2 resonance. Ergodic Theory and Dynamical Systems 4 (1984), 553-568 Duistermaat, J.J.: 1983, Bifurcations of periodic solutions near equilibrium points of Hamiltonian systems. In: Bifurcation Theory and Applications, Montecatini 1983, ed. L. Salvadori, Leet. Notes Math. 1057, (1984) Giorgilli, A. and Galgani, L.: 1978, Formal integrals for an autonomous Hamiltonian system near an equilibrium. Cel. Meeh. 17 (1978), 267-280 Guckenheimer, J.: 1984, Multiple bifurcation problems of codimension two. SIAM J. Math. Anal. 15 (1984), 1 49 Guckenheimer, J. and Holmes, P.J.: 1983, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. New York, Heidelberg, Berlin: Springer Guckenheimer, J. and Knobloch, E.: 1983, Nonlinear convection in a rotating layer: amplitude expansions and normal forms. Geophys. Astrophys. Fluid. Dyn. 23 (1983), 247-272 Gustavson, F.G.: 1966, On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron. J. 71 (1966), 670-686 Holmes, P.J.: 1980, Unfolding a degenerate nonlinear oscillator: a codimension two bifurcation. Annals of the New York Academy of Sciences, 357, Nonlinear Dynamics (1980), 473-488 Holmes, P.J.: 1981, Center manifolds, normal forms, and bifurcation of vector fields. Physica 2D (1981), 449-481 Kelley, A.: 1963, Using changes of variables to find invariant manifolds of systems of ordinary differential equations in a neighborhood of a critical point, periodic orbit, or periodic surface. History of the problem. Technical report no. 3, Depth, of Math., Univ, of California (1963) Knobloch, E.: 1984, Bifurcations in doubly diffusive convection. In: Chaos and Statisti­ cal Methods, ed. Y. Kuramoto, New York, Heidelberg, Berlin: Springer (1984) Kummer, M.: 1976, On resonant nonlinearly coupled oscillators with two equal fre­ quencies. Comm. Math. Phys. 48 (1976), 53-71 Kummer, M.: 1978, On resonant classical Hamiltonians with two equal frequencies. Comm. Math. Phys. 58 (1978), 85-112 Kummer, M.: 1981, On the construction of the reduced phase space of a Hamiltonian system with symmetry. Indiana Univ. Math. J. 30 (1981), 281-291 Kummer, M.: 1981, On the existence of invariant analytic two-manifolds in a resonant system with two degrees of freedom. In: Classical Mechanics and Dynamical Systems, cds. R.L. Devaney, Z.H. Nitecki, New York: Marcel Dekker (1981) Langford, W. and looss, G.: 1980, Interactions of Hopf and pitchfork bifurcations. Work­ shop on Bifurcation Problems, cd. H. Mittelman, Birkhauser Lecture Notes, Basel (1980) Langford, W.: 1983, A review of interactions of Hopf and steady-state bifurcations. In: Nonlinear Dynamics and Turbulence, eds. G.I. Barenblatt, G. looss, D.D. Joseph, Boston: Pitman Meer, J.C. van der: 1982, Nonsemisimple 1:1 resonance at an equilibrium. Cel. Meeh. 27 (1982), 131 149 Meer, J.C. van der: 1986, The Hamiltonian Hopf Bifurcation. New York, Heidelberg, Berlin: Springer-Verlag Meron, E. and Procaccia, I.: 1986, Theory of chaos in surface waves: the reduction from hydrodynamics to few-dimensional dynamics. Phys. Rev. Lett. 56 (1986), 1323 1326 Meyer, K.R.: 1974, Generic bifurcations in Hamiltonian systems. In: Dynamical Systems Warwick 1974, Leet. Notes Math. 468, cd. A. Manning (1974) Meyer, K.R.: 1974, Normal forms for Hamiltonian systems. Cel. Meeh. 9 (1974), 517-522

