Contributions to the Theory of Nonlinear Oscillations (AM-45), Volume V 9781400882649

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Annals o f Mathematics Studies Number 45

ANNALS OF MATHEMATICS STUDIES Edited by Robert C. Gunning, John C. Moore, and Marston Morse 1. Algebraic Theory of Numbers, by

H erm ann W e y l

3. Consistency of the Continuum Hypothesis, by 11.

Introduction to Nonlinear Mechanics, by N.

16. Transcendental Numbers, by

Ku r t G odel

and N.

Kr ylo ff

17. Probleme General de la Stabilite du Mouvement, by M. A. 19. Fourier Transforms, by

B o g o l iu b o f f

C a r l L u d w ig S ie g e l

S. B o c h n e r

and

L ia p o u n o f f

K. C h a n d r a se k h a r a n

20. Contributions to the Theory of Nonlinear Oscillations, Vol. I, edited by S. 21. Functional Operators, Vol. I, by

L efsc h et z

J o h n von N e u m a n n

22. Functional Operators, Vol. II, by

J o h n vo n N e u m a n n

24. Contributions to the Theory of Games, Vol. I, edited by H. W. 25. Contributions to Fourier Analysis, edited by A. d e r o n , and S. B o c h n e r

Zy g m u n d ,

W.

27. Isoperimetric Inequalities in Mathematical Physics, by G.

Kuhn

and A. W.

and G.

P o lya

28. Contributions to the Theory of Games, Vol. II, edited by H. W.

Kuhn

30. Contributions to the Theory of Riemann Surfaces, edited by L.

Ah lfo r s

E dw ard

J.

C.

E.

Sh a n n o n

and J.

L efsc h et z

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B ochner,

M cC arth y

36. Contributions to the Theory of Nonlinear Oscillations, Vol. Ill, edited by S. 37. Lectures on the Theory of Games, by

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et al.

33. Contributions to the Theory of Partial Differential Equations, edited by L. and F. J o h n 34. Automata Studies, edited by

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In press Kuhn

39. Contributions to the Theory of Games, Vol. Ill, edited by M.

and A. W.

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A. W.

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40. Contributions to the Theory of Games, Vol. IV, edited by R.

D uncan L u ce

and A.

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F u rsten berg

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CONTRIBUTIONS TO THE THEORY OF NONLINEAR OSCILLATIONS VOLUME

J. P. L aSALLE J. JARNIK J. KURZWEIL

V

L. MARKUS G. REEB P. MENDELSON

J. CRONIN

J. ANDRE

J. K. HALE

P. SEIBERT

L. CESARI A. STOKES

C. COLEMAN R. P.

de

FIGUEIREDO

EDITED BY L. Cesari, J. LaSalle, and S. Lefschetz

PRINCETON, N EW JER SEY PRINCETON UNIVERSITY PRESS

1960

Copyright © 1 9 6 0 , by Princeton University Press All Rights Reserved L. C. Card 6 0 - 1 2 2 2 5

Printed in the United States of America

PREFACE This volume is the fifth in a series of contribution to the Non­ linear Oscillations published by the Annals of Mathematics Studies. As usual there are quite a variety of topics covered. The first paper by LaSalle is on a central problem on optimal control — a problem which is agitating mathematicians both in the United States and in the Soviet Union. This is followed by a contribution from Harnlk and Kurzweil on a generalized differential equation depending on a parameter. This is a continuation of a series of papers published on the topic by Kurzweil. We have next an elegant paper by Jane Cronin on the perturbation theorem of Poincare. This is followed by two papers by Hale and one by Cesari. The first paper concerns a discussion of the critical points for a linear system with periodic coefficients. The second paper of Hale deals with the important question of the stability of periodic solutions of periodic and autonomous differential systems. Simple criteria for stability are proved by the use of a convergent process of successive approximations. The paper by Cesari gives existence theorems for periodic solutions of periodic and autonomous differential systems satisfying a Lipschitz con­ dition. These theorems are proved by application of fixed point theorems in functional spaces and from this approach an interpretation Is given of the method of successive approximations used In the paper by Hale. Stokes, in his paper, applies a fixed point theorem In linear spaces to the study of certain stability and boundedness properties of the solutions of differ­ ential equations. By the use of such methods he Is able to reduce the study of systems to that of related first order equations. The next paper by Marinis deals with a curious application of abstract algebra to the classification of linear differential systems. The paper by Reeb deals with a property of the totality of bounded solutions of certain dynamical systems. Mendelson!s contribution deals with the stable motions in a neighborhood of critical points in topological dynamics. It is connected with a topological method introduced and promoted by Wazewski. The joint paper by Andr6 and Seibert deals with piecewise continuous differential equations. It Is part of the noteworthy work done by those authors in v

organizing the earlier work of Mrs. Flugge-Lotz. Colemanrs paper is a study of asymptotic stability of a 3-dimensional system where the leading terms are a homogeneous function of degree m. In the final paper of this volume Figueiredo studies the self-sustained oscillations of a second order system. All but the papers by Jane Cronin, Jarnik and Kurzweil, Reeb, and Figueiredo were carried out at RIAS and supported in part by contract number AF i*9(638)-382. The work by J. Cronin and A. Stokes was supported in part by the Office of Naval Research. Editors L. Cesari J. P. LaSalle S. Lefschetz

vi

CONTENTS Page Preface By L. Cesari, J. I. II. III. IV.

V.

VI.

VII.

VIII.

IX.

X.

XI.

XII. XIII.

P. LaSalleand S. Lefschetz

v

The Time Optimal Control Problem By J. P. LaSalle Continuous Dependence on a Parameter By J. Jarnik and J. Kurzweil Poincare*s Perturbation Method and Topological Degree By Jane Cronin On the Behavior of the Solutions of Linear Periodic Differential Systems Near Resonance Points By Jack K. Hale On the Stability of Periodic Solutions of Weakly Nonlinear Periodic and Autonomous Differential Systems By Jack K. Hale

1 25 37

55

91

Existence Theorems for Periodic Solutions of Nonlinear Lipschitzian Differential Systems and Fixed Point Theorems By Lamberto Cesari 115 The Applications of a Fixed Point Theorem to a Variety of Non-Linear Stability Problems By Arnold Stokes

173

Quadratic Differential Equations and Non-Associative Algebras By Lawrence Markus

185

Sur Une Propriete De L*Ensemble Des Trajectoires Bornees De Certains Systernes Dynamiques By Georges Reeb

215

On Lagrange Stable Motions In the Neighborhood of Critical Points By Pinchas Mendelson

219

The Local Theory of Piecewise Continuous Differential Equations By J. Andre and P. Seibert

225

Asymptotic Stability in 3-Space By Courtney Coleman

257

Existence and Uniqueness of the Periodic Solution of an Equation for Autonomous Oscillations By Rui Pacheco de Figueiredo

269

Appendix By E. Pinney

2 85

CONTRIBUTIONS TO THE THEORY OF NONLINEAR OSCILLATIONS VOLUME V

I.

THE TIME OPTIMAL CONTROL PROBLEM J. P. La Salle §1.

INTRODUCTION

It has been an Intuitive assumption for some time that if a con­ trol system is being operated from a limited source of power and if one wishes to have the system change from one state to another in minimum time, then this can be done by at all times utilizing properly all of the power available. This hypothesis is called Mthe bang-bang principle.'1 Bushaw accepted this hypothesis and in 1 952 in [2 ] showed for some simple systems with one degree of freedom that of all bang-bang systems (that is, systems which at all times utilize maximum power) there is one that is optimal. In 1953 I made the observation in [7] that the best of all bang-bang systems, if it exists, is then the best of all systems operating from the same power source. More recently, more general results have been obtained by Bellman, Glicksberg, and Gross in [1 ], and later — but seemingly independently — by Krasovskii in [6] and Gamkrelidze in [k ]. We confine our attention to the time optimal problem for control systems which are linear in the sense that the elements being controlled are linear and as a function of time the steering of the system enters linearly. The differential equation for such systems is (1 )

x(t) = A(t)x(t) + B(t)u(t) + f(t)

where of the an (n tem of

x and f are n-dimensional vector functions (x(t) is the state system at time t), A is an ( n x n ) matrix function, and B is x r) matrix function. The vector equation (1 ) represents the sys­ differential equations

dx^(t)

n

r

dt

1

2

LA SALLE

Our ability to control the system lies in the freedom we have to choose the "steering" function u. We assume that the admissible steering func­ tions are piecewise continuous (or measurable) and have components less than 1 in absolute value (|u^(t)| < 1). The number 1 is selected for convenience. We could just as well assume - a^. < uk (t) < b^ where a^ and b^ are any positive numbers. Given an initial state xQ and a moving particle z(t), the problem of time optimal control is to hit the particle in minimum time. Let x(t, u) be the solution of (1 ) satisfying x(o) = xQ . An admissible steering function u* is optimal if x(t*, u*) = z(t*) for some t* > o and if x(t, u) / z(t) for o < t < t* and all admissible u. In [1 ] Bellman, Glicksberg, and Gross consider the system (2 )

x(t) = Ax(t) + Bu(t)

and restrict themselves to the problem of starting at xQ and reaching the origin in minimum time. The (n x n) matrix A is constant and its characteristic roots are assumed to have negative real parts. B was assumed to be a constant non-singular (n x n) matrix, and it is this restriction that is quite unrealistic. Gamkrelidze in [4] considered the same problem, removed the restriction that B be non-singular, and showed for systems which are later in this paper called "normal" the existence and uniqueness of an optimal steering function. The form of the optimal steering function is the same as that given in [1 ], and for normal systems one can conclude that the optimal steering is bang-bang (|u^.(t)| = 1 ). Krasovskii, in [6 ] considers the more general control system (1 ) and the more general control problem of hitting a moving par­ ticle. Using results of Krein on the L-problem in abstract spaces, he proves the existence of an optimal steering function for, what are called in this paper, "proper" control systems. We prove the same existence theorem without restricting ourselves to proper systems. Krasovskii makes the further assertion that the optimal steering function is unique and simple examples show that this is not true even for proper systems. To date, therefore, the most general bang-bang principle has been proved by Gamkrelidze for normal control systems of the form (2 ) and for the special problem of reaching the origin in minimum time. We shall show for the general control problem that if an admissible steering function can bring the system from one state to another in time t then there is a bang-bang steering function that can do the same thing in the same time. This extends my result in [7 ] and at the same time extends the bang-bang principle. This does not mean that all optimal steering functions are bang-bang. Even for proper control systems there are

TIME OPTIMAL CONTROL

3

examples where the objective can be reached in minimum time using a steering function which during part of the time has some zero components. As in the special case considered by Gamkrelidze, it is shown that normal systems have unique optimal steering functions and for such systems there is a true bang-bang principle: the only way to reach the objective in minimum time is by bang-bang steering. In Theorem 5 a result is established that should be of some practical importance in the synthesis problem, which is the problem of determining the optimal steering u* as a function of the state of the system. This result shows that for some control systems optimal steering can be determined by what amounts to running the system backwards. In Section k we discuss the con­ trollability properties of proper control systems. The examples have been given at the end of the paper In Section and they serve to Illustrate the ideas and the application of the theory. §2.

THE GENERAL PROBLEM

The problem, described in the Introduction, for the system (1 ) of reaching a moving particle in minimum time will be called the general problem. For the control system (1 ) the state x(t, u) of the system at time t is given*by

(3 )x ( t, u) = X(t)xQ + X(t)

J

t Y(t )u(t )dT + X(t) o

t

J X‘1(T)f(T)dT o

X(t) is the principal matrix solution of X(t) = A(t)X(t), and Y(t ) « X”1(t )B(t ). We want at some time t to have x(t, u) = z(t); that Is, to have (*0

w(t) =

J

t Y(T)u(OdT

,

o where £

W (t ) = X”1(t )Z (t ) - XQ -

J

X~1(T)f(OdT

.

o We assume throughout that A(t), B(t), f(t) and z(t) are continuous for 0 < t < 00. The proof of the following lemma was pointed out to me by L. Pukanszky. LEMMA 1. Let M be the set of all real-valued measurable functions a(t) on [0, 1] with

k

LA SALLE |cr(t )|1. Let M° be the subset of functions In M with |of(t )| ® 1. Let y(t) be any n-dimensional function in L^fo, 1] ). Define

1 K =

a(t)y(t)dt;

a e mJ

o and K° =

J

1

Qf°(t )y(t )dt;

a0 € M°j-

o Then PROOF.

K°

is closed and

K = K°.

For each measurable set n(E) =

J

E

in

y(t)dt

[0, 1 ] define

.

E Let R^ denote the range of this vector measure. Let cE (t) be the characteristic function of E, and let a°(t) = 2c-g(t) - 1. Then a°(t) € M°, and each a°(t) € M° can be so represented. Then clearly K° = 2R^ - f , where i y =J

By Llapounov'3 Theorem ([8 ], [5 ]) K° is closed and convex. Let z =

J

y(t)dt

.

is closed and convex, and therefore

of(t )y(t )dt

be any n-vector in K. What we wish to show is that z is a limit of vectors in K°, and hence is in K°. It "will then follow that K° = K. Let p(t) = i(a(t) + 1 ) and z = £ ( z + y). Note that o < p(t) £ 1 and

1 z =J Define

p(t)y(t)dt

.

TIME OPTIMAL CONTROL

5

where ej

= { t;

l

< p(t) < 1 \

m

-mJ

Then

15- “

UI

1 1 Iy(t)'dt =i f Iy(t)'dt L1I

lid

-p(t )) y(t)dt i*

j=i Ej

which shows that

1

j=i E j

z^ -- > z

as

m -- > ».

°

Letting

m Fj - [J % i=j

-

we obtain m

■: I / 7 z as

e R^; therefore m -- > «>. This completes the proof

For our purposes here we wish to restate this lemma in the following more general fashion. LEMMA 2. Let ft be the set of all r-dimensional vector functions u (t ) measurable on [0, t] with |u±(t )| < 1. Let ft0 be the subset of functions u °(t ) with |u ?(t )| = 1. Let Y( t ) be any (n x r) matrix function in L ([o, t]). Define

A(t) =

Y(T)u(T)dT;

u e ft j-

and A°(t) = { /

Y ( O u ° ( O d T;

u° e n° }

.

o Then

Y( t ),

A°(t)

PROOF. Let and define

Is closed and

A(t) = A°(t).

y 1(x), y2(0, ..., yr (T)

be the column vectors In

6

LA SALLE t

'J

A j (t ) = -j

y J'(T )Uj(t )dr;

u e £2 j-

o and

t AS (t) = { f

yJ’^ ) u j ^ ) d T;

u° € n° }

o Since

J

t

' Y(t)u(t)

r

dT= M

J

t ( T )Uj ( t

)dx

,

0

we see that A(t) = A1(t) + Ag(t) + ... + Ar (t)

and A°(t) = A°(t) + A°(t) + ... + A£(t)

.

By Lemma 1 = for> each J = •••>**, and hence A(t) = A°(t) Each A?(t) isbounded, and it follows from Lemma 1 that each iscompact. Therefore A°(t) is closed, and this completes the proof. It is convenient to point out also two elementary properties of convex sets that we will use later. LEMMA 3« Consider sets A(t) of En, t £ t*, with the following properties: a. Each A(t) is convex. b. Corresponding to each e > o there is a 5 > 0 such that d(p, A(t)) < e for each p e A(t*) and all t* - 8 < t < t*. [d(p, A(t)) is the distance of p from A(t).] Then, if q is in the interior of A(t*), there is a t1 < t* such that q is an interior point of A(t1 ). PROOF. Let q be in the interior of A(t*), and let N be a neighborhood of q of radius e > 0 inside A(t*). Suppose for each t < t* that q is not an interior point of A(t). Then for each t < t* there is a support plane Pt through q such that there are no points of A(t) on one side of Pt ([3])- Because of the neighborhood N about q that lies in A(t*) we see that for each t < t* there is a

TIME OPTIMAL CONTROL point p in A(t*) contradicts (b).

whose distance from

A(t)

7 is at least

b.