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Moroz, I.M. and Holmes, P.J.: 1984, Double Hopf bifurcation and quasi-periodic flow in a model for baroclinic instability. J. Atmo. Sciences 41 (1984), 3147-3160 Moroz, I.M. and Leibovich, S.: 1985, Oscillatory and competing intstabilities in a nonlinear model for Langmuir circulations. Phys. Fluids 28 (1985), 2050-2061 Moser, J.: 1958, New aspects in the theory of stability of Hamiltonian systems. Comm. Pure Appl. Math. 11 (1958), 81-114 Moser, J.: 1973, Stable and Random Motions in Dynamical Systems. Princeton: Princeton University Press (1973) Schmidt, D. and Sweet, D.: 1973, A unifying theory in determining periodic families for Hamiltonian systems at resonance. J.Diff. Eq. 14 (1973), 597-609 Sell, G.: 1979, Bifurcation of higher dimensional tori. Arch. Rat. Meeh, and Anal, 69 (1979), 199-230 Sternberg, S.: 1961, Finite Lie groups and the formal aspects of dynamical systems. J. Math. Meeh. 10 (1961), 451-474 Takens, F.: 1973, Singularities of vector fields. Publ. IHES, 43 (1973), 47-100 Takens, F.: 1974, Forced Oscillations and Bifurcations, Applications of Global Analysis. Comm, of Math. Inst., Rijksuniversiteit, Utrecht (1974) Ushiki, S.: 1984, Versal deformation of singularities and its application to strange attrac­ tors. In: Chaos and Statistical Methods, ed., Y. Kuramoto, New York, Heidelberg, Berlin: Springer-Verlag (1984) Weinstein, A.: 1973, Normal modes for non-linear Hamiltonian systems. Inv. Math. 20 (1973), 47-57 Whittaker, E.T.: 1902, On the solution of dynamical problems in terms of trigonometric series. Proc. London Math. Soc. 34 (1902), 206-221 Whittaker, E.T.: 1917, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 2nd ed., Cambridge: Cambridge Univ. Press (1917) Williamson, J.: 1936, On the algebraic problem concerning the normal forms of linear dynamical systems. Am. J. Math. 58 (1936), 141-163

Pasadena, California

Table of Contents Introduction to the English Edition by Dr. Steven Wiggins............................

V

Part I The Local Method of Nonlinear Analysis of Differential Equations

Table of Contents..................................................................................................

3

Basic Notation......................................................................................................

7

Introduction..........................................................................................................

9

Chapter I.

§ 1. § 2. § 3.

Foundations of the Local Method ...............................................

14

Linear Inequalities inthe Plane.............................................................. Zeros of an Analytic Function............................................................... Level Curves of an Analytic Function .................................................

14 37 64

A System of Two Differential Equations.....................................

96

Chapter IL

§ 1. § 2. § 3. § 4.

Simple Points and Elementary Singularities ....................................... 96 Generalized Normal Forms................................................................... 124 A Nonelementary Singular Point ......................................................... 136 On Distinguishing Between a Center and a Focus............................. 153

Chapter III.

§ 1. § 2. § 3. § 4.

The Normalizing Transformation......................................................... The Integration and Classification of Normal Forms......................... Analytic Integral Sets............................................................................. The Normal Form and Methods of Averaging...................................

Chapter IV. § 1. § 2.

The Normal Form of a System on n Differential Equations .... 166

166 189 202 210

On the Newton Polyhedron ....................................................... 217

A System of Differential Equations....................................................... 217 Other Problems Using the Newton Polyhedron................................. 231

X

Table of Contents

Chapter V. § 1. § 2.

Applications of the Normal Form in Mechanics....................... 235

On the Motion of a Gyroscope in a Cardan Suspension................... 235 On the Oscillations of a Satellite in the Plane of an Elliptical Orbit. . . 254

References.............................................................................................................. 264

Part II The Sets of Analyticity of a Normalizing Transformation Preface.................................................................................................................... 275

Table of Contents.................................................................................................. 277 Basic Notation...................................................................................................... 279

Introduction.......................................................................................................... 281

§ 1. § 2. § 3. § 4. § 5.

The Seminormal Form........................................................................... Questions of Convergence..................................................................... A Hamiltonian System........................................................................... Families of Periodic Solutions............................................................... Integral Manifolds with Small Divisors...............................................

284 295 313 328 335

Author’s Comments (1986).................................................................................. 343

References.............................................................................................................. 344

Subject Index........................................................................................................ 347

Part I The Local Method of Nonlinear Analysis of Differential Equations

Table of Contents Basic Notation..............................................................................................

7

Introduction.................................................................................................

9

Chapter I.

§

Foundations of the Local Method.........................................

1. Linear Inequalities in the Plane......................................................... 1.1. Basic Definitions (thm. 1; examples 1,2)................................... 1.2. Cones (thm. 2, examples 3, 4, 5)................................................. 1.3. Degenerate Cases (example 6)................................................... 1.4. Polygons (example 7)................................................................ 1.5. Subsets of the Integral Lattice (thms 3, 4)................................. 1.6. The Dual Open Polygon (example 8)....................................... 1.7. Frommer’s Method (example 9) ............................................... 1.8. Affine Transformations.............................................................. 1.9. Unimodular Transformations (examples 10, 11; exercise 1) ... . §2 . Zeros of an Analytic Function.......................................................... 2.1. A Simple Point (thm 1; example 1)........................................... 2.2. The Order of Magnitude of a Function................................... 2.3. The Order of Magnitude of a Plane Curve (exercise 1)........... 2.4. Asymptotics of a Function on a Curve..................................... 2.5. Finding the Truncation (example 2; exercise 2)....................... 2.6. Power Transformations............................................................. 2.7. The Critical Point...................................................................... 2.8. The First Method (examples 3, 4; exercise 3)........................... 2.9. The Second Method of Resolution of a Singularity (examples 5, 6; exercise 4)..................................................... 56 2.10. The Method of Weierstrass...................................................... §3 . Level Curves of an Analytic Function ............................................. 3.1. Statement of the Problem.......................................................... 3.2. Coordinate Transformations (thm. 1)....................................... 3.3. A Nondegenerate Critical Point (thm. 2; example 1)...............

14 14 14 17 20 22 24 26 27 29 32 37 37 39 40 43 44 45 47 49

62 64 64 64 66

Table of Contents

4

3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10. 3.11.

Chapter II.

Sets of Convergence of Power Series of Class (examples 2, 3; exercise 1)..................................................... Properties of Class Y Series (thm. 3; lemma 1)....................... Transformations of Class (thm. 4) ..................................... The Vertex Case (thm. 5; exercise 2)......................................... An Edge of the Newton Open Polygon ................................... Synthesis and Examples (examples 4, 5; exercise 3).................. On Normal Forms (thm. 6)....................................................... On Partioning Spaces (example 6; exercises 4, 5).....................

69 74 76 78 80 83 89 90

A System of Two Differential Equations.............................

96

1. Simple Points and Elementary Singularities.................................... 1.1. Introduction .............................................................................. 1.2. The Neighborhood of a Simple Point (thm. 1)......................... 1.3. Linear Systems (examples 1, 2) ................................................. 1.4. An Elementary Singular Point (example 3; exercise 1)............ 1.5. The Principal Theorem on the Normal Form (thm. 2)............ 1.6. Classification and Integration of Normal Forms ................... 1.7. Power Transformations of Differential Equations .................. 1.8. The Convergence of the Normalizing Transformation (example 4)............................................................................. Ill 1.9. Real Systems (thm. 3)................................................................ 1.10. On a Smooth Normalization..................................................... 1.11. Integrating Real Normal Forms (exercise 2) ........................... §2 . Generalized Normal Forms.............................................................. 2.1. The Second Theorem on the Normal Form (thm. 1; lemma 1) . 2.2. Integrating the Generalized Normal Form............................. 2.3. The Third Theorem on the Normal Form (thm. 2; lemma 2; exercise 1)................................................. 132 §3 . A Nonelementary Singular Point.................................................... 3.1. Statement of the Problem........................................................... 3.2. The Case of a Vertex................................................................... 3.3. The Case of an Edge................................................................... 3.4. Synthesis ..................................................................................... 3.5. Examples 1, 2, 3 (exercise 1)....................................................... 3.6. Comparison with Frommer’sMethod....................................... §4 . On Distinguishing Between a Center and a Focus......................... 4.1. Introduction ............................................................................... 4.2. Trigonometric Theory (thm. 1; example 1)................................ 4.3. The Study of Normal Forms..................................................... 4.4. On the Computability of theCoefficients of the Normal Form . 4.5. Integration of the Truncated System (3) and Computation of the Constant a (example 2; exercise 1)................................. 158

§

96 96 97 101 103 104 107 110

113 116 117 124 124 129

136 136 137 139 142 143 152 153 153 153 157 157

Table of Contents

4.6. 4.7. 4.8.

Normalization in Complex Coordinates ................................. Classification (exercise 2).......................................................... The Neighborhood of a Periodic Solution...............................

5

161 162 164

The Normal Form of a System of n Differential Equations .