This

LEMMA Let M be a convex set in ER containing the origin with the property: given any number K and any non-zero vector t) in En there is a vector y in M such that (tj, y) > K. Then M = En . PROOF.

Suppose that

M

is not

En * Then there is some non­

zero 11 that is not in M, and hence an is a boundary point of M for some a > o. Then a|i has a support plane P ( [ 3 ]) • Let t] be a non-zero vector normal to P and directed toward the side of P that contains no points of M. Then (t), y) is bounded above for y in M — a contradiction. Therefore M = En Turning our attention back to the control problem, we see that the set ft in Lemma 2 is the set of admissible steering functions and ft0 is the set of bang-bang steering functions. The set A(t) is related by equation k to the set of states that can be reached in time t by the admissible steering functions, and A°(t) is similarly related to the set of states that can be reached in time t by the bang-bang steering functions. Lemma 2 then states that anything that can be done by an admissible steering function can also be done by a bang-bang function. Suppose that there is a steering function u e ft such that x(t^ u) = z(t1 ); the particle is then hit at time t ^ and

w(t 1 ) =

J

tl Y(t )u(t )dT

.

o The lemma then states that there will be a u° e ft0 with the property that x(t.,, u°) = zCt-j); bang-bang steering can accomplish the same thing in the same time. As a direct consequence of this we obtain the following pair of theorems. THEOREM 1 . If of all bang-bang steering functions there is an optimal one relative to ft0, then it is optimal (relative to ft). THEOREM 2 . If there Is an optimal steering function, then there is always a bang-bang steering function that is optimal.

8

LA SALLE

The first of these theorems extends the result in [7], and the second is a general bang-bang principle. Although we did not state the intuitive hypothesis this way, the feeling actually is that not only should there always be bang-bang steering that is optimal but no other type of steering should be optimal. If at some time all of the available power is not being used, then it should be possible to improve the per­ formance by using, properly, the additional power that is available. Per­ haps this is true in some more general sense, but under our restriction on the amplitude of each component of the steering function this is not true without placing further restrictions on the control system. We return to this question in the next section. We can now extend the results that have been obtained previously on the existence of and the form of optimal steering. The set A(t) is, as we have said, related by equation ( k ) to the set of all states of the system that can be reached in time t. The existence is a simple conse­ quence of the fact that A(t) is a closed set, and the form (5 ) below for optimal steering follows from the convexity of A(t). [For r-dimensional vectors a and b, a = sgn b means that a^ = sgn b^, i = 1, ..., r; sgn bj = 1 if bj > 0 , sgn b^ = -1 if b. < 0 ,and is considered to be undetermined if b . = 0 . In using this notation a is always a column vector and b is a row vector.] THEOREM 3* If for the general problem there is a steering function u in ft such that x(t, u) = z(t) for some t > 0 , then there is an optimal steering function in ft. Moreover, all optimal steering functions u* are of the form u*(t) = sgn [rjY(t)]

(5) where

tj

is some non-zero n-dimensional vector.

PROOF. Our assumption that x(t, u) = z(t) for some t > 0 and some u e ft Is equivalent to assuming that w(t) € A(t) for some t > 0. Let t* be the greatest lower bound of all positive t's with this property. Let t

o Then there is a non-increasing sequence t converging to t* and Uh € ft such that w(tn ) = y(tR, uR ) e A(tR ). Now for some number K

9

TIME OPTIMAL CONTROL it is easy to see that

Therefore w(t*) is the limit of points in A(t*), and since A(t*) is closed, w(t*) is in A(t*). This proves the existence of a steering function u* in ft such that w(t*) = y(t*, u*). By the definition of t*, u* is optimal. We now want to show that u* is of the form (5)- In order to do this we note first that w(t*) cannot be an interior point of A(t 1 ) for o < t.j < t*. Suppose this were true. Let N be a neighborhood of w(t*) contained in A(t 1 ). Then, since A(t1 ) C A(t) for all t 1 < t, N 1 is contained in A(t) for all t > t 1 • The trajectory w(t) is continuous and this implies w(t2 ) e A(tg ) for some t2 < t*. This contradicts the definition of t*, andtherefore, w(t*) cannot be an interior point of A(t1 ) for t 1 o t

(7 )

- xQ =

J

Y(t)u(t)dx

.

o For this special problem we can show that If there is a steeringfunction of theform (5 ) that brings the system from the initial state xQ to the origin in finite time then it is an optimal steering function. THEOREM 5 . If for some t > 0 and some n-vector t) there is a solution u = u of (7 ) of the form (5 )

u( t ) = sgn [t]Y(t ) ], and if tjY( t ) ^ o on an interval of positive length, then it is an optimal steering function for the special problem.

TIME OPTIMAL CONTROL

PROOF. satisfying

By Theorem

3 the existence of a u and a t > 0

J

t

- x 0 = y (t, u) =

Y(t)u(t)dt

o implies the existence of an optimal u* then y(t*, u*) = - x , and

and a minimal time

t* < t.

But

t* (t), y ( t ,

u)

- y(t*,

u*))

= J

T]Y(t) [ u ( t ) - u * ( t ) ]dt

o t +

J

TiY(t)u(t)dt

=

0

t*

Since

u

is of the form (5), we can conclude that T]Y(t) =

0

on

[t*, t]

and that u?(t) = ^(t) on [0, t*] wherever the jth T)Y(t) 4 0. Therefore, by the assumption that TjY(t)^ of positive length, it follows that t* = t. Hence u steering function. This raises in a quite natural way "proper" control system which is discussed in the next

component of 0 on an interval is an optimal the concept of a section.

Although the above result is quite elementary, it is of con­ siderable practical significance and does offer a means of solving the synthesis problem. It has been mathematically convenient up to now to treat the steering function u as though it were a function of time. What one actually wants, is to know the optimal steering function u* as a function of the state x of the system, and this Is the "synthesis problem". If the control system is autonomous (equation 2), then we can replace t by - t in equation 2, use a steering function of the form (5), start at the origin, and look at the solution. Theorem 5 states that the particular steering function used is optimal for all states of the system through which the solution passes. The time at which the solution passes through a state of the system is then the minimal time for the system to go from that state to the origin. For normal sys­ tems the optimal steering is unique, and this procedure of reversing the system determines the optimal steering as a function of the state of the system. It is even true for some systems which are not normal that the synthesis problem can be solved in this way. This procedure leads to

LA SALLE

12

the determination of the switching surfaces, which are surfaces where certain of the components change sign. This method is illustrated in Section 5, and the switching surfaces (curves in the examples considered) are easily obtained by quite elementary reasoning. In Example 3 of Section 5 the synthesis problem is solved for a system that is not normal. §1+. PROPER CONTROL SYSTEMS AND CONTROLLABILITY In this section we introduce the concept of a proper control system and establish some controllability properties of such systems. We say that a control system is proper if ^Y(t) = o on an interval of positive length implies t) = o. This is equivalent to saying that the row vectors y.,(t), y2(t), • Yn (t) of Y(t) are linearly independent functions on each interval of positive length. It is clear that every normal system is proper. If the steering function has only one component (r = 1), then the concepts of proper and normal are equivalent. How­ ever, it is not in general true that every proper system is normal, as is shown by Example 3 in Section 5 • It is a direct consequence of Theo­ rem 3 that in proper control systems optimal steering u* has the property that at any given time some component of u* assumes an extreme value. Think now of removing all of the constraints on the admissible control functions, and consider any two states x1 and x2 and any two times t1 and t2. If for each pair of states and pair of times there is a steering function such that starting at x1 at time t1 * the system is brought to the state x2 at time t2, then the system is said to be completely controllable. THEOREM 6. Proper control systems are completely controllable. PROOF. We may certainly assume that t1 = 0 and t2 > 0. From (k) we see that complete controllability is equivalent to the property that for each t > 0 the set M^_ of all vectors t I'

Y(t)u(t)c1t

,

o

using all possible steering functions Let v (t ) = t)Y(t ), and define

u,

is the whole phase space En«

TIME OPTIMAL CONTROL

13

t y - /

« - > sgn v(t )dT

o

Then r (n> y) =

t

Y, I

lvj(x)ldT

•

3-i ° Since the control system is proper, .l

i

|v .(T)|dT > o

if t) 4 0 - Therefore corresponding to each direction r\ there is a vector y in M^ such that ( t ), y) > o. M^ is obviously a linear manifold, and therefore M^ = En[Itis of interest to note that the theorem is also a consequence of the fact that for proper control sys­ tems the linear transformation F onEn defined by

F ( T| ) =

J

t Y (t)Y ' (Ori'dT

o is non-singular for each t > o. The prime denotes the transpose. Thus the system is completely controllable with steering functions restricted to those of the form u ( t ) = Y'COr)1.] Complete controllability assumes that there is no restriction on the admissible control functions, and if unlimited power is available to a proper control system the above theorem says that it is always possible to move the system from any one state to any other state as quickly as we please. For each r-vector v define ||v|| = |v1 | + |v2 | + ... + |vp |. If we replace the concept of being proper by the requirement that 00

J It T|Y( t ) ||d t > 0 o for each r\ 4 °, then it is easily seen that with no constraints on the steering functions there is a time T > 0 such that the system can be started at any state xQ at time t = 0 and can be steered to any other state in time T. This fact does not appear to be of any particular importance, and for autonomous systems this condition is equivalent to

LA SALLE

the condition that the system be proper. We wish now to return to the assumption that the system has limited power and to the special problem. We shall say that a control system is asymptotically proper if 00

IIT)Y( T )||dT = 00

J

O

for each ™ 4 o. For the special problem we are interested in systems with the property that given any initial state x Q there is a steering function in ft that brings the system to the origin in finite time. We shall say that such a system is controllable. Of course, by Theorem 3 we know that if a system is controllable then for each initial state x Q there is a steering function in ft that is optimal for the special problem. THEOREM 7 « Asymptotically proper control systems of the form (6) are controllable. PROOF. For systems of the form (6) we know from (*0 that the system is controllable if corresponding to each xQ in En there is a u in ft with the property that t - xQ

= y(t,

u)

= j'

Y ( t )u(t )dT

o for some t > 0. Let A = {y(t, u); u e ft, t > 0}. Clearly A convex set, and what we wish to show is that A = Er.

is a

Taking u (t ) = sgn [t]Y(t )], we have t (n>

u ))

=f

)lldi:

o

Since the system is asymptotically proper, we know that (t], y(t, u)) — > as t -- > oo for each r\ 4 °. By Lemma k, A = En and the system is controllable. We now examine some special properties of autonomous control systems (equation 2). Let cp(\) = det(A + \ I ) = A,n + c ^ 11”1 + ... + cn; cp(- X) Is the characteristic polynomial of A, and we know that cp(— A) = o. Hence v(t) = T]Y(t) = Tie“AtB satisfies the nth order linear differential equation with constant coefficients

TIME OPTIMAL CONTROL

15

v ^ ( t ) + c ^ 11 1^(t) + ... + Cnv(t) = o , where v^k ^(t) is the kth derivative. Thus v(t) = 0 on an interval of positive length is equivalent to v(o) = t^B = o, v'(o) = - t]AB = 0, ..., v(n_1 ^(o) = (- 1 )n”1T]An“1B =0. It then follows that an autonomous control system is proper if and only if one of the following; hold (M = B(Er ) is the range of B and b 1, ..., bI* are the column vectors of B) k,

1 0 < k < n - 1, 2.

M +

3 . The ...,

• For each non-zero vector such that i-jA^B / 0.

i] in Er

there is an integer

A(M) + ... + A^n"1^(M) = Er. set of vectors b1, ...> br, Ab1, ..., Abp,..., 1^b1*contains a set of n linearly independent

a

vectors. The requirement that a system be normal is, in general, much stronger. The system is normal if and only if no component of v(t) = r]Y(t) is identically zero on an interval of positive length. Therefore an autonomous system is normal if and only if for each integer j, o < j < r , the vectorsb'V Ab^,..., A^ W are linearly independent. Let us now assume that an autonomous control system is both proper and stable [the characteristic roots of A have non-positive real parts]. For r\ 4 0 we know that one component of v(t) = T]Y(t) = T)e"^B is not identically zero. We may suppose that v^t) 4 °* Then r-i ^(t) =

)

a .t P .(t)e J cos(Pjt + 8 j)

,

J= 1

a. > 0 and J say t > a,

P*(t) J

are polynomials.

Hence for

t

sufficiently large,

|v1(t)| > |p(t) + q(t)| , where p(t) is almost periodic, is not identically zero, and It then follows that T lim - f T -> oo T J

TA |q(t)| < -

T |p (t ) + q(t)|2dt = lim - f T-> oo T J

|p(t)|2dt = 2c > 0

16

LA SALLE

Consequently, for

J

T | v1(t)

a.

|dt

>

T

J

sufficiently large,

T

i |p(t)

q(t)|dt

+

>

T- 2

a,

J

T |p(t)

+

q(t)|2dt

>

cT2

a

T h erefo re 00

J

|V -j(t )Idt =

00

,

o and the system is asymptotically proper.

What we have shown is that:

If an autonomous system is both stable and proper, then it is asymptotically proper and therefore controllable. Let us now see what can be said about the existence of a solution and the determination of optimal steering for the general problem of starting at xQ at time t = o and hitting a moving particle z(t). The control system is (1 )

x(t) = A (t )x(t ) + B(t)u(t) + f(t)

We wish to know whether or not there is a that [equation (*0 ] w(t1) =

J

u e n

and a

t1 > o

such

fci Y(t)u(x )d

o where *1

w(t 1 ) = X- 1 (t1 )z(t1 ) - xQ - J o

X"1(t )f(t )dT

;

that is, whether or not w(t 1 ) e A(t1 ) for some t 1 > o. Think for the moment that t 1 is fixed and ask whether or not, for some t > o, w(t 1 ) € A(t). This is then the special problem for the control system (6 )

x(t) = A(t)x(t) + B(t)u(t)

of starting at - w(t1) and reaching the origin in finite time. Suppose that the synthesis problem for this special problem can be solved. Then

TIME OPTIMAL CONTROL for each point - w (t1 ) of the curve - w(t), we would know the optimal steering function u and the minimum time t 1 for starting at - w(t1 ) and reaching the origin. If t 1 < then w(t 1 ) e A(t 1 ) C A(t1 ) , and we would know by Theorem 3 that the general problem has an optimal steering function u* and a minimum time t*; we would know also that t* < t 1. Thus, we have the following relation between the general prob­ lem and the special problem: The general problem x(o, u) = x , x(t, u) = z(t) has a solution if and only if in the special prob­ lem it is possible, starting at some point - w(t1) of the curve t - w (t ) = - X” 1 (t)z(t) + xQ + X” 1 (t )f (t )dt , o

J

to reach the origin in time

t2 < t1.

For proper control systems we can say a bit more. Supposethat the general problem has a solution. Let u* be the optimal steering function and let t* be the minimum time. Let u* be the optimal steering for the special problem of starting at - w(t*) and reaching the origin. Let t* be the minimum time. Then t* < t*. Now, using thesteering function u* during the time interval [0 , t*] and coasting (u(t) = 0 ) during the interval [t*, t*], we hit the particle at time t*. But we know for proper systems that optimal steering is neveridentically zero on an interval of positive length, andtherefore it must be that t* = t*. Thus, we have shown that For proper systems, if u* is optimal steering for the general problem and t* is the minimum time, then u* is optimal for the special prob­ lem of starting at - w(t*) and reaching the origin; t* is the minimum time for doing this. Thus, If the system is proper and if the synthesis problem for the special problem can be solved, one can move along the trajectory - w(t), t > 0 , use optimal steering from each point, and locate the first point where one can go from - w(t) to the origin in time t. The steering function that does this would then be optimal steering for the

18

LA SALLE

general problem. Let m(x) be the minimum time to go from the initial state x to the origin. What we wish to find is the smallest t > 0 satisfying m(- w(t)) = t.

§5 • EXAMPLES The purpose of these examples is to illustrate the concepts that have been introduced and to indicate how the theory can be applied to solve the synthesis problem. Although similar examples have been given before, the more complete theory simplifies the reasoning and the computation leading to a solution. We had pointed out previously that the restriction |ujJ < l was a matter of convenience and that this can be replaced by the more general condition - a^ < u^ < b^, a^ > o, b^ > o. This is illustrated in Example 1. Example 2 was solved in [1 ], and we consider the same example to show how a more complete theory leads to a simpler solution. In Example 3 a system which is proper but not normal is con­ sidered. The optimal steering is not unique, not necessarily bang-bang, but yet the synthesis problem, which is to define optimal steering uniquely as a function of the state of the system, can be solved. EXAMPLE 1. The differential equation of a controlled, undamped harmonic oscillator is x + x = u. We consider the constraint system is

-a0.

x =y y = -x +u

A - (_ J ;), B - (j), X(t) - ( l ™ I -

T)Y(t) = t^ cos

.

1 }

t + T)2sin

An equivalent

Y(t) = X-1(t)B =

i)

,

t = A cos(t + 8)

The system is proper, and since r = 1, is also normal. It is stable and is therefore controllable. We know therefore that for each initial point (xQ, yQ ) there is a unique optimal control function of the form ,.s „ rA ^ f u(t) = sgn* [A cos(t +6)] =

b, if A cos(t + 8) > 0 if A 003(t + 5) < 0

TIME OPTIMAL CONTROL which brings the system to the origin in minimum time. Conversely, we know that any steering function of this form which brings the system to the origin is optimal. We replace t by - t , obtain

& = Q t

X

-

u

,

and consider steering functions of the form u(t) = sgn*[A cos (6 - t)]. When u = c the trajectory in the phase plane is a circle with center at (c, o) and as t increases (t decreases) the direction on the trajectory is counterclockwise. The arrows in the figures are in the direction of increasing t. Thus, the trajectory can leave the origin, as indicated in Figure 1 , going counterclockwise on either the circle with center at (b, o) or the circle with center at (-a, o). De­ pending on the choice of t], the steering function can change sign at

FIGURE 1

20

LA SALLE

any time before a half-revolution and cannot go past a half-revolution without changing sign. The switching curves are therefore the chain of semicircles shown in Figure 1. Above these semicircles u = - a and below them u = b. Thus starting in the first quadrant, as illustrated, optimal steering opposes the motion and does not change sign until past the point of maximum displacement except at those exceptional points where the semicircles join. Because of the simple nature of the tra­ jectories we know that there is a unique optimal trajectory through each point of the phase plane. This is also a consequence of the fact that these are the only possible switching curves and that the system is both controllable and normal. EXAMPLE 2. This example was solved in [1] by reasoning, which we can now see, was more complex than is necessary. In Its normalized form the system is " 2y1 + u1 + u2 y2

= - y2

ul

+

+

2u 2 ,

|u,|


j=k

jVi

i=k

i=k

Consequently

£i * ( "^i ' sin 21+11 i=k

i=k

- 2C i=k Let us choose

i=k in such a manner that

i=k

DEPENDENCE ON A PARAMETER

i=k with

k -- > oo

t 4 0

(cf. (13))«

Then

|xk (t) |-- > oo with

k -- > oo for

(cf. ( 1 2 )). REFERENCES

[1 ]

GICHVLAN, I. I.,Concerning a theorem of N. N. Bogoljubov, Ukrain. Math. Journal, k , 2 , 215-219 (in Russian).

[2 ] KRASNOSELSKIJ, M. A. and KREJN, S. G., Averaging principle in non­ linear mechanics, Uspechi mat. nauk., 1 o, 3 (1 9 5 5 ), 147—152 (in Russian). [3] KURZWEIL, J. and VOREL, Z., Continuous dependence on a parameter of solutions of differential equations, Czechoslovak math, journal, 7, (8 2 ), k (1957), 5 6 8 - 5 8 3 fin Russian). [4]

KURZWEIL, J., Generalized ordinary differential equations and con­ tinuous dependence on a parameter, Czech, math, journal, 7 (8 2 ), 1957, 3, 418-4^9.

Ill-

POINCARE'S PERTURBATION METHOD AND TOPOLOGICAL DEGREE

Jane Cronin §1.

INTRODUCTION

We study the periodic solutions of a system of differential equations

( 1. 1 )

§£ = F(x, t, n) Qu

where x, F are n-vectors, appropriate differentiability and periodicity conditions are imposed on the components of F, and \i is a real parameter. Poincare’s perturbation method is used and the results extended by using topological degree. The technique developed gives a general approach to the existence problem for the degenerate case (the case for which the variational equation has periodic solutions) and new existence theorems are obtained if the degree of degeneracy exceeds one. (In prob­ lems of mechanical or electrical oscillations, the degree of degeneracy is frequently greater than one.) The technique consists in using topo­ logical degree to make an !in the large1 study of the bifurcation system (Verzweigungsgleichungen) without imposing any local uniqueness conditions on the solutions of the bifurcation system. The results obtained are generalizations and refinements of the results in a previous paper [7], (Numbers in brackets refer to the bibliography at the end of this paper.) although the present paper is independent-of [7 ]- Existence theorems are obtained for a wider class of equations; autonomous systems are studied; and it is shown that for the totally degenerate case (degree of degeneracy equals dimension of the system), the topological degree is a kind of lower bound for the number of distinct periodic solutions. The literature on this subject is extensive. General treatments are described briefly in [7 ]« Bass [2] has given a large bibliography including not only general treatments but papers on applications. The work of Friedrichs [8, 9, 10], Coddington and Levinson [3 , b] and 37

38

CRONIN

Lefschetz [11, 12] are most closely related to our approach. We base our treatment on the results of Coddington and Levinson concerning properties of the bifurcation system. As in Lefschetz's work, we need not impose local uniqueness conditions on the solutions of the bifurcation system, and our results may be regarded as an extension of the criterion for the existence of real periodic solutions given by Lefschetz. In Section 2, we describe how the bifurcation system is set up. Periodic solutions of the original differential equation correspond to solutions of the bifurcation system. We study solutions of the bifurcation system by determining the topological degree of the mapping defined by the bifurcation system. By using the properties of the bifurcation system obtained by Coddington and Levinson [3, b ] , we show in Section 2 that the problem of determining the topological degree can be reduced to that of determining the topological degree of a mapping in Euclidean q-space where q is the degree of degeneracy. In Section 3, we apply the method to obtain some existence theo­ rems for periodic solutions of non-autonomous differential equations. In particular we show that if the function F(x, t, n) in (1.1 ) is such that F(x, t, n) = Ax + nf(x, t, n) where A is a constant matrix and f(x, t, ii) = f 1 (x, (i) + f2 (t, n ) where the components of f 1(x, o) be­ have like polynomials for large x, then, except for cases for which the topological degree is not defined, equation (1.1) has, for sufficiently small n, at least one periodic solution near the initial solution. In Section k , the exceptional case in which the topological degree is not defined is discussed. In Section 5, the case In which the topological degree is even is briefly considered. In Section 6, we prove that the topological degree is, for the totally degenerate case, a kind of lower bound for the number of distinct periodic solutions. Finally In Section 7> application of topological degree to an autonomous differ­ ential equation is described. I am indebted to Professor S. Lefschetz for a number of ex­ tremely helpful discussions of this material. Most of the work on this paper was done under the sponsorship of the Office of Naval Research contract Nonr-l 8 5 8 (01+). §2.

THE BIFURCATION SYSTEM

In [8], Friedrichs treated the equation

( 2.1 )

x = F(x, t, ii)

(• = d/dt)

POINCARE!S PERTURBATION METHOD

39

where x, F are n-vectors; the components of F have continuous deriva­ tives with respect to x, t, and n; these derivatives have continuous derivatives with respect to x; F(x, t, 1 1 ) is periodic in t with period T(|i) where d2T/d|i2 is continuous; and (2 . 1 ) has for n = 0 a solution xQ (t) of period T(o). The problem is to determine if for sufficiently small j1 , equation (2 .1 ) has a solution x(t, n) periodic of period T(n). The classical result of Poincare ([13], Vol. 1, Chapter IV) states that if the variational equation x = A(t)x, where

x=xQ (t ),n=o has no non-zero solutions of period T(o), then for sufficiently small n, equation (2 . 1 ) has a unique solution x(t, n), periodic of period T(n), such that lim

x(t, n) = x (t)

•

-* o Here we treat the degenerate case in which the variational equation has q linearly independent solutions, periodic of period T(o), where 1 < q < n. The number q is called the degree of degeneracy of the problem. Friedrichs shows that if

then the problem can be reduced to the study of the equation (2 .2 )

x = A(t)x + nf(x, t, n)

and then derives a bifurcation system for (2 .2 ).

Coddington and Levinson

([3], [4]), by imposing the further condition that A(t) is a constant matrix and then putting this constant matrix in a canonical form, derive a more explicit form for the bifurcation system. (Coddington and Levinson also assume that T(n) is a constant function. This is not a restriction In the generality because Friedrichs shows in [8] that the general case may be reduced to the case T(n) a constant.) Now we describe the bifurcation system derived by Coddington and Levinson. The system is obtained by using the variation of constants formula to describe the solution and then imposing the condition that the solution be periodic of period 2it (i.e., assume T(o) = 2 it. ) The system

CRONIN

l+o

is: (2.3)

(e2nA - E)c + n

J

2n

{e(2 n_s)Af[x(s, n, c), s, n])ds

(this is equation (1.12), p. 23, of [3 ] or equation (3*20), p. 360 of [1+]) where E is the identity matrix, c is an n-vector such that x(t, 11, c) is a solution of (2.2) with initial value c, i.e., x(o, 11, c) c, and x(t, ii, c) has period 2it. Thus (2.3) is a system of n equations in of the n unknowns which are the components V The prob­ lem of finding periodic solutions for (2.2) is reduced to that of solving (2.3) for c1, . . cR. In order to study the properties of (2.3), Coddington and Levinson assumed that matrix A already has the following real canonical form

A =

where the elements not shown are zeros. Each J= = 1 (a. even) rows and columns of the form matrix of

sj

E2

Sj

Aj E2

where all elements are zero except

S.

Sj

and Eg, and

k,

is a

POINCARE!S PERTURBATION METHOD

where N. is a positive integer. A matrix A* may have only two rows and columns in which case A- = S.. Each matrix B. has p . rows and cl J J J columns, j = 1, ..., m, and is of the form

/

0

1

0

....

o

0

\

'

0 *. ••. . .

\

0

.

. ...

0

/ o

o

\ \ \

where Bjmay have only one row and column in which case B . consists of the single element o. The matrix C has (n - zj=1Pj) rows and columns and has no characteristic roots of the form IN for any integer N including N = 0. Matrix C need not be in canonical form. If (c^, ..., c^, ..., c ) is an n-vector, the indices i corre sponding to the last two rows of any A. or to the last row of any B. are called exceptional indices. They are the indices with the following form: i = a1 + a2 + ... + (aj

- 1 )

i = a1 + c*2 + *** + aj where

j = 1, ..., k and 1 = 0^ + ... + 05^. +

+ ... + Pj

where j = 1, ..., m. The indices i corresponding to the first two rows of any A* or to the first row of any are called singular indices. They are the indices with the following form: i = 1, 2, a1 +

1,

+2,

a1 + a2 +... + c l^

+

...

+

+

|3 ^ +

1,

...,

a1 + «2 +•••+ o:k_1

+2, a1 + ... + ..*,

c l^

+

...

+

There are (2k + m) exceptional indices and (2k + The number q = 2k + m is the degree of degeneracy Throughout our study, we assume that q > 0, i.e., least one A- or one B- in the canonical form of J

J

+1 +

Pi

+

+1,

, ••• +

^tn-1

+

m) singular indices. of the problem. that there is at matrix A.

Now let (ej, ..«, be the (n-q)-vector whose components are the components (in the same order) of (c^ cn ) which have

CRONIN non-exceptional indices and let (c", c”) be the q-vector whose components are the components of (c^ cn ) which have exceptional indices. Let subscript j denote a singular index and j1 a non­ singular index. It is shown in [3] and [ k ] that system (2 .3 ) can be re­ placed by the system

2* (2 . k )

N(c1 ',Cn-q)+11

o

(j! )

2ic c ) } Sf ^ ]dsl

e (2«-s)Af.[X (3> ^ o

= 0 J (j)

where N is a non-singular (n-q) x (n-q) matrix acting on vector (c!j> •••> c^_q ) and

J

|

2n e^2""3 ^[x(s, n, c), s, n]dsj-

o

(j' )

denotes the vector composed of the

(n-q)

components of

2k

e ( 2 *~s ^Af [ x ( s ,

u, c), 3, n]dsj-

o which have non-singular indices.

Similarly

2Jt | J' o denotes

e^2"-3^ [x(s, ji, c), s, nldsj(j )

the vectorcomposed of the

q

components which have singular indices.