166

§1 . The Normalizing Transformation..................................................... 1.1. Statement of the Problem.......................................................... 1.2. Linear Systems .......................................................................... 1.3. The Existence of the Normalizing Transformation (thm. 1) ... . 1.4. Transformations of the Normal Form (thm. 2; example 1)...... 1.5. Distinguished Transformations (thm. 3)................................... 1.6. Small Parameters (thm. 4; example 2)....................................... 1.7. On the Calculation of the Normal Form (thms. 5, 5'; example 3)......................................................... 178 1.8. Invariance (thm. 6) .................................................................... 1.9. Real Systems (thm. 7; lemma 1; example 4) ............................. 1.10. Linear Automorphisms (thm. 8; example 5)............................. §2 . The Integration and Classification of Normal Forms..................... 2.1. Power Transformations (thm. 1; lemma 1)............................... 2.2. Classification (thm. 2; lemma 2)................................................. 2.3. One-dimensional Normal Forms (thms. 3, 4)........................... 2.4. The Secondary Normalization (example 1)............................. § 3. Analytic Integral Sets........................................................................ 3.1. Statement of the Problem.......................................................... 3.2. Convergence of the Normalizing Transformation (thm. 1)...... 3.3. On Sets........................................................................................ 3.4. The Set (thm. 2; example 1).................................................. 3.5. The Set $ (thm. 3)...................................................................... 3.6. Properties of the Sets j/ and $................................................. 3.7. A Refinement of Theorem 2 (thm. 4; example 2)..................... 3.8. The Part of the Spectrum not on the Imaginary Axis.............. 3.9. The Neighborhood of a Torus................................................... §4 . The Normal Form and Methods of Averaging............................... 4.1. Local Coordinates...................................................................... 4.2. The Normal Form (thm. 1) ...................................................... 4.3. Calculating the Coefficients of the Normal Form (thm. 2; example 1) ............................................................... 213 4.4. A Hamiltonian System (example 2)........................................... 4.5. On Problems in Mechanics which Can Be Solved with the Aid of the Normal Form ..................................................... 215

166 166 167 168 171 174 176

Chapter III.

Chapter IV.

182 185 187 189 189 191 195 200 202 202 202 203 204 205 206 206 208 209 210 210 211

214

On the Newton Polyhedron.................................................

217

§1 . A System of Differential Equations...................................................

217

Table of Contents

6

§2

1.1. Introduction .............................................................................. 217 1.2. Simple Points and Elementary Singularities (thms. 1, 2).......... 218 1.3. The Generalized Normal Form (thm. 3)................................... 221 1.4. The Polyhedron and the Normal Cones (example 1)............. 221 1.5. A Non-elementary Singular Point............................................. 225 1.6. Basic Mathematical Problems................................................... 229 . Other Problems Using the Newton Polyhedron............................. 231 2.1. The Hadamard Open Polygon ................................................. 231 2.2. The Number of Solutions of a System of Algebraic Equations . 232 2.3. The Separation of Branches of an Algebraic Curve ............... 232 2.4. Linear Partial Differential Equations....................................... 234

Chapter V.

Applications of the Normal Form in Mechanics.................

235

On the Motion of a Gyroscope in a Cardan Suspension.................. 1.1. Reduction to a Normal Form (thm. 1)..................................... 1.2. Integration of the Normal Form (thm. 2)................................. 1.3. Stability with respect to xr and x2 (thm. 3)............................. 1.4. Calculating the First Terms in f(h) (thm. 4)............................. 1.5. Statement of the Problem of a Gyroscope in a Cardan Suspension............................................................... 242 1.6. The General Case |u0| < 1 (thm. 6)........................................... 1.7. The General Case |w0| = 1 (thm. 7)........................................... 1.8. The Case A + B° - C° = 0...................................................... 1.9. The Balanced Gyroscope.......................................................... 1.10. The Spherical Pendulum .......................................................... § 2. On the Oscillations of a Satellite in the Plane of an Elliptical Orbit.. 2.1. The Formulas of the Asymptotic Method............................... 2.2. The Problem of the Oscillations of a Satellite......................... 2.3. Oscillations in the Orbital System of Coordinates................. 2.4. Oscillations in the Absolute System of Coordinates...............

235 235 237 238 239

References.....................................................................................................

264

§ 1.

244 247 249 250 252 254 254 257 259 262

Basic Notation x = dx/dt X = (x1,...,x„) Y = (y1,...,y„) A = U1,...,AJ Q = (41,...,^) Scalars are denoted by small letters, vectors and matrices by capitals (only in Chapter V are capital letters used for scalars). Vectors are written as rows, but are considered to be columns in matrix multiplication. 0 A—the complex conjugate of A. W = xJ^X) = x^/qX0, Fq ~ (flQ’flQ’ • • • F = (f19...Jn)^YFQXQ. R"—the space of exponents R2—the conjugate (dual) space to R". Sets in these spaces are denoted by boldface Latin capitals (L, M,...) and by the letters F, A. L—a linear subspace of R” consisting of all real solutions Q of the equation = 0. O = {2: 2 > —the non-negative orthant (quadrant, octant,...) Z"—the integer lattice in R". D = D(/) = {Q: fQ / 0}—the support of the series f = ^fQXQ. A—the convex hull of the set D. r—the closure of A. dr—the boundary of F F/d)—a face of the polyhedron F. F—Newton open polygon, a part of the boundary dF.

8

Basic Notation

V}d)—the tangent cone to the face 7}(d). Ujd)—the normal cone to 7}(d). Ujd)(e)—a set in R", corresponding to the face 7}(d) and to the number e > 0. Ro or Cq—the basic space with coordinates X, Y or Z,... Capital script letters denote sets in R£ (or CJ) and classes of functions and power series: —a neighborhood of the point X = 0, —a part of corresponding to the face ZJ(d), —the class of power series, the supports of which lie in the set V/d). A tilde is placed below the capital Greek letters, Z, H, M, N, E to distinguish them from the Latin letters, Z, H, M, N, E, which are similar. All asterisks have the same meaning, regardless of type size.

Introduction 1. Classification of Equations In many cases, problems in mechanics, physics, and other natural sciences can be reduced to the solution of a system of equations 7J(x1,...,xM) = 0 ,

j=l,

(1)

where the fj are polynomials in the variables,

X=

(2)

or else are close to polynomials. Depending on the nature of the variables xz and coefficients a, different types of equations are obtained—types which are studied by different mathematical sciences. Generally speaking, the classification of these types takes the following form: I. The coefficients a are integers or rational numbers, as are the desired solutions xz. These are the problems of number theory. IL The coefficients a and solutions xz are real or complex numbers. These problems arise in algebra—specifically, in algebraic geometry and in the theory of analytic sets. III. The coefficients a are real or complex functions of some variable t, the unknowns are functions of this variable, xY = y^t),..., xt = yz(t) and derivatives of these functions, Xj = dsyt/dts{j > I). The problem consists of seeking those functions yjt),, yz(t) which satisfy system (1). Such problems are treated by the theory of ordinary differential equations. IV. The coefficients a are functions of several variables t15..., tk, the unknowns are functions of these variables, xr = yi(t19..., tk)9 ..., xz = yt(t 19... ,tk) and their derivatives: dsyi . .

■

;>u-s,+ ' + S1.

The problem of finding functions yz which satisfy (1) belongs to the theory of partial differential equations.

10

Introduction

Each type can be further divided into linear and non-linear problems. System (1) is called linear if each of the variables xf appears in the polynomials of (2) only in the first or the zeroth power; that is, if, in each term of the polynomial of (2), the qk are either zero or one and their sum is not more than one. If this is not the case, then system (1) is non-linear. Linear problems are normally much easier to solve than non-linear ones. Further, we shall distinguish between local and global problems. A local problem consists of investigating those solutions X of system (1) which lie in an arbitrarily small neighborhood of some known solution X°, for one or another definition of neighborhood. It is important that the size of the neighborhood not be essential; it is enough to find all solutions in some neighborhood of the solution X°. A global problem, on the other hand, consists of seeking all solutions to system (1) which belong to some fixed set—for example, for a problem of the second type finding all complex solutions X. Global problems are usually more difficult than local problems. To date some general approaches to non-linear problems have been developed, particularly in the local case. Non-linear problems are identified as being either quasi-linear or essentially non-linear. Quasi-linear problems are solved either by taking into account nonlinear “perturbations” (the classical approach) or through a change of coordinates which simplifies (1) and transforms it into a so-called “normal form” (this is the modern approach). In essentially non-linear problems, one must know how to choose a first approximation to system (1) (i.e., a truncation) that leads to linear inequalities in the exponents qr,..., qn of expression (2). These inequalities are solved by means of a geometrical construc­ tion (“Newton’s polyhedron”). By means of power transformations, non-linear in X but linear with respect to the and lnxf, one can succeed in simplifying the truncation. By this means, one can reduce an essentially non-linear problem to several quasi-linear problems. Remarkably, essentially non-linear problems are thus closely connected with certain linear geometry in the “space of the exponents.” 2. Fundamental Ideas of the Local Method We will explain these ideas by considering those solutions of a system of ordinary differential equations

dX/dt = 0(X)

(3)

which lie in a neighborhood W of the stationary solution X = (xn ..., x„) = 0. We consider in parallel the cases of real and complex X and Here, the function d>(x) is assumed to be analytic or sufficiently smooth in the neighborhood We make no further restrictions: we even consider cases of degeneracy and of resonance. We shall solve the problem posed above by constructing a special local coordinate transformation which will change (3) into an integrable system. Such