The left side of system (2.4) describes a continuous mapping (call it e M ) of real Euclidean n-space into itself. Let c" denote the vector c in which the ( n - q ) components with non-exceptional indices have been set equal to zero. Then f

r

e (a«-s)Af[esA0M>

0]dgl

Lo defines by c M Q*

a

(j )

continuous mapping of real q-space into call the system {

f

e (2rt-3)AfresAc", s, o]ds}

- o

itself which

we denote

{ f

e^2""3 ^

POINCARE1S PERTRUBATION METHOD

43

the reduced bifurcation system. Friedrichs [8] and Coddington and Levinson, [3 ] and [4 ] assume the existence of a solution cM of the reduced bifurcation system such that the Jacobian of the reduced bifurcation system is non-zero at c". From this, the existence of a solution of (2.3) and hence of aperiodic solution of (2.2) is obtained. We use instead the notion of topological degree. LEMMA 2.1. For given r, there is an s > 0 such that if |n| < e, the topological degree of oM at the origin and relative to a solid (n-1 )-sphere with radius r and center at the origin is equal (except possibly for sign) to the topological degree of c4i 0 at the origin (in q-space) and relative to a solid (q—1)-sphere with radius r and center at the origin. PROOF. The proof follows from the definition of topological degree and the invariance under homotopy of the topological degree. (see [1, Deformationssatz, p. 4 2 4 ].) Lemma 2.1 shows that in order to demonstrate the existence of periodic solutions x(t, n, c) of (2.2) it is sufficient to showthat the topological degree of cM 0 is non-zero. For then the degree of cM is non-zero; hence for given small n, system (2.3) has at least one solution c0 = (c°, ..., c°).This implies that (2.2) has a solution x(t, \x, cQ ) of period 2 n suchthat x(o, \1, c Q) = (c°, ..., c°). From the funda­ mental properties of topological degree, the initial values (c.,, ..., cn ) are continuous in \i in the following sense: If for given nQ, (c°, ..., c°) is an isolated solution of (2.3) which has non-zero topo­ logical index, then given e > 0, there is a 5 > 0 such that if |n1 - n | < 5 , there is at least one periodic solution x(t, 11, c1) with initial value (c-/, ...,) such that |c^ - c?| < e for i = 1, ..., n; if (c°, ..., c° ) is not an isolated solution of (2.3), i.e., (c°, ..., c°) is a limit point of a set S of solutions t(c*, c*)} of (2.3) for |i = n , then suppose that the topological degree of an open set N containing S but no other solutions of (2.3) is different from zero. Then given e > 0, there is a 5 > 0 such that if 111 - |iQ|< 5 , then there is at least one solution x(t, n, c) with initial value (c^ ..., cn ) such that for some (c^, ..., c^) e S, we have |cV - c^| < e for 1 = 1 , ..., n. From the usual existence theo­ rems for differential equations, it follows that the solutions x(t, n, c) are themselves continuous in \± in the same sense.

kk

CRONIN §3-

EXISTENCE THEOREMS FOR THE NON-AUTONOMOUS CASE

We obtain existence theorems for periodic solutions of the equa­ tion (3.1)

x = Ax + |if(x, t, ii )

where A is a constant matrix; \± is a real parameter; and f, df/dx^ are continuous in (x, t, ii) for small 11 1 1 and all (x, t); and f has period 2 k in t. The periodic solutions of the variational equation x = Ax correspond to the characteristic roots of A that are of the form IN where N is an integer. ASSUMPTION 1 . Suppose that A has just one pair of character­ istic roots of the form iN, - iN where we assume the integer N is non­ zero. Then A may be put in the standard form ( 0

-N 0

N

A =

\

0

0

Ao

where AQ is not necessarily canonical but is real and has no character­ istic roots of the form IN. We assume that A is in this standard form. ASSUMPTION 2. The components f(x, t, ii) have the form

f 1(x, t, m.) and

f2 (x, t , \i)

(i = 1 ,

t, n) = Hj_(x > •••> xn> n) + K._(t,

of

2)

where ..., xn are the components of x, and H^, are functions of the indicated variables which have continuous second derivatives in these variables. The functions K 1(t, 11 ) and K2 (t, \i) have period 2 k in t. Also there exist polynomials P ^ x ^ x2 ) and P2 (x.j, x2 ) such that H, (x , X ,0, ...,0,0) lim — — ■ — ---------- = 1 r-* 00 P1 (x1,x2 ) where

2

r

2

(i = 1 , 2, )

2

= x 1 + xg .

Two real independent periodic solutions of the variational equation are: x^1 ^(t) = (cos Nt, sin Nt, 0, ..., 0 ) x^2 ^(t) = (- sin Nt, cos Nt, 0,

..., 0 )

POINCARE’S PERTURBATION METHOD

45

Following Coddington and Levinson, we may write the reduced bifurcation system as: 2 it

(3-2 )

J'

|^cos(Ns)H1

- sin(Ns)H2(c1x^1^ + c2x^2\

0 )| ds

2*

J

|cos(Ns)K1 (-s, O') - sin(Ns)K2 (-s, 0 )| ds = 0

2 it

(3*3)

/

j^sin(Ns)H1 c ^ ^ ^ - s )

+ c2x ^ ( - s ) ,

0

o + cos(Ns)H2 (clx ^ 1 ^ + c2x^2 ^, o)j-ds +

2xr +

J* -jsin(Ns )K1(-s,

o) + cos(Ns)K2 (-s, 0 )j- ds = o

o (Here for convenience we have not indicated the components of c,x^1^ + c2x ( v2 ) separately although H 1 and H2 are actually functions of these components. We use this convention with P1 and P2 also.) Let k1 and k2 denote the constants which are the second integrals on the left sides of equations (3*2) and (3 *3 )* Let c2 ) and ^ ( c ^ c2 ) denote the first integrals on the left sides of (3*2) and (3*3), and let ^(Ci, c2 ^ 2 1> c2*) denote the polynomials obtained if in ^ and 2 , the terms H ^ c ^ ^ 1^ + c2x^2 ^, o) and H ^ c . ^ 1^ + c2x^2 ^, o) are replaced by P ^ c ^ 1^ + c2x^2 ^) and

P2(clx^1^ + c2x ^ ) .

ASSUMPTION 3 * If and £>2 are of degrees n1 and n2 respectively in c1 and c2, let Q1 and Qg be the homogeneous polynomials of degrees n1 and n2 in*P ^ and P 2respectively. Let M : (c^ c2 ) -- > (c*, c2 ) be the mapping defined by Q*i (o ^, c2 )= c ^

Qg (c1, c2 )= c2 We assume that the topological index at

.

(o, o)

of M

is defined.

46

CRONIN From standard arguments, we have: M 1 : (c.,, Cg ) -- > (cj, Cg ) be

LEMMA 3-1- Let defined by:

(c1, c2 )+ k 1 = c^

* V C1' °2 )+ k2 = °2 The topological and relative to center (o, o ) at (o,o) of

•

degree of mapping M 1 at (o, o) any sufficiently large circle with is equal to the topological index M.

THEOREM 3 .1 . Let Assumptions 1 , 2 , 3 be satisfied. Then for all sufficiently small 11 , Equation (3 * 1 ) has at least one solution of period 2 k

.

PROOF.

Let

o 4 i0 : (c^

c2 ) -- > (cj, c2 ) be defined by:

1(c 1, c2 ) + k 1 = c*

^ ( c ^ c2 ) + k2 = c2

.

It is sufficient by Lemma 2 . 1 to show that the topological degree of cM Q at (0 , 0 ) and relative to a circle with center at (0 , 0 ) is different from zero. We show first that relative to any sufficiently large circle, the topological degreesof M 1 and c 4 i0 at (0 , 0 ) are thesame. From Assumption 3and the fact that P . and P p arepolynomials,it follows 2 2 2 that given m >0 , there exists r 1 > 0 such that if c 1 + c2> r1, then c2 ) +

e > 0

Further there exists

and

k 1 ]2+ [pgCc,, c2 ) + k2 ]2

and

rp > 0

> m .

such that if|*nn | < e, |t)p | < e,

2 c1 + 2c2 >2 r2, then 2

J

it

-jcos (Ns )[P1 (c^ ^ 1 ^ + c2 x^2 ^)](l + r|1 )

o - sin(Ns )[P2 ( c ^ 1 ^ + c2 x ^2 ^)](1 + r\2 )j» ds + k 1 J

+

POINCARE!S PERTURBATION METHOD

+

47

2* ^sin(Ns )[P1( c ^ 1 ^ + c2x^2 ^)](1 + n i ) o

j'

+ cos (Ns )[P2 ( c ^ ^ 1 ^ + c2x ^ ) ] (i + r|2 )| ds + k2 2 > l m

By Assumption 2, there exists (C ^X ^

+

C 2X ^ 2 \

where |rj^| < e for i = (cjCt), Cg(t)) defined by:

cj(t) =

r^ > 0 such that if 0)

=

(1

then

T)1 )Pi ( c iX ^ 1 ) + c 2 x ^ 2 ^)

Hence the homotopy h: (c^ c2, t) -- >

2.

2n | c o s ( N s ) [ P 1( c ^

J

+

2 2 2 c1 + c2 > r^,

1}

+ c 2x ( 2 ) ) ] ( i + tri 1 )

o - sin(Ns)[P2 ( c ^ 1^ +

c

2 x ^2 ^ ) ] ( i

+ tj\ 2 )|ds + k]

2

C2 (t ) = J j^sin(Ns )[P1(c^^1^ + o - cos(Ns )[P2 ( c ^ 1 ^ +

c

c

2 x ^2 ^ ) ] ( i

2x ( 2 ) )]( i

+ tr]1)

+ t^2 )|ds + k2

shows that the topological degrees of M] and cM0 at (o, o) and relative to any sufficiently large circle with center (o, o) are the same. A trivial computation based on the fact that 2 jt

J

sinra(x) cosn(x) dx 4 o

o only if m and n are even shows that *p 1 and ^ 2 are polynomials of odd degree in c1 and c2- Then the topological index at (o, o) of M is odd [5 ] and hence the topological degrees of M1 and e4 i Q are different from zero. This completes the proof of Theorem 3.1. As a second example, we consider Equation (3 * 0 under the following assumptions. ASSUMPTION 4 . Matrix A has just one pair of characteristic roots IN, - IN where N is a non-zero integer and A has 0 as a characteristic root. Also A has the form

48

CRONIN

0

\

N 0

0

-N 0 0

0

0 0

0

Ao

where AQ is not necessarily canonical but is real and has no character­ istic roots of the form IN. ASSUMPTION 5 * The components f1(x, t, n), f2(x, t, n) and f^(x, t, n) of f(x, t, n) have the form: f±(x, t, ii) = H ^ x ^

(i = i, 2y 3)

xn, n) + K^t, n)

where x^ . xR are the components of x; and and are functions of the indicated variables and have continuous second deriva­ tives in these variables; and K 1 , Kg, have period 2* in t. There exist polynomials x2, x^), x 2 , x ^) , x 2 , x^) such that H. (x ,X , X , 0 ,.. --- -p T | - k i n —

iim

Also the degree of a 4 °) in P^-

P^

is k,

0) = 1

Ci = 1, 2, 3)

odd, and there is a term

K ax^

•

t (with

Three real independent periodic solutions of the variational equation are: ) = (cos(Nt), sin(Nt), 0, x.(1 ;

o)

x^2 ^ = (- sin(Nt), cos(Nt), 0, x
c’, c2,c^bethe ASSUMPTION 6. Let M : (c^ cg, c^) mapping defined by c^ = Q^(c.,, c2, c^) fori = 1, 2, 3. Weassume that the topological index at (0, 0) of M is defined. By the same type of argument as for the preceding example, we obtain: THEOREM 3*2. Let Assumptions 4 , 5, 6 be satisfied. Then for all sufficiently small \±, Equation (3*1 ) has at least one solution of period 2jt. More complicated examples may be treated in a similar manner. §4 . EXCEPTIONAL CASES By Assumption 3, the topological index at (0, 0) of mapping M is defined. Now suppose that this assumption is not satisfied, I.e., suppose the topological index is not defined. This means that Q1, have a common real factor. Hence by varying one coefficient in Q (I.e., varying H1) arbitrarily slightly, we obtain a mapping for which the topological index Is defined. When such a change is made, two differ­ ent cases arise. First suppose that after Q1 is changed slightly in some definite manner, the topological index j of the resulting mapping is defined and suppose that j Is odd or that |j| > 2. Then by using the analysis of [5 , Section 3 ] It can be seen that regardless of how Q or Qg is changed (provided the magnitude of the change is sufficiently small) the topological Index will be non-zero and there will be at least one periodic solution. Now suppose that after Q,1 is changed slightly in some definite way, the index j is zero. Then no conclusion can be drawn about whether there is a periodic solution. Further suppose that j - 2 or - 2 . Then, again by using the analysis of [5 , Section 3 ] it can be shown that

50

CRONIN

a different small change in Q will yield a mapping of index zero. Again no conclusion about the existence of periodic solutions can be drawn. Another exceptional case which may occur is this. In the first example of Section 3, functions ^(c-j, c2 ) or ^ ( c ^ c2 ) may be identically zero. If ^(c-j, c2 ) = 0 and k1 4 0, the bifurcation system is inconsistent and there are no periodic solutions with initial values continuous in \i for small \±, If H ^ c ^ c2 ) = 0, and k1 = 0, no immediate conclusion can be drawn. It is worth noting that if we consider a problem in which

11=0 then as is shown in [8], neither

1 or

*f2 is identically zero.

§5 . IF THE TOPOLOGICAL DEGREE IS EVEN The condition (in Assumptions 2 and 5 ) that the f^(x, t, \±) can be written as the sum of aterm that depends only on x and n and a term that depends only on t and 1-1 impliesthat the topological degree is odd and therefore non-zero. If we do not impose this con­ dition, the topological degree may be odd or even. By the results of [5 ] and Lemma 3.1, if there are two equations in the reduced bifurcation system, the topological degree can be computed in all cases and the computation consists simply in approximating the real roots of certain polynomials. (See [7], Theorems 1 and 2.) If there are three or more equations in the reduced bifurcation system, the degree can be computed in many special cases. If the topological degree is zero, Equation (3 * 0 may or may not have periodic solutions as the example in [7 , Section 5 ] shows. §6.

THE NUMBER OF DISTINCT PERIODIC SOLUTIONS

For the totally degenerate case (q = n), the topological degree also yields an estimate of the number ofdistinctperiodic solu­ tions. We illustrate this by considering the first example of Section 3 with the additional hypothesis that n = q. As before, the reduced bifurcation system is: (c1, c2 ) + k1 = 0 (c1, c2 ) 4- k2 - 0

POINCARE’S PERTURBATION METHOD

51

LEMMA 6 .1 . Let M be a continuous mapping defined on the closure o of an open set o C Rn, real Euclidean n-space, and suppose M is differentiable in o. Suppose the topological degree of M at point pQ and relative to o is d 4 °* Then there is a neighborhood U of pQ and a set E of n-dimensional measure zero, E C U , such that if p e U - E, then M~1 (p) is a finite set con­ sisting of at least |d| points. This lemma is proved in [6 , p. 213]• Now suppose that the topological degree at (o, o) and relative to a solid circle S with center (o, o) and radius r of the mapping defined by the reduced bifurcation system is d for all sufficiently large r. Let mapping M^ be defined by c’ = ^ (c1, c2 ), Cg = c Then if r is sufficiently large, the topological degree of M^ at (- k 2 ) and relative to S is also equal to d. Applying this fact and Lemma 6 . 1 to the bifurcation system, we see that if kg are changed arbitrarily slightly, there will be at least |d| distinct solutions of the bifurcation system. Since

o and similarly for kg,

we obtain:

THEOREM 6 .1 . Suppose K 1 and Kg are independent of ia and suppose the topological degree at (o, o) of the mapping defined by the left side of the re­ duced bifurcation system is equal to d if the topological degree is taken relative to any sufficiently large solid circle with center at (o, 0 ). Let [i have a fixed value, say n . Then given tj > 0 , there exist functions K^(t), K 1 2 (t), both of period 2 it, with continuous second derivatives, such that max

0£ t£ 2n

|KM (t) - K, (t)| < i)

IX

(i = 1, 2 )

and such that if \± = 11 in Equation (3 .1 ) and K 1 (t), Kg (t) are replaced by Kn (t), K 1 2 (t),

52

CRONIN

then the resulting equation has at least distinct solutions of period 2 *.

|d|

§7- THE AUTONOMOUS CASE Topological degree can also be used to study autonomous sys­ tems, but because the bifurcation system is quite different in the autonomous case, a different approach must be used. We describe the application to an example. Consider the equation: (7-1 )

x = Ax + nf(x, n)

.