Introduction

11

a transformation exists for the entire neighborhood only in comparatively simple cases. In complicated cases, the neighborhood being investigated must be divided up into parts, each with its own local coordinates with respect to which the system is integrable. Generally speaking, the construction of these coordinate changes and decompositions is accomplished step by step, and not all at once. At each step we construct a finer decomposition, and in each part of the neighbor­ hood we introduce a coordinate system within which the system is simpler. Working under very general assumptions, we can arrive after a finite number of steps at a system which is integrable in each part of the neighborhood. In systems arising in mechanical, physical, and astronomical problems it is usually sufficient to go through one (occasionally two) such steps. The foundation of this local method rests on two basic ideas: 1. The normal form. If system (3) has a non-zero linear part, then we can transform it into a normal form, which easily reduces to a system of lesser order with a vanishing linear part, and is often immediately integrable. Various cases of such a normalizing transformation have been considered by Poincare, Picard, Horn, Dulac, Birkhoff, Cherry, Siegel, Moser, Sternberg, and others. A definitive form, a method of reduction of order and general properties of the normal form have been described by the author [Bruno 1964-1976]. 2. The reduction of complex singularities. If the expansion of the vector­ function 0(X) in powers of the contains no linear terms, then we may select a finite number of “truncated systems” dX/dt = 0/>(X)

(4)

and correspondingly divide the neighborhood °U into parts ^JJ)(e) such that each part ^Jd)(e) is a curvilinear cone touching the stationary point, while the truncation (4) is the first non-trivial approximation to system (3) in ^d)(e). This is effected by the construction of a certain polyhedron in the exponent space. Further, for a simple (in some sense) truncation (4), we can construct a normal form of system (3) in the part ^Jd)(g)—that is, we can simplify system (3) in that region. For a more complicated truncation (4) we must make a birational (power) transformation, blowing up the stationary point into a manifold JI and the region ^d)(e) into a neighborhood W of this manifold. It is then necessary to find the stationary points in Jt and to study their neighborhoods, which will be parts of the neighborhood W. In each of these parts the system will be in some sense simpler than the initial system, and we can employ anew our simplifying and reducing constructions. This method of reducing singularities is analogous to the “multiple sigma-process” of algebraic geometry and has its origin in a Newton polygon. For a two-dimensional system of differential equa­ tions (3), various truncated systems (4) have been selected and studied by Briot and Bouquet [Briot, Bouquet, 1856], Horn, Frommer, and others. This has been done for multi-dimensional systems by the author [Bruno, 1962, 1965]. The local method, naturally, has two parts. The first—the algebraic part— consists of finding an algorithm for constructing the required formal power

12

Introduction

series. The second part consists of interpreting these series by means of analytic functions (Poincare, Dulac, Siegel, Pliss, and the author [Bruno, 1971]) or smooth functions (Birkhoff, Sternberg, Hartmann, Grobman, and others), or in estimating the accuracy of approximate integration by means of these methods (Birkhoff, Siegel, Moser, and others). 3.

The Contents of this Monograph

The main objects of study of the present work are local non-linear problems of the second and third types, though their study sometimes reduces to that of linear problems of the first or fourth types. In addition, our attention is primarily directed towards analytic systems (1), where the fj are analytic functions of their arguments. Chapters I and II consider two-dimensional systems in great detail. Chapters III and IV treat multi-dimensional systems in slightly less detail. Chapter V applies the local method to concrete mechanical problems. § 1 of Chapter I is devoted to the solution of linear inequalities, and in connection with this discusses geometry in the plane Rf and the dual plane R2. The techniques developed there will be the basis of the “geometry of exponents”. § 2 of Chapter I presents methods of solving an analytic equation

/(X1,X2) = 0 in the neighborhood of the critical point xr = x2 = 0. § 3 applies the local method to the study of the level curve /(x1,x2) = c in the neighborhood of the critical point. This is the simplest problem with which we can demonstrate the whole apparatus of the local method. Chapter II treats the system of differential equations

dxjdt = i(x1,x2) ,

dx2/dt — (p2(xi,x2) ,

in the neighborhood of a singular point using the local method. Chapter III systematically presents the theory of the normal form of the system of differential equations (3) in a neighborhood of the elementary singular point X = 0. §4 of the chapter extends this theory to a neighborhood of an invariant manifold. § 1 of Chapter IV treats the application of the local method to the solution of system (3) in the neighborhood of a non-elementary singular point. §2 gives a review of recent investigations using the geometry of exponents (Newton’s polygon or polyhedron). § 1 of Chapter V considers the problem of the influence of nutational oscilla­ tions on the rate of precession of a heavy gyroscope in a Cardan suspension. This example illustrates the ideas of §§ 1, Chapters II and III. §2 of Chapter V