Assume A has just one pair of characteristic roots of the form IN, - IN, where N is a non-zero integer, that A has zero as a character­ istic root, and that A has the standard form described in Assumption b. Assume further that n = q. The bifurcation system may be written: 2Jt - vc2 + jf jcos(Ns )f1 (c. ^ 1 ^ + c2 x^2 ^ + c^x^, o) o - sin(Ns )f2 (c.,x^1 ^ + c2 x^2 ^ + c^x^, 0 )j- ds = 0

vc1 +

J

2 jt

sin(Ns)f1 + cos(Ns)f2j- ds = o

o

vc^ +

J

2 Jt

jf^Cc^

1 ^ + c2 x^2^ + c^x^,

o)} ds = o ,

o where f~$ are the components of f, /o \ f-(x, (a), fp, d x have the same meaning as before and

and

x ^1 \ and

v = iim i M [ l - ^ o Mwhere t ( h ) is the period of the solutions of (7*1 )• The physical prob­ lem represented by the autonomous case is such that the corresponding mathematical problem can be posed as: set onec^ = oand solve for the remaining c^1s and v. (See [3, pp* 30-31] and [b, pp. 3 6 ^-3 6 6 ].) Let us assume that

f 1 (x, 0 ), f2 (x, 0 ), f^(x, 0 ) are polynomials

POINCARE*S PERTURBATION METHOD

in the components of x. (7.2 )

53

Then the bifurcation system may be written: - VC2

+ P1(ci, C2, c3)

(7.3)

vc3 + P3(Cn, Cg, c3 ) = 0

(7 .4 )

where P.,, P2, P^ are polynomials in c^ c2, c^. We set c^ = 0 and solve (7.2 ), (7*3), (7*4) for c^, cg, and v. Multiplying (7 .2 ) by c1 and (7*3) by c2, and adding, we obtain the system: (7.5)

(7.6) If

0 ^ ( 0 ^

Cg,

0)

+ CgPgCo^

Cg,

0)

=0

P3(c1> °2> °) = 0

tx is variedslightly, Equations (7*5) and (7-6) become

(7.5)'

c-jR^c.,, Cg, n) + CgRg(c1, c2, n) = o

(7.6)'

RjCc^ c2, d) = 0

where R1, R2, R^ are continuous functions of c^ c2, and \i.Suppose the topological degree of the mapping defined by the left sides of (7-5) and (7»6) is non-zero. Then for a sufficiently small n, (7*5)’ and (7*6 )T have a solution (c°, c°). When \i is varied slightly in (7 .2 ), (7.3), and (7*4) (but keeping c^ = 0 ) we obtain: (7 .2 )’ (7 .3 )1 (7*4 ) !

— vCg+R^C^, C2, (i) = 0 vc1

+

R2 (c1,

C2 ,

0

|_i) =

R3 (c 1, c2, ii ) = 0

.

Now suppose one of the c°, c°, say 0 °, is different from zero. (This will occur if, for example, P^(c1, c2, 0 ) has a constant term.) We set Rg(c°,c°,n)

Then

c2, vQ are a solution of (7 .2 )’, (7.3)*, and (7*4)’.

54

CRONIN

REFERENCES [1 ] ATiEXANDROFF, P., and HOPF, H., Topologie 1, Berlin, 1935• [2] BASS, R. W., Extension of frequency method of analyzing relayoperated servomechanisms,~ Section ill, Pinal report, Contract DA-36 -o3 4-ORD-1273 RD, Johns Hopkins Institute for Cooperative Research (1 955 )• [3] CODDINGTON, E. A., and LEVINSON, N., "Perturbations of linear sys­ tems with constant coefficients possessing periodic solutions", Contributions to the theory of nonlinear oscillations, Vol. II, Annals of Mathematics Studies No. 29, Princeton, 1952 • [4] CODDINGTON, E. A., and LEVINSON, N., Theory of ordinary differential equations, New York, 1 953• [5] CRONIN, J., "Topological degree of some mappings", Proc. Amer. Math. Soc., 5 (1954), pp. 175-178. [6] CRONIN, J., "Branch points of solutions of equations of Banach Space, II, Trans. Amer. Math. Soc., 76 (1 954), pp. 2 0 7 -2 2 2 . [7]

CRONIN, J., "Note to Poincare’s perturbation method", Duke Mathematical Journal, 26 (1959), pp* 2 5 1 -2 6 2 .

[8 ] FRIEDRICHS, K. 0., Advanced ordinary differential equations (mimeo­ graphed ) New York University, 1 949. [9] FRIEDRICHS, K. 0., "Fundamentals of Poincare’s theory", Proceedings of the symposium on nonlinear circuit analysis, New York, 1953, pp.' 5 6 -6 7 . [10] FRIEDRICHS, K. 0., Special topics in analysis (mimeographed), New York University, 1 954. [11] LEFSCHETZ. S., "Complete families of periodic solutions of differential equations1, Comment. Math. Helv., 28 (1 954), pp. 341-345. [12] LEFSCHETZ, S., Differential equations: geometric theory, New York, 1957. [13] POINCARE, H., Les methodes nouvelles de la mecanique celeste, Vol. I, II, III, (1 8 9 2 -1 8 9 9 )- Reprinted by Dover Publications, New York, 1957-

Polytechnic Institute of Brooklyn

IV.

ON THE BEHAVIOR OF THE SOLUTIONS OF LINEAR PERIODIC DIFFERENTIAL SYSTEMS NEAR RESONANCE POINTS Jack K. Hale §1 . INTRODUCTION

In the last few years, systems of differential equations of the form n (i*i)

y] + ^ y j = e

(Pjkyk>

j = 1,

n

,

k=l where £ is a real small parameter, the constants a . are positive, and the real functions cp ( t ) are periodic of period T = 2 */(jd and L-integrable in [0 , T], have been investigated by many authors [1 , 3 a, 4a, 5 , 7, 8 , 9 , 1 0 b, 1 Oe, 1 2 , 13]• (See also the book [3b]). For n = 1 , classical results of 0. Haupt [9] assure that all solutions of (i.i) are bounded for |e| sufficiently small and an arbitrary periodic function cpn provided 2 a1 ^ 0 (mod ). For the study of the system (1 .1 ) for |e| small and n^> 1, L. Cesari [3a] in 1940 considered a method of successive approximations, which was successively developed by L. Cesari, J. K. Hale and R. A. Gambill [6 , 4b, 1 0 c, 1 Oe] and applied to questions of existence and stability of periodic solutions of weakly nonlinear differential systems (cf. the book [3b] and [ 3 c ] ) . By using this method it was first proved [3 a, 1 0 b] that all solutions of (1 .1 ) are bounded provided 2 a . = o, ai - ak ^ 0 0 0 J ^ k, j, k = 1 , 2 , ..., n, and the matrix ®(t) = [cpjk(t )] is either symmetric or even in t. Under the same re­ strictions on the numbers aj more general boundedness theorems have then been proved [5 a, 1 Oe] by the same method, but they all involve some type of "symmetry" conditions on the matrix ®(t). Again using the same method it has been proved [3 a, 5 b] that some "symmetry" condition is necessary to assure that all solutions of (i.i) are bounded for e 55

56

sufficiently small.

HALE

For systems more general than (1 .1 ), see [1 , lOe].

Analogous boundedness theorems involving "symmetry" conditions have been proved by M. Golomb [7] by a different method. Finally, J. Moser[13]^ J* M. Gel!fand and V. B. Lidskii [ i b] have recently shown that it is sufficient to require 2 a. f 0 , a . + a, ^ 0 (mod cd), j / k, J J K j, k = 1 , 2 , ..., n, when a> is a symmetric matrix. In the present paper a procedure based on the same method developed by Cesari, Hale and Gambill is given for the study of the be­ havior of the solutions of (i.i) (and more general systems) near the "resonance points" 2 a. = sod, a. + a, = s 0 (Remark5-3)* which agrees with the conclusions of J. Moser [ 1 3 ] • Some examples are given in §5 to show that G(s, a) may be < 0 for some matrices $ if $ satisfies (C) and is not symmetric. If Theorem (5-4) is applied to system (1 .1 ) satisfying (B), (C), then all of the AC solutions of (1 .1 ) are bounded for e sufficiently small. In §6 , the above method is applied to systems of Mathieu type equations of the form (i.i) where each cp("t) = djk cos 2 t,each djk is a constant and 2 a^ f 0 , a^ + a^ f 0 (mod 1 ), j ^ k, j, k = 2 , 3 , ..., n, a^ (o) = m, where m is a positive integer. Sufficient conditions are given (Theorem 6 .1 ) to insure that there are unbounded absolutely con­ tinuous solutions of (1 .1 ) in every neighborhood of the point (m, o) in the (a-j,e ) -plane, where m is a positive integer. A corollary of Theorem (6 .1 )is the well known fact [1 1 ] that there are unbounded solutions of the Mathieu equation x" + a2x + e(cos 2 t)x = o, in every neighborhood of the point (m, o) of the (a, e)-plane for every positive integer m.

LINEAR PERIODIC SYSTEMS

Using determinants, the behavior of the solutions of the "resonance points" has also been discussed by E. Haacke [3] for the case where $(t) is even in t. the above type seem to be easier to obtain using the since it does not involve determinants.

57

(1 . 1 ), n > 1 , near Mettler [1 2 ] and W. General theorems of method of this paper

Another application of the method discussed here concerns the stability of periodic solutions of weakly nonlinear differential systems. The linear variational equations associated with such a periodic solution is a linear differential equation with periodic coefficients of the form (1 .1 ) and, in many cases, some of the basic frequencies a. are "in resonance" with o>. The asymptotic stability of periodic solutions of weakly nonlinear periodic differential systems has been discussed re­ cently by H. R. Bailey and R. A. Gambill [2] using essentially the same method. For a more complete discussion of the application of this method, in particular, Theorems (2 . 1 ) and (3 . 1 ), to the stability of periodic solutions of both weakly nonlinear autonomous and weakly nonlinear periodic differential systems, see [1 Og]. §2.

DESCRIPTION OF THE METHOD

Let C denote the family of all functions which are finite sums of functions of the form f(t) = eq>(t), - » < t < + °°, where a Is any complex number and q>(t) is anycomplex-valued function of the real variable t, periodic of period T = 2jt/(t) e C^ and M-j C|i+h, [i+& ~

i = n + 1, •••, n; h = n + 1, • • • > n •

We also suppose that the equations in (3-13) are reordered so that A = diag (p1, • • • >

(3«15)

pn+(i )

where each p. is one of the numbers X and the p. \ (3• 1) • It is very easy to prove the following lemma.

satisfy condition

LEMMA (3 . 1 )• If the alternative method of successive approximations is applied to the auxiliary equation of (3 *1 3 ); (3*15) with the d. in (3*7) such that dgj-i = ~d2 ‘, j = 1, 2 , ..., v, then (3 *1 6 )

^2 j_1

d, e) = Vc .(t, d e),

j = 1, 2, •••> v ,

for every t , d where Vj is defined by (3«9)» Therefore, the determining equations (3•1 0 ) become T - £ V2 - ^(t , d e) = p^

j=i,2,

..., v,

(3.17) d = (d-j, •••, d2 v ), d2 j = - d2

,

j = 1, 2, •••, v •

For applications to the stability of periodic solutions of non­ linear differential equations (see [lOg]), it is convenient to have the

LINEAR PERIODIC SYSTEMS following theorem. THEOREM (3«1 )• Consider the system of differential equations (3 *1 1 ) with ^2j-1 = (3 .1 8 )

^2j-1 ^ ^2j

X2 j- 1 “ ^2 k-l = mjkCDi^ mjk an inteSer or zero, j, k = 1 , 2 , ... \* f

1 , 2 , ..., 2 vj k

(mod coi), j=

= 2 v + 1 , ..., N ,

X . , = ~ (- a. + i? n- ) , 7 a = )2 > 0, ^J 1 2 J J J J J j = 1j 2, •••, M-, ^|i+j = ^ = ^ + •••> and let H = (hjk )>k = 1 , 2 , ..., v, be the v x v matrix defined by

where

T

h3*m i

/

c (t)e 2J—1,2k-1 ' '

(3 .1 9 )

j1

V..4(J+at>

j, k = 1 , 2 ,

v ,

where the functions c2 j_i 2 k-l are by (3*14). If PQ is a simple root of the equation |H - PI| = 0 and the corresponding eigenvector has no zero components, then there are two character­ istic exponents t , t of (3 . 1 1 ) given by (3.20)

t

for

= Pl + e PQ + 0(e2 )

e sufficiently small.

PROOF. For written in the form

£ / o the determining equations (3*17) can be

(H - pi)d =

o(e)

,

where £ p = td = (d^ d^, ..., d2 y _ 1 ) and H = (hj^) is the v x v matrix given by (3•1 9) • It then follows immediately from the implicit function theorem that Theorem (3*0 is true for £ sufficiently small.

68

HALE

§4.

SOME BASIC LEMMAS

Consider the system of linear differential equations, ii y j: +

a jy j =

e

I

n (cp3k y k

+

♦jkyp

+

k=l

8

3

I

=

1, 2,

. . . , , ,

k=n+i

u.i) n

n

y • = 6 £ ^j^k + V k ’ + e I V k ’ J= “ + '> •••> n k=i k=[i+i

>

where e is a real parameter, each a . is a positive real number (and if o-j = crj(£) is an analytic function of eat e = 0, then crj > 0 for £ 4 each cpjk(t), ^^(t) is a real function, periodic in t of period T =2 tt/o), L-integrable in [0, T]. Suppose the matrices ^ ~ ^9 jk^ ^ = ^9 2 9 •••> k = i, 2, •••, y ('I’j]^)^ j; k = 1, 2, ..., n, are partitioned so that c> = ($jk )* j= 1> 2 , 3 ,k = 1 ,2 , Y = ^Yjk^ k = 1 , 2 , 3 ^ where $n , are p x p matrices, ® 2 2 9 ^22 are (^ ” ?) x (^ “■ P) matrices and is an (n - 1-1 ) x (r matrix. Throughout the present section, we always assume that -

(k . 2 )

for all

*jk(t) = ( - O k+j ®jk(-t), ^jk(t) = (-l)k+J+1 ?jk(-t)

j, k. LEMMA (4.1 ). If aj = crj (£ ) is an analytic function of £ at £ - 0, cTj(o) f- 0 (mod aai), j = 1, 2 , ..., n, and the matrices 0 , y satisfy ( 4 . 2 ) , then there exists an £Q > 0 such that there are n - m. linearly independent periodic solutions of (4.1) for I 1£ I t< \ £o . PROOF.

The transformation y^+j = z y

j = 1> 2, ..., n - ji,

j y • = 2 1 (z .), J i= J v n-n+j. + z n+j.)* n-n+j. - z n+j" leads to an equivalent system of first order equations

y-=(2ia.)**(z

(4 .3 )

1,2,

*

9

....

9

u,

z’ = A z + £ C ( t ) z ,

where A = diag(o, ..., 0 , ic^, ..., icr^, - l o ] , ..., j = 1, 2, 5;

c = ^Cjk^

69

LINEAR PERIODIC SYSTEMS

°jk " “ °j+2,k' J' ~ 2 ’ 3 ’ k "

Cjjk - Cjjk+2> k - 2,3, j = 1>2, ..., 5;

C11 - f33> C21 = Y1 3 ; °31 = *23; 2C1,k+1 = ®3kAk

+ *3k' ^ = 1,2; 2Cj+1^ +1 =

= $j,kAk1 + ^jk’ ^ k = 1,2; A1 = dia8(ic,i>*••Jiap )j k 2 = dias(itJp+1, With this partitioning of the matrix C it is convenient to also partition the vector z = (z^ y ..., z ^ ) , where z(j) ^as a dimension compatible to the matrix multiplication. We now apply the above method of successive approximations of §2 directly to (4.3) with x^0^ = (d, 0, 0, o, o), d = (d^ ..., dn_^), where d^ dn_^ are arbitrary real numbers. The solution obtained in this manner will obviously be periodic. If the successive approximations x^r ^ are partitioned the same as z, i.e., (r) , (r) (v)^ x v = (x (-|)^ • • • > x (5 )^ then we first show by induction that H (r) = diag(s(r), ..., S^J) = 0

,

(k.k) x^j(-t) = x^j(t), xjgjc-t) = - x|^](t), x ^ ( - t ) = xj^(t) ,

r = o, 1^ 2, ...