Introduction

13

treats the problem of the existence of periodic oscillations of a satellite in the plane of an elliptical orbit; this illustrates the methods of §§ 3 and 4 of Chapter III. The treatment of Chapters I, II, and V and §§ 1, 2 of Chapter III is detailed (especially Chapters I and II). The last half of Chapter III and all of Chapter IV, in contrast, are more review and problem oriented. Chapters I and II are based on lectures given by the author [Bruno, 1973a] and are written in the form of a text book. These chapters contain detailed proofs, many figures, examples, and exercises. They should be understandable to students who have completed two years of university study. Chapters III and IV are based on work by the author [Bruno, 1971, 1972a] and a series of his articles. The presentation is such that it should be possible to read each chapter independently. For a short outline of the local method in the author’s own translation see the author’s lectures [Bruno, 1987]. A recent development of the local method can be found in the following works by the author: On stability in a Hamiltonian system. Mat. Zametki, 40, No. 3, 385-392 (1986) [Russian]. Math. Notes, 40, No. 3, 726-730 (1986) [English] Bifurcations of periodic solutions in a symmetrical case of a multiple pair of imaginary eigen­ values. In: The Numerical Solution of Ordinary Differential Equations. Akad Nauk SSSR Inst. Prikl. Mat, 161-176, 239 (1988) [Russian] The normal form of a Hamiltonian system. Usp. Mat. Nauk, 43, No. 1, 23-56 (1988) [Russian]. Russ. Math. Surv, 43, No. 1 (1988) [English] The normalization of a Hamiltonian system near a cycle or a torus. Usp. Mat. Nauk, 44, No. 1 (1989) [Russian]. Russ. Math. Surv, 44, No. 1 (1989) [English] On small divisors. Banach Center Publications. Warsawa: PWN, v. 23 (1989) [English] On the question of stability in a Hamiltonian system. Ibid.

[Translator’s note: the author’s name is often spelled “Bryuno” in translations, but “Bruno” is the spelling used throughout this monograph.]

Chapter I Foundations of the Local Method

§1. Linear Inequalities in the Plane 1.1. Basic Definitions

By R2 we will denote a real two-dimensional vector space. That is, R2 is the real plane, with Cartesian coordinates qA and q2. The points of this plane are two-dimensional vectors Q = If Pi> P2 and c are fixed real numbers, then the solutions of the equation Q1P1 + Q2P2= c

(1)

lie along a straight line in R2 perpendicular to the vector P = (pi9p2). Now let us vary the magnitude of c and consider the lines defined by (1) for different values of c. These will be mutually parallel lines. If c = 0, the line will pass through the origin, Q = 0; if c > 0, the line will be displaced in the direction of P, and the displacement will increase with increasing c; finally, when c < 0, the line will be displaced in the opposite direction (figure 1). Each of these lines divides the plane into two half-planes: the positive half-plane, in which 4iPi + Q2P2 > c> and the negative, in which qxpv + q2p2 < c. Note that if we use the scalar product = qxpx -F q2P2, then equation (1) can be written in the form = c . From now on we will consider the vectors P and Q, the two factors in this scalar product, to be points in two distinct planes:

Q 6 R2 ,

P E R2 .

The reason for this separation will become clear as we proceed. Let zf be a point set in R2; it is convex if, along with any two of its points Qx and Q2, the set contains the entire line segment joining the two points. The set J is called the convex hull of a set D if J is the smallest convex set containing D. For example, the convex hull of two points and Q2 is the line segment which joins them; the convex hull of three points is the triangle with its vertices at those three points. This fact can be expressed by the formula

1.1. Basic Definitions

15

^ = {e:Q = = 7](d) n D ,

consisting of those points of D which lie on the face /}(d). We will call the face /](d) proper if it has the same dimension as the convex hull of Djd); otherwise, we call /j(d) improper.

Example 1. D = { Q3L where Qr = (1,0), Q2 = (0,1), and Q3 = (1,1) (figure 2). The support line for the vector Pr = ( — 1, — 1) is given by the equation = -l(since = = -l,= -2< -l).Thus,

Fig. 2

17

1.2. Cones

the supporting half-plane is defined by the inequality — qr — q2 < — 1. Similarly if P2 = (—1,0), the support line LPz is given by the equation qY = 0 and the supporting half-plane by the inequality — qY 0. Finally, the support line LP3 and the supporting half-plane for P3 = (1,1) are qr + q2 = 2 and + q2 < 2, respectively. The polygon Pis the triangle with vertices g15 Q2, Q3. Thus r^} = Qj , J = 1,2,3 ;

the edges /}(1) are the line segments connecting ingly,

D'0’ = Qj , {61,62} ,

with

Correspond­

j = 1, 2, 3 ; DI1’= {62,83} ,

D’1’ = {63,61}-

Here, all faces are proper. Example 2.