.

(r ) The induction is on the functions x (j)* The assertion is clearly true for r = 0. Assume the assertion true for r = 0 , 1,2, ..., v - 1 . Then the matrix C(t)x^P ^(t) satisfies the property

C(-t)x^(-t) = E C(t)x^(t), E = diag(-l, -1, 1, -1, 1) r = o, 1, 2, •••, v — 1 . Sincethe first vector component of this matrix: product is an odd func­ tion of t it follows immediately from (2.10) that = 0, r = 1, 2, ..., v, for every d = ..., dn_^). To complete the induction, we have from (2.11 ), since B = A, x (v)(-t) =

e-At j V AaC(a)x(v-l)(a)da =

= - e-At J ' eAaC(-a)x^v-1)(-a)da = =

- e ~A t f e Aa EC(a)x(v-1)(a)da

.

HALE

TO

This is clearly assertion (4.4) for r = v, since C. v = - C . ,, j>& J j = 2, 3 , k = 1, 2, ..., 5; therefore, (4.4) is true for all r. Since the dj, j = 1, 2, ..., n - n, are arbitrary, one can find n - ji such linearlyindependent solutions and the lemma is proved. Suppose the aj = aj(£) (^#1 ) are analytic functions of at £ = 0, aj(°) > 0, j = 1, 2, .. “ ak(°) =^jk^' m ik an integer or zero, j, k = 1, 2, ..., q, and the transformation (3*12) is applied to (4.1 ), to obtain the equivalent first order system (4.5 )

z1 = Az

+

£ C(t )z

where A = diagdci,, - ia . , ..., l aH I I j, k = 1, 2, ..., n + n, where

,- l a

9 ’Vr

C2j-1,2k-1

Ta^ +

9

, 0,

(4*6)

..., o) and

C jK =C.tJ,

*lr

C2j-1, 2k =

‘ +jk' k = ^ 2> •••» ^5

nj C2j, t = ~ C2j-1 , l >

C2j-1,h = *j,h’ h = M +

i5 = 1, 2, ..., n; j = 1, 2, •••, M-;

9hk Ch,2k-1 = Ch,m =

9hk + '•'hk' Ch,2k =

m = n + 1> .

" +hk’ k = '> 2 ’

n; h = n + 1,

n

|j; .

The auxiliary equation of (4.5) is chosen as (4 .7 ) where 0,

...,

e

z1 = Bz + e C(t )z B = diagd^, 1t2, ..., iT2q, iaq+i>“ icyq+i' • • • > i(V 0 ), and the t ' s are real numbers such that X2J-1 = • 72 j ’ T2j-1 - T2k-1 = m jk®'

k = 1, 2,

" icV

q ,

(k.6)

T2j_1 ^ °> T2j-1 i ffk^°) ^ 0 (m0d 00^ j = 1» 2> k = q + 1, ..., n and the are defined above by the numbers that 2t ^ may or may not be a multiple of cd.

aj

;

at £ = 0 . Notice

LINEAR PERIODIC SYSTEMS

71

LEMMA (4.2). If the matrices associated with (4.1) satisfy (4.2) and if the alternative method of successive approximations is applied to (4.7), (4.8) with the zero^ approximation given by I t ,L t /n > 1T1 1T_ L (0) = (d.e d0 e 2q , o, . . o) xvu; (d^ 1 ,, with dj = b y j = 1, 2, ..., 2 min(p, q); d . = l~b j = 2p + 1, ..., 2q; where each bj is a non zero real number, then the numbers V y j = 1, 2, ..., 2q, defined by (3*9) are purely imaginary. PROOF. For simplicity in notation we only prove the lemma for P ^ q> but it will be clear how to discuss the other situation. Applying the method (3-7), it is clear that x ^ ( t ) has the form 2q (*.9)

x(r)(t) = £

+oo

e1^

£

j=l (r ) is an n + where duction that (4.10)

m

eikcut

,

k=-*>

dimensional column vector.

We show by in­

B ^ } = C°1 (Bjki^’ 1 BS k b Bjk3}' r = °' 1' 2’ *•*' (r*)

and all j, k, where ^ = 1^2, 3, are real column vectors of dimension 2p, 2m. - 2p, n - m > respectively. The assertion is clearly true for r = o. Assume that (4.10) is true for r = o, 1, ..., v - 1 and observe that the Fourier series for +00 c ( k) e ^ mt

C(t) -v £ k=oo

in (4.5) is such that = (a-. .c P ^ ), h, j = 1, 2, 3, where a, . = 1 11>J /t-n if h + j is odd; = i, if h + j is even, and each is a real matrix with Cjj , j = 1, 2 , 3 , of dimensions 2p x 2p, ( 2 m - 2p) x ( 2 m - 2p) and (n - m ) x vn - m )> respectively. Therefore, 2(1

C (t )x^r ^(t) = ^ j=1

,»

e

+

J*

+ °°

^ k=-oo

= ^ ^ v-1,

HALE

72

where

= col(iDj^^ ^jk^' ^jk3 ^ ¥hePe each of the real vectors

k £ ’ £ = 1' 2' has the same dimension as From (3 .6) and the assumption on the numbers d - in the statement of the lemma, J

(ii-.1l)

(v) S\ is purely imaginary,

j = 1, 2, ..., 2q; r = 1, 2, ..., v .

J

It then follows immediately from (3-7) and the above results that (4.10) is true for r = v and, therefore, (4.10) is true for all r. Finally, (4.11 ) also holds for all r and the statement of the lemma is true. REMARK (4.1 ). Lemma (4.2) holds also in a slightly more general situation, namely when q of the cr’s differ by a multiple of cjd at 8 = 0 , but the ones which satisfy this property are not necessarily the first q. The notation for the statement of this result as (4 .2 ) is extremely complicated, but it should be clear that the same conclusion is true, the only thing necessary is the proper choice of the numbers d y REMARK (4.2). condition

When

m. = n

T2 j_i ^ 0 (mo^

in Lemma

in (4.1), it is obvious that the raay be eliminated.

REMARK (4.3)* The determining equations for the 2q + 1 real numbers b.j, ..., b2(^ and t1 [only t^, since the t • are related by (4.8)] in Lemma (4.2) are the 2q real equations

(4 .1 2 )

Tj — e I [V

^ > £ )] =

(“O^*

= ^’

^j*

29 ***’ 2^ *

b = (b.,, ..., t»2q^ ^ the imaginary part of a complex number w. If, in addition to condition (4.8), we assume that 2 tj ^ 0 (mod a>), j = 1, 2, ..., 2q; and choose h2j._1 = - b 2J., j = 1 , 2, ..., p, b2j_i = b2j, j = p + 1, ..., q

in Lemma (4.2), then we know from Lemma

(3 .1 ) that V2 j_ 1 = ^2j, ^ = (4 .1 2 ) are equivalent to the equations (4.13)

T2j-1

[^2 j— 1 ^T

^

^

= ^j> J = ^

for the q + 1 real numbers b ^ b^, ..., t>2q-i will be important for the applications.

determining equations

^ * • • • j Q.

anc^

T1 -

>

This remark

REMARK (4.4). In the previous remark, equations (4.12), (4.13) were considered as determining equations for b and but we could just as well have considered t 1 as fixed and determined the numbers o , b so as to obtain a solution of (4 . 5 )• This will be important in de­ termining the stable and unstable regions around "resonance points”.

LINEAR PERIODIC SYSTEMS

§5.

73

SOME BOUNDEDNESS THEOREMS

As an application of the lemmas in §4, we state a few theorems concerning the behavior of the characteristic exponents of (4.1 ) and also some theorems concerning the boundedness or unboundedness of the AC so­ lutions of (4.1 ) in (- oo, + oo). THEOREM (5*1 )• If the matrices , ¥ associated with system (4.1 ) satisfy (4.2) and the numbers o j = CTj(e) are real analytic functions of s at 8 = 0 , with cij(o) f o (mod a>), j = ^,

(5*1 )

2,

..

,

n,

then there exists an eQ > o such that there are always n - \± linearly independent periodic so­ lutions of (4.1 ) for 181 < e . If one of the numbers o- , say , is such that (5.2)

2(^(0)

4

o, ti(°) = Hi­ proof. The first part of the theoremis a restatement of Lemma (4.1 ). The second part of the theorem is an immediate consequence of Lemma (4.2) and Remark (4.3). For since q = 1, the implicit function theorem assures us that Equation (4.13) has a real solution t1 analytic in e at e = 0 with t1(0) = a 1. COROLLARY (5*1)• satisfied and (5 .3 )

2(Xj(0) 4 0,

crj(0) 4

If the conditions of Theorem (5 .1 ) are

+ ak (0) (mod a>), j 4

k,

j. k = 1, 2, ..., n;

then all of the AC solutions of (4.1) are bounded for |e| < e0, eQ > 0. PROOF. This result follows immediately by applying Theorem (5.1) to each of the numbers, cr., j = 1, 2, ..., n.

HALE

Corollary (5*0 coincides with a previous result obtained by the author [1Oe ]. For 11 = n - 1, this corollary has also been ob­ tained by the author [lOf] without using successive approximations. For m- = n, this result has also been obtained by M. Golomb [7]. Consider the system of differential equations (4.1) satisfying (4.2) and suppose that a^(o) - ex (o) = sou where s is an integer or zero and each a- is a real positive analytic function of s at e = 0 with J

(5.4)

2 cTj(o) 4 0, crJ(0) + jk(t), ^ ( t ) are given by

n

djki must ^e either purely im­ If

bjjo = djj0 = bjk3 = djk3 = 0, j 4 k, j, k = 1, 2

,

then sj1^ = Sj1^ = o. Calculating the functions xj1 j = 1, 2, n + \i, from (3*7) and (4.6), it is then very easy to show that

(5.11 ) S 7kji

V, = T, v2 = T - SCO, Vj = O y

and the index

\

1

, }

V

j = 3,

°jkji/'i^ k'

u ,

£ + i • in (5 .1 2 ) does not take on the values in (5 .1 0 ). J

In order to make the statement of the following theorem meaning ful, first observe that the conditions (4.2) and Lemma (4.2) imply that each of the numbers is ^eal if t is real and either p ^ 2 or (s ) (s) (s) (s) p = 0 , and a \-\ > ^22 are real> 12 a2l are Pupe1^ imaginary if (s ) t is real and p = 1 , since the are independent of the numbers dj in Lemma (4.2).

THEOREM (5 .2 ). Consider the system of differential equations (4.1 ) with the functions ^jk^^ satisfying the condition (4.2) and having the Fourier series +00

0 for e = 0 , then there exists an eQ > °such that all of theAC solutions of (4.1 ) with the as above are bounded for |s| < e Q; (11) If H(s, a1) 4 0, [G(s, a,, a)]2 +H(s, a, ) < 0 for 6 = 0 , then some of the AC solutions of (4.1) with the a- as above are unbounded for every s 4 0 . J

PROOF.

From the remarks preceding the statement of the theorem

ico + t - v

LINEAR PERIODIC SYSTEMS

77

it is sufficient to discuss the four characteristic exponents which are close to the numbers

icr., -

^or

e ^

^he determining equations

for these characteristic exponents are given by iF«(x,a,b,e) = It + — [ 1 41 11

(t ) + — b

(t )] 12

la.1 +

O(e^) = 0

(5-13) F2 ( T , a , b , e ) = ^ a j g ^ r ) b

- G( s ,

a) - a

are given by (5 .1 2 ).

where the

+ o(e) = 0

Any solution

(5 *1 3 ) yields two characteristic exponents

t, a, b

of Equation

It, - It.

Suppose that the conditions of (i) are satisfied. teger

p

associated with system (4.1 ), (4.2) is

^ 2,

If the in­

then, from the

remark preceding the statement of the theorem, all of the numbers are real.

Furthermore, from Lemma (4.2), the functions

are real if r, a are real and b = 1 . The numbers aQ = CG(s, c^, + [G2 (s, c^, ) + H(s, a.,)]*} • [20^

a)

that

a

F.j (t, aQ, 0 ) = F2 (tq, aQ,

functions with respect to from zero.

0 ) =0

t, a

at

F 1, F2

tq ^(a1 )] " 1

and the Jacobian

t = tq, a = a , e =

a jk ^ a 1 ) in (5 .13)

= a1 (0 ), are such

of these two 0

is different

Consequently, Equations (5-13) have two real distinct solutions

t = t(e) a = a(e)

analytic in

£

at

s = 0

and

For p = 1, take a purely imaginary, b = 1 and for p = 0 , take a pure­ ly imaginary, b = - i, and apply the same reasoning. The reasoning for the proof of (ii) of the theorem is exactly the same as above except that we do not have to apply Lemma (4.2) but apply the implicit function theorem directly to the functions F^, F2 in Equations (5• 13 )• The two t ! s obtained are obviously such that one I t has a real part positive and the other has a real part negative. There­ fore, the theorem is proved. REMARK (5 . 1 )• In the statement of Theorem (5*2) some cases have beenexcluded, in particular, the case where H(s, a^ ) 4 0 and the other expression in Theorem (5*2) is equal to zero. Reasoning as in the proof of (i) Of Theorem (5 *2 ), one sees that two of the remaining characteristic roots are still purely imaginary, but one cannot decide using the above method whether the solutions are bounded or -unbounded. The other cases may be treated by going to higher approximations in the method. Theorem (5*2) gives some insight into the behavior of the lutions of (4.1 ) near the resonance point ^ ( 0 ) - cr2(o) = sou. More

so­

78

HALE

specifically, if cr1, ..., cjn are fixed and say independent of e, if the conditions of Theorem (5.2) are satisfied and H(s, ) 4 °> then all of the AC solutions of (4.1 ) are bounded for s sufficiently small o along the curve a2 = a1 - sa> + s a in the (cr2, e) plane if the dis­ criminant D(a) = [Or(s, a^, cr)]2+ H(s, ) > o and some solutions are unbounded along this curve for every s 4 0 if D( 0, then D(cx) > 0 for every a and the AC solutions of (4.1 ) are bounded along every curve or2 = - so> + s2 0 for e = 0 implies that the AC solutions of (4.1 ) are bounded in a sufficiently small neighborhood of the point (a1 -so), 0 ) in the (a2, e ) plane: (ii) H(s, a1 ) < 0 for s = 0 implies that some of the AC solutions of (4.1 ) are unbounded in every neighborhood of the point (a1 - sa>, 0 ) of the (o2 , e) plane. REMARK (5 .2 ). Theorems (5 .2 ) and (5*3) deal only with the case a^(o) - ct2(0) = so). The same type of analysiscould be used to discuss the resonance points cr^o) + cj2(o) =s o d . The essential element of the argument was (5.11) and Lemma (4.2). It should be clear that a statement similar to Lemma (4.2) is true for the more general situation where crj(°) + ak (0) = nj^o), n an integer or zero, j, k e T1, crj(°) t

A

2k

v2 = t - so>, vk = ak, k = 3, 4,

,

52?)(oi

qnd H(s, a1) > 0 for all s if ^(a1 ) 4 0* Actually, all that we have required to obtain boundedness at ai ” a2 = Sa) ^ik = ^k19 cp2k = 9 ^2 ^ k = 1> 2' •••> n# To obtain boundedness at all of the points crj - cr^ = so), j ^ k, j, k = 1, 2, ..., n, the matrix $ must be symmetric. In case $ is not symmetric, thfen some solution may not be bounded for an e 4 0. For example, consider the system (4.1), (4.2) with [i = n = 2 , Y = o, c.j = a2(o), a1 independent of e, 2p

COS

t 2

COS

t

$ = 2q cos t 2r cos t Prom (5-12 ), a|2 ^(a1 ) = P(p + r), a^^(o1 ) = p q(p + r ), p = a“1(4a^ - 1 )”1 for £ = 0. Therefore, from Theorem 5-3 the AC solutions are bounded in

80

HALE

a neighborhood of the point (c^, 0) in the (o^, e) plane for e sufficiently small if p + r ^ o, q > o, and some solutions are unbounded in every neighborhood of this point if p + r ^ o , q < o. As another example, consider system Y = 0, c^ - o2 (o) = 1, or1 independent of e,

(4.1 ), (4,2) with n = n = 2,

2p COS t 2 COS 2t $ = 2q cos 2t 2r cos 2t From (5.12), = p( 1, q > 0, and some solutions are unbounded in every neighborhood of this point if p 4 °> a1 > 1, q < 0. The following theorem concerns the behavior of the solutions of (4.1 ), (4.2) with |i = n when one of the a*s approaches zero as e approaches zero. THEOREM (5-4). Consider the system of differential equations of order 2n, yn = Dy = e 0 (t )y + £ Y(t)y*

(5.15)

where £ > 0 , D = diag(ecr^, a2 , cr^), 0 Y are n x n real matrices whose elements cp^j(t), tj_j(t) are periodic in t of period T = 2it/a), L-integrable in [0, T] and have mean value zero; $ = (4>jk )> * = 3> k = 2> where 4^, 'fjk are matrices of the same dimension and (5 .1 6 )

*jk(-t) = ( - 1 )k+J'oJk(t), *j k ( -t ) = ( - 1 )k+ j+ 1f j k (t), j> k = 1, 2 . If*

(5 .it) 2ak 1 0,

^

-

2*

**■>

CTj + ak 1 0, j / k (mod

id ),

j = k, j, k =

2,

3,

[if the a. are analytic functions of £ at e = 0, then in (5 -1 7 ) is replaced by ^(o)], then there exists an eq > 0 such that all of the absolutely

n ,

81

LINEAR PERIODIC SYSTEMS

continuous (AC) solutions of (5*15) are bounded for 0 < e < eQ (notice this is not true for e = o). Furthermore, the characteristic exponents of (5-15) are analytic functions of nTb at 8 = 0. PROOF. By using an elementary argument, we show that there are 2n -2linearly independent bounded AC solutions of (5*51 )• This same result could also be proved by the method of successive approximations. The characteristic exponents are only determined up to a multiple of col, but without loss of generality we may assume that the k = 3, 4, •••, 2n are such that T2 j-1 ^0^ = 1(7y T2j^0^ = “ j = 2, 3, ..., n. From the Floquet theory condition (5*17) assures us that these functions ^(e), k = 3> 4, ..., 2n, are analytic in e at 8 = 0 and T2j—1^^ — (5 -1 9 )

J —

y 3> •••j n;

4 0, t. 4 Tk(mod coi), j 4 k, j, k = 3, 4, 0
there exist, from (5 .1 6 ), a nonsingular matrix P such that Py(-t) is also a solution of (5• 15 )- This, together with (5 .1 8 ) implies that each t^, k = 3 , 4, ..., 2n, is purely imaginary andthis obviously implies the existence of 2n - 2 linearly independent solutions of (5 •1 5 )• It does not seem possible to obtain the other two characteristic exponents by such an elementary argument so we apply the above method of successive approximation. The transformation of variables y - = (2ia . )-1(Z2j- 1 + z2j^ = 2_1^z2j-1 “ = 2> •••, n, y, = zt, yj = = -ie, leads to the equivalent system of first order equations (5-19) where 1, 2,

z1 = Az + \C(t, x)z A = diag(o, 0, ia2 , - l a 2 , 2n,

ion, - i. If x^r ^(t) has the Fourier series (5 .2 5 ) where that

x(r)( t ) . Z B < r ) eM (r )

is a

2n

dimensional column vector, then we show by induction

LINEAR PERIODIC SYSTEMS

33

4 r) = col (Bjfr), i B ^ }, i B ^ \ B ^ }), k = 0 , + 1 , + 2 , (5 . 2 6 ) P = 0, 1, 2, ..• •

;

(I*) (I*)(l*) where B^ 1 , B£ 2 are real scalars, ^ 3 a 2|i “ 2 dimensional real vector and B^J^ is a 2 (n - \±) dimensional real vector. The assertion is clearly true for r = 0 . Assume that it is true for r = o, 1 , 2 , ..., v - 1 . Then from the definition of the c in (5 .2 0 ) and the assumptions on the c p \ | r o f the theorem, it is clear that C(t)x^(t) ~ z D^p ^eikait where (5.27)

D J = 2> 3, . n, for |s| sufficiently small. Therefore, |pj| = 1, j = 3, 4, ..., 2n, and Bj = 1, |Aj| < 1, j = 2, 3 , ..., n. Since Hj'=1 Bj = 1, B1 = 1 and

f(p) = n (p2- 2Ajp+1) * n

(6.4)

j=i

85

LINEAR PERIODIC SYSTEMS

The remaining characteristic multipliers p2 will have absolute value less than one if and only if |A1| < 1 and have absolute value greater than one if and only If |A| > 1. Consequently, the transition curves in the (cr.j, s )-plane from stability to instability will occur when A 1 = + 1 or A 1 = - 1 . But this implies that p1 + p2 = + 2 or p1 + p 2 = -2, I.e., there Is a periodic solution of (6 . 1 ) of period x or 2 jt . These comments are generalizations of the well known facts for the Mathieu equations [1 1 ]. The purpose of the present section is to determine the nature of these transition curves for system (6 . 1 ) satisfying condition (6 .3 ). To obtain these curves transform system (6 .1 ) into an equiva­ lent system [see (3 *1 2 )]. = Az + ie(e2it + e"2it)Dz

(6.5) where

- 1CT

A = diagd^, - ic^,

d 3k

(6 .6 )

and

d jk j, k = 1 , 2,

" d jk

" d jk

The alternative method of successive approximations is now applied to the auxiliary system Bz + i£(e2lt + e 2lt)Dz

(6 .7 )

where B = dlag(im, - im, ia0 - ict_, iu . - icr ), m integer, and the zeroth approximation is given by

a positive

(6 .8 )

o)

z(0) = x(0) = (eirat, be-imt, o,

and b is a real number. Prom Lemma ( k * 2 ) , we know that the functions Vj obtained from the successive approximations are purely imaginary. Con­ sequently, the determining equations for e 4 0 are equivalent to the equations m - el [V1 (m,

, b, e)] =

(6 .9 ) I [V1 (m,

c r1 ,

b, e)] + I [V2 (m,

b, e)]

=

0

.

Since m is fixed, equations (6 .9 ) may be used to determine cr1 and b as functions of e for |e| sufficiently small and the corresponding

HALE

86

function a1 = ^(s) will be a transition curve in the (o ^ e)-plane near the point e = 0 , a1 = m. Therefore, the only thing that remains to be done is to find the first nonzero terms of V 1 and V2 containing b and solve equations (6 .9 ). ) First, it is clear that Xj(£ '(t), j = i, 2 , £,

contains only terms of the form ..., + £. Let

xj

=

V

[ a (^

/ 1

for all

ei(m+2P)'t^ e-i(m+2p)t^ p = 0 , + 1 ,

+z

(i) ( t )

2 n,

+

PJJ

0{Pl). JJ

,

P=-i

(6 .1 0 ) for all

j = 1 , 2 , ..., 2 n £,

) a'( £ i,

where

P> J

(£ ) • P> J

cj;

are constants.

We now show by induction that

c(i) - b aP(i) c(*} Cp , 2 j ,2j-1' P» 2 j - 1 = ba(*} J?>2j’

n;

j = 1, 2,

(6 . 1 1 ) + £; £ = 0, *[, 2,

P = 0 , + 1,

m - 1

(£ ) where the constants aj;(£ ) a l 0 . are independent of b. For k = 0 , ), cj ?;0 ) the constants a'( ;0^ . satisfy (6 . 1 1 ) for all p, j. If the constants P> J

P> J

a^L • satisfy (6 .1 1 ) for all p, j and £ = P* J P> J 0 ) 0 then the terms ; Sg in (3 .6 ) are given by

0,

( ()

s(i} = i £

d1k

k=1

Y

(ap^2k-i +

4 :* '

1 , 2 , ..., v - 1 ,

’

p=-1,1

(6 .1 2 ) S(£) - ib"1 VZ, d l k VZ, (c(i_l) + c P(i_l))S(^ b2 ■ 1D P>2k-1 ,2k ; - " b1 k=1 p=-i,1 £=

( 0)

1

y

>

2, . a . , V

y

( 0)

if v < m and 1 are independent of b. From the definition of x( .v) in (3.7), it now follows immediately that the coefficients J

cM satisfy (6 .1 1 ) for all P> J for all v < m.

P *J

v < m.

Therefore, (6.12) is also satisfied

87

LINEAR PERIODIC SYSTEMS Prom (3 .6 ) and (6 .1 1 ),

s (m)

1

=

i

V

d

I"

y

,

( m - 1 ),( m - 1 ) s

L lk [ 2, (aP,2k-l + ®j),2k '

„ (m -l)

+-(m-i),2k-i+

1 ) - 1 ),2 k

] J

(6 . 1 3 ) s{"> + s then it is a simple calculation to show that

z

n

n

(6.14,) /m

m-1 11

k i> ••*,lcm-ii=1 i = 0

V where

a.

= m

2dk k ak Kr\e+i Ki+1 (m-2i-2 )2- s> least one djk is 4 °> and > M, k = 2 , 3 , ..., n, M a positive integer. There are unbounded solutions of (6 .1 ) in every neighborhood of the point (m, 0 ) in the (cr1,e )-plane for every positive integer m < M + 2. PROOF. djk is 4 0.

In this case (6.14) implies

7m 4 0

if at least one

In case 7m = 0 , then there still may be two distinct solutions to the determining equations (6 .9 ). It should be clear from the proof of relations (6 . 1 1 ) and (6 .1 2 ) that If the first S$.^ which de^ pends on b occurs In the mth 1 approximation, then relations (6 .1 2 ) are valid for v = m 1 - 1 , and (6 .1 3 ), (6.14) will hold for m = m 1 and the determining equations (6 .1 5 ) for m = m 1 will have two different solutions. Therefore, Theorem (6 .1 ) will hold for m = m1• In case is in­ dependent of b for every 1 , then the determining equations (6 .9 ) will always have only one solution = 0 ^ (e) which is independent of b. Therefore, there are two linearlyindependent periodic solutions along this curve and the AC solutionsare bounded in a neighborhood of the point (m, 0 ) in the (c^, e)-plane. BIBLIOGRAPHY [1 ] BAILEY, H. R. and CESARI, L., Boundedness of solutions of linear differential systems with periodic coefficients, Archive Rat. Mech. Ana. 3(1958), 246-271. [2 ] BAILEY, H. R. and GAMBILL, R. A., On stability of periodic solutions of weakly nonlinear differential systems, Journ. Math. Mech. 6 (1 9 5 7 ), 6 5 5 -6 6 8 . [3] CESARI, L., (a) Sulla stabilita delle soluzioni del sistemi di equazionl differenziali linear! a coefficient! periodic!, Atti. Accad. Italia, Mem. Cl. Fis. Mat. Nat. (6) 1 1 (1 940), 6 3 3 -6 9 2 ; (b) Asymptotic behavior and stability problems In ordinary differ­ ential equations, Ergbn. d. Math. N.F. Heft 1 6 , Springer 1959;

LINEAR PERIODIC SYSTEMS

89

(c) Existence theorems for nonlinear Lipschitzian differential systems and fixed point theorems, this Study. [4]

CESARI, L. and HALE, J. K., (a) Second order linear differential systems with periodic L-integrable coefficients, Riv. Mat. Univ. Parma 5(1954 ), 5 5 -6 1 ; (b) A new sufficient condition for periodic solutions of weakly nonlinear differential systems, Proc. Amer. Math. Soc. 8(1957), 757-764.

[5 ]

GAMBILL, R. A., (a) Stability criteria for linear differential systems with periodic coefficients, Riv. Mat. Univ. Parma 5(1954), 1 6 9 -1 8 1 ; (b) Criteria for parametric instability for linear differ­ ential systems with periodic coefficients, Riv. Mat. Univ. Parma. 6(1 955 ), 37-43.

[6 ]

GAMBILL, R. A. and HALE, J. K., Subharmonic and ultraharmonic solutions for weakly nonlinear differential systems, Journ. Rat. Mech. Ana. 5(1956), 353-394.

[7]

GOLOMB, M., Expansion and boundedness theorems for solutions of linear differential systems with periodic or almost periodic co­ efficients, Archive Rat. Mech. Anal. 2 (1 9 5 8 ), 284-308.

[8 ]

HAACKE, W., Uber die Stabilitat eines Systems von gewohnlichen linearen Differentialgleichungen zweiter Ordnung mit periodischen Koeffizienten, die von Parametern abhangen (I und II) Math. Z. 56(1952), 65-79, (1952), 34-45.

[9]

HAUPT, 0., Uber lineare homogene Differentialgleichungen zweiter Ordnung mit periodischen Koeffizieten, Math. Ann. 79(1 91 9)^ 2 7 S.

[1 0 ] HALE, J. K., (a) Evaluations concerning products of exponential and periodic functions, Riv. Mat. Univ. Parma, 5(1954), 6 3 -8 1 ; (b) On boundedness of the solutions of linear differential systems with periodic coefficients, Riv. Mat. Univ. Parma, 5 (1 9 5 4 ), 137-167; (c) Periodic solutions of nonlinear systems of differential equations, Riv. Mat. Univ. Parma, 5 (1 954), 2 8 1 -3 1 1 ; (d) Sufficient conditions for the existence of periodic solutions of systems of weakly nonlinear first and second order differential equations, Journ. Math. Mech. 2 (1 9 5 8 ), 1 6 3 -1 7 2 ; (e) Linear systems of first and second order differential equations with periodic coefficients, 111. Journ. Math. 2, 5 86 -5 9 1 , (1 9 5 8 ); (f) A short proof of a boundedness theorem for linear differential systems with periodic coefficients, Archive Rat. Mech. Ana. 2 (1959), 429-434; (g; On the stability of periodic solutions of weakly nonlinear periodic and autonomous differential systems. This Study. [11 ] McLACHLAN, N. W., Theory and application of Mathieu functions, Clarendon Press, Oxford, 1 947• [1 2 ] METTLER, E., Allegemelne Theorie der Stabilitat erzwungener Schwingungen elasticher Korper, Ing. Arch. 1 7 (1 949), 41 8 -4 4 9 . [1 3 ] MOSER, J., New aspects in the theory of stability of Hamiltonian systems, Comm. Pure Appl. Math. 9(1958), 8 1 -1 1 4. [14]

GEL!FAMD, J. M. and LIDSKII, V. B., On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients, Uspehi Mat. Nauk (N.S.), 1 0 (1 9 5 5 ), 3-40; Am. Math. Soc. Trans. (2 ) 8 (1 9 5 8 ), 1 4 3 -1 8 2 .

v.

ON THE STABILITY OF PERIODIC SOLUTIONS OF WEAKLY NONLINEAR PERIODIC AND AUTONOMOUS DIFFERENTIAL SYSTEMS J. K. Hale

§1.

INTRODUCTION

Consider the system of differential equations (1 .1 )

x" + Ax = e f(x, x1, s, t)

where e is a small real parameter, x = (x^ ..., xn ), A = diag(a2,...,

°> - a CT1 sin t, o, .