D is the interior of the unit circle:

D = {Q.ql + q2 < 1} .

Then J = D, and Pis the closed unit disk:

{Q: q2 + q22

1} •

The boundary dr of T is the unit circle qj 4- q2 = 1. Each of its points is a face /}(0). All D}0) are empty (the dimension is — 1); therefore all faces are improper. Note that an edge 7](1) will be either a line (thus unique), a half-line (of which there can be no more than two), or a line segment.

1.2. Cones

A set K c R2 is called a cone (see Goldman and Tucker, 1956) if, along with the point P, it contains the ray cP (for all c > 0). Thus, the cones in R2 are: the origin P = 0, any ray, any sector, and the whole plane. A cone K is convex if it is a convex set. Thus, a sector is a convex cone only if its interior angle is no larger than n. Let LP be the support line of a set D for a vector P; then for all c > 0, LcP = LP. That is, all vectors P' for which LP is the support line lie along the ray P = {cP,c>0}. Let us denote by U the set of all vectors P e R2 for which there exists a supporting half-plane L^-) for the set D. Theorem 2. The set U is a convex cone.

Proof. Let two vectors Pt and P2 lie in the set U. We will show that U contains every vector P = aiPi + a2P2, where a1? a2 > 0. The set T has two

18

Chapter I. Foundations of the Local Method

supporting half-planes, intersection:

and Lj>2); consequently, it is contained in their (3)

PcL^nL^ .

If Pi and P2 are not parallel, the support lines LP1 and Lp2 divide the plane Ri into four parts; one of these is LjP n L^P. This set has support lines for all vectors P lying between the rays {cPt} and {cP2}. By virtue of (3), Palso has support lines for all such P. If Px and P2 are parallel, on the other hand, then they lie either along the same ray or else along the same line. Either of these sets is convex. The theorem is proved. □

We will call the set U the normal cone of the set D (or of the set P). Let be a face of some closed convex set P The set Ujd) of all vectors P e R2, for which the support line LP intersects P precisely at the face /}(d) is called the normal cone of the face rfd}\ Ujd) = {P: rfd} = VPcyr} .

It is easily seen that Ujd) is a convex cone. The union of all the normal cones Ujd) of the faces /}(d) is the normal cone U of the whole set P. If /}(d) = LP n P and Q e LP, then f = for all 2 l < for all Q e r\Ifd> . ( }

Therefore, the vectors in the normal cone Ujd) are determined by the system of equations and inequalities (4). The first of these conditions implies that the line = contains the face the second implies that this line is the support line to the set P But then it must be the support line to the set D, so that the second condition of (4) can be rewritten as < f°r all 2 e D\D/>. If the boundary subset Djd) is non-empty, we can take Q e Djd). Finally, if /}(d) is a proper face, the first condition of (4) need only be applied to the points of Djd). Thus, for some point Q e Djd)

u(a)= L = J I ’ <

for all Q e D/> 1 for all Q e D\DJ‘,)J '

1 ’

Example 3 (continuation of example 1). D = {{2i> 2z> 2s}; 2i = (1,0), Q2 = (0,1), Q3 = (1,1)} (figure 2). Here, the cone U will be the entire plane R2. For the edge Px(1), we have (in agreement with (5)) the normal cone

U(1)=L. = J 1 I ' 0.

Example 7. D = {Q19Q29Q3,Q4}, where Q. = (3,0), Q2 = (0,3), Q3 = (1,1), g4 = (1,2) (figure 6). The points Q19 Q29 Q39 are the vertices of a triangle r. We let jT1(0) = 21? r2(0) = Q3, and T3(0) = Q2. Then 7~i(1) is the edge connecting Qr and g3, the edge r2(1) connects Q3 and Q2, and the edge F3(1) connects Q2 and Qr (figure 6). Further the sets Djd) are

Dl0) =

,

^1) = {Q1,Q3} ,

m°> = q3 ,

o30) = q2 ,

D^ = {e3,e2},

D^ =

We next find the normal cones

U?’ = {P:

{22,24.21} •

We have (figure 7):

= > } = {P: p2 = 2P1

< 0} ;

= {P: = > (rr.P)} = {P: 2p2 = P1 < 0} ;

UV’ = {P: = > } = {P: Pi = P2 > 0} ;

24

Chapter I. Foundations of the Local Method

u*, 0* = {P: < , < = {P: < W>\p\