.

o)dt < o ,

j = 1, 2, ..., n

,

then this periodic solution of (1.4) is asymptotically orbitally stable in [tQ + oo], 0 < e < e Q, e Q > 0. This result was obtained Dy A. Andronov and A. Witt [1] for n = 2, by evaluating the characteristic polynomial of a matrix of order four. The result above for arbitrary n is proved

n,

HALE

9^

by the above method and the properties of the characteristic exponents without evaluating any determinants. In §3 and §4, the nonlinear functions in the differential equa­ tions are assumed to satisfy a certain condition of analyticity. It is shown in §5 that it is sufficient to require that these functions possess continuous second derivatives. Throughout the present paper, the following notations are used: i = V-1; a is the complex conjugate of a; R(a) is the real part of a; 1(a) is the Imaginary part of a. Also eQ is used to denote a sufficiently small positive number. §2.

DEFINITIONS OF STABILITY

Consider a system of differential equations (2.1)

x ! = q(x, t )

where q = (q^, ..., qn ) is a continuous vector function of the real vector x = (x1, ..., x ) and the time t and suppose that (2 .1 ) satisfies a uniqueness condition in a region R of the (n+1)-dimensional (x, t) space. A solution x (t) = x(t, x , t ), tQ < t < 00, of (2 . 1 ) with x*(tQ ) = x is said to belong to R if [x (t), t] is an interior * point of R for every t ^ tQ. A solution x (t) = x(t; xQ, tQ)is said to be asymptotically stable (to the right) ifthere exists a 5 > 0 such that (i) every solution x(t; x t Q ) exists for all t > tQ and belongs to R if ||x1 - xQ|| < 6; (ii) ||x(t; x.,, tQ )-x(tj xQ,t0)|| -- > 0 as t -- > 00 if ||Xl - xQ|| < 5, where ||x|| = Zjs=1 |xj|. Suppose now that .q(x, t) in (2 . 1 ) is periodic in t and the solution x (t) is also periodic and the second partial derivatives of the functions qy j = 1, 2, ..., n, with respect to xk, k = 1 ,2, ..., n, in the region R of (x, t) space.Then the linear variational equation * for the solution x is given by (2.2) where

y' = Q(t)y, Q(t)= (dqj/dxk )x=x#, j, k = 1, Q(t)

2,

n,

is periodic in t. We need the following known theorem.

THEOREM 2.1. If q(x, t) satisfies the above conditions and the characteristic exponents of (2 .2 ) have negative real parts, then the periodic solution x (t) of (2 .1 ) is asymptotically stable to the right ([Liapunov, 9]> see also [5, p« 3 1 ^])«

STABILITY OF PERIODIC SOLUTIONS

95

If system (2.1) is autonomous, i.e., x 1 = q(x), then any periodic solution x (t) defines a closed curve C in the n-dimensional x-space, En* If d(x, c) denotes the distance of a point x in En from the curve C in En, then the solution x (t) is said to be asymptotically orbitally stable (to the right) provided, given e > o, there exists a 5 > 0 such that every solution x(t) with d(x(tQ ), C) < 5 for some tQ implies d(x(t), C) < e for t ^ tQ and d(x(t), C) — > o as t -- > + oo* THEOREM 2.2. If x*(t) is a periodic solution of the autonomous system x ! = q(x) where each component of qj has continuous second derivatives in a region R of E^, and n - 1 of the characteristic ex­ ponents of the linear variarional equation for x (t) have negative real parts, then the solution x* ✓ (t)\ is asymptotically stable to the right. [5, p. 323]. §3.

PERIODIC DIFFERENTIAL SYSTEMS

Consider the periodic system of equations + tTjXj = e fj(x, x', e, t), j = 1, 2,

Xj +

IX ,

(3.1 ) xl + PjXj = e fj(x, x 1, e, t), j = ii + 1, . . n; tQ £ t < + °o

,

where s > 0is a real parameter, x = *(x.j, ..., xn ), x* = (xj, ...,x^), each fj is real and analytic in a neighborhood U of the origin of the (x, x 1, e) space; the power series expansion of each f . in U has co­ efficients periodic in t of period T = 2*/cd, L-integrable in [0, T], and there exists a function T}(t), L-integrable in [0, T], such that (3.2)

x *' e ’

for tic j= j=

in U. The parameters a^, o* ar^ real, analy­ 0 < e < eQ, 6Q > 0; 7j(e) = (4tf- a 2. )* > 0, b < s0. Furthermore, we define numbers p., by the relations

all (x, x 1, e) functions of e, 1 ,2, |i, 0 < 1 ,2, .••, n + n, p2 j - i

=

a j

+

J* = 1' 2> •••> n

±7^

=

pzy

J

=

1> 2>

•••>

(3 . 3 ) j = ii+l,

n

•

p^+j

>

=

-

Pj

>

96

HALE

The system of periodic differential equations (3 • 1 ) satisfying all of the above conditions shall be referred to as system (3*1 )• Notice the vectors x, x 1 used above do not have the same dimension. Suppose there exists a real periodic solution, X(s, t) = Xn ), of system (3«1 ), analytic in e for 0 £ s < e Q, eQ > 0

(X1 , with Xj0

=

x.(0, t)

=

aj cos (r^.cut

+ , m = m 1 ...mv. Many methods have been given for obtaining such periodic solutions (for example, see [4a, Section 8 .5 ] and [4b]). (3.1) n

(3.5) then the linear variational equation for yj

is

(3 .6 ) > •••7 n

j

where (3 .7 )

f L = f ix (x, X 1, e, t), f? , = f ix, (X, X', e, t) Jxk Jxk Jxk JX k

and X = (X1, Xn ), X' = (X], yk, y£ In (3 .7 ) is periodic In t

X^). Each of the coefficients of of period mT = 2«m/a>.

The transformation of variables “ (2l7j)_ (z2j-1 + Z2j}' yj = (2±V (3.8)

2j-1Z2j-1 (p:

STABILITY OF PERIODIC SOLUTIONS

97

leads to the equivalent system of first order equations (3.9) where C =

z! = A(e)z + e C(t, s)z A = diag(p.,, ..., Pn+|i)> the (Cjk ),

j, k = 1 , 2,

p.

are defined by (3*3), and

n + n ,

C2j-1,2k-1 = ^2lrk^

+ P2k-1fJXk ^ °2j-1,2k = " °2j-1,2k-1

’

k = 1, 2, •••, M-j c2j-l,n+h =

h = ti + 1,

nj C2j-1,/ = " c2 j 1 '

(3»T0)

2’ *•*’ n + ^

j = 1j2, ..., p.j

cn+h,2k-1 “ ^2i7k^

^ ^ k + p2k-l^hx-^* °}i+h,2k “ ” °[i+h,2k-l

,

k = 1 , 2 , •••, M-j * cu+h,n+i " ^hx^' 1 ~ ^ + ] >

n; h - n + 1,

n

,

* * f . , f . , are defined by (3*7 )• jxk Jxk In the sequel, we need the following known results.

and the functions

LEMMA 3«1• (J* K. Hale [8, Theorem 2.1]). Consider the system of differential equations (3 •9 X with C(t +mT, e) = C(t, e), T = 2it/o), m = m^.-m .Suppose that co* =co/m, pj(°) - Pk^0^ =

mjk 311

or zero,

i , k = 1; 2, ..., p;

(3.11 ) pj(°) 4 Pk^0^ ^mod

j=

2> •••'

and define the p x p matrix diag(d1, •••, ^p)> H = ( h ^ )> where d. = lim «

e —> o

(3 .1 2 )

G = H - D, D = k = 1 , 2 , •••, p,

£*”1 [p.(e) - p.(o)] J

, h.n jk, = (mT)

k = p + 1 , ..., n + |i,

J

pT

o

J

i(mk -m. )o)'t e J

°)dt, j, k = 1, 2,

p.

HALE

98

If \ is a simple characteristic root of the matrix Or and the corresponding eigenvector has no zero com­ ponents, there is a characteristic exponent x(e) of (3*9) which Is analytic in e for 0 < s < s , and (3.13)

t(e) = p i (0) + e

\Q+

0(e2 )

.

COROLLARY' 3*1 • Consider the system of differential equations (3• 9) with C(t + mT, e) = C(t, s), T = 2 it/co, m = m 1 ...mv* If cjd1 = co/m and pj(°) ? p ^ 0)(mod a) 1 i ), k ^ j, k = 1 , 2 ,

(3*14)

..., n + ia ,

where j is a fixed integer, 1 < j < n + |i, there is a characteristic exponent of (3 .9 ); t* = t«(e), J analytic in e for 0 < a < e , with T (3*15)

“ pj ^ +

e T ""1

J

°^dt

o

J

+

°(e2) *

LHVMA (3 *2 ) (J. K. Hale, [8 , Theorem 3-1]). Consider the system of differential equations (3 *9 ) with C(t + mT, e) = C(t, e ), T = 2 */a), m = m 1 ...my. Suppose cDf = a)/m, j—i "" p2 j■, p2 j—1 ^^^ ^ p£j(0 ) (mod a)1 1 ), ^ = 111jk^1' rajk 8X1(1 integer or zero,

P2 j- 1 (0) = P2 k- 1

j, k = 1 , 2 , ..., p ;

(3 .1 6 ) pj(°) f Pk(°) (raod

2,

J=

and define the p x pmatrix G = D = diag(d1, •••> ^p)> = ^jk^ where d.

J

=

lim

e _> o

e " 1 [p

. (e)

J~

-

p

.

,(o )],

J

2 p; k = 2 p + 1 ,

H - D, =

j

= 1,

•••>

2,

...,

P>

p

,

(3.17)

h.k = (mT)-

fP11 czj_uzls_,(t, 0) eitm^-tn .. )a>*tdt j, k - 1, 2, •••, p •

n + n,

'

STABILITY OF PERIODIC SOLUTIONS

99

If \ Q is a simple characteristic root of the matrix G and the corresponding eigenvector has no zero com­ ponents, then there are two characteristic exponents t, t of (3«9) analytic in e for 0 < s < e Q and (3 .1 8 )

t(e) = Pl(0) + e X0 + 0(e2 ) .

From the Floquet theory [5, p. 78] the characteristic exponents of system (3*9) are obtained by finding the eigenvalues of a matrix of order 2n. We shall illustrate by means of a few theorems how Lemmas 3«1 and 3*2 can be of assitance in reducing the order of this determinant. Morespecifically, if there are p of the characteristic exponents close to the origin inthe complex plane, then they will be obtained from the eigenvalues of adeterminant of order p. If the remaining characteristic exponents are Isolated for all e, 0 < e < eQ, then they will be ob­ tainable immediately from Corollary 3•1• If there are say 2pcharacter­ istic exponents which are two by two complex conjugate none ofwhich are close to the origin in the complex plane, and p are equal for e = 0, then they will be obtained from a determinant of order p. THEOREM (3*1 )• Consider the system (3*1 ) and suppose that (3*4) is a periodic solution of (3*1). Suppose s > 0, of1 = a2 ... = aq = 0, ofj > 0 , j= q + 1, ..., n; Pj > °> j = M- + 1>•••> n; all s, 0 < £ < eQ, and

P2 j - 1 85 1 kj “^ j = p2 j ^ 0^ 3 = 1 > 2,

. . . , v; v V>

where the functions under the integrals are evaluated at (XQ, X£, 0, t), ^2j—1 ” P2k-1 = nijk an integer or zero, j, k = 1, 2, ..., v, are distinct, have negative real parts and the corresponding eigenvectors have no zero components. PROOF. From Corollary 3-1, it follows immediately that the characteristic exponents Tj(e), j = 1, 2, ..., n + n, Tgj-i (°) = T2 j(°)

=

i [- aj(°) = i7j(o)], j = q + 1, ..., n; -rn+j^0 ^ = " pj^0^ J' = 11 + 1> ' " > n > have negative real parts for o £ e < sQ. Furthermore, from the same corollary and (3 .1 0 ),

mT

R(t 2j_-l ) = R(t2J) = e(2mT)'1

J

fjx'/V

0 , t)dt + o(e2 )

J

for j = v + 1, ..., q. Therefore, from (i) of the theorem, the real parts of these characteristic exponents are negative for 0 < e < sQ. The other characteristic exponents ..., t2v are determined from Lemma 3-1 for p = 2v. The matrix G of the theorem is precisely the matrix G of Lemma 3.1. Therefore, from (ii) and (3 .1 3 ), it follows that these charac­ teristic exponents also have negative real parts for ° < e < eQ. Theorem 2 . 1 now implies that Theorem 3 . 1 is true. REMARK 3*1 • Theorem 3*1 has been previously obtained by H. R. Bailey and R. A. Gambill [3] for the case where v = p. = q = n; i.e., for systems of second order equations, xV + ajxj = e ^j(xi> •••> xn 9 xl9 ***,xn,e

STABILITY OP PERIODIC SOLUTIONS

j = 1, 2, . n, where each a- = ctj(e ) satisfies the relation aj(o) = j = 1, 2, n. An interesting example is also given in [3] for v = ia = q. = n = 2 illustrating the application of this theorem. REMARK 3*2. If any one of the roots of |G - pi| = 0 in (ii) have a positive real part or if any of the functions in (i) are positive, then the periodic solution (3*4) of (3•1 ) is unstable in [tQ + »] for every e 4 °* THEOREM 3.2. Consider the system (3*1 ) and suppose that (3-4) with v = 1 is a periodic solution of (3*1). Suppose 6 > 0 and the or., £•, a,- satisfy J J J the conditions of Theorem 3*1 with v = 1; i.e., P1(0) = i kjjCD/m1 = p2(o),

( 3.21 )

(i)

pj(°) i Pk(°) (mod cDfi) j = 1, 2; k = 3, 4, ...,

n+

n

,

pj(°) i Pk(°) (mod a)1i) j 4 k, j, k = 3, 4, ...,

n+

\i

;

where cd* = a)/m1. The periodic is asymptotically stable to the if the following conditions are mT f fjx .(X0, X£, 0 , t)dt < 0 , where

(ii)

XQ, X^

solution (3*4) of (3-1) right for 0 < s £sQ satisfied: j = 1 , 2,

are defined by (3*4) for

The numbers A, B m-T (2m1T)A = J

v = 1;

defined by

fiXl,dt »

2

miT

r

™iT

(2miT)2B = ( J

f“ix .dt)

o

(3.22; (0)

J

+-0^(0 )ff 1xat] o

n^T

~1

q ;

flx cos 2a1(0)t dt -

J

n^T flx, sin 2o1(o)t dt

m. T +/ 1 f 1 o lim e“1 [cr^s) - cr^o)] ,

m, T

f 1x

o d1 =

sin

2a,(0)t

dt

, COS 1

2 a 1 (0 )dt j

HALE where all of the functions are evaluated at (XQ, X^, 0 , t), satisfy the property A 4 B, A2 i B, B > 0. REMARK 3*3* This theorem has been proved by H. R. Bailey and R. A. Gambill [3] and L. Mandelstam and N. Papalexi [1 0 ] for the case where q = v = |i = n =1 ; I.e., a second order equation x + a x = e f(x, x 1, s), where x, f are scalars and a = or(s), a ( o ) = ko)/m. PROOF. The system under consideration in Theorem 3 * 2 is a special case of the system in Theorem 1 for v = 1 . Condition (i) of Theorem 3 .2 , for j = 2 , 3 >•••, q Implies condition (i) of Theorem 1 is satisfied for v = 1 . For v = 1 , the characteristic equation of the matrix G in Theorem 3*1 is given by X 2 - 2 AX + B = 0> where A, B are given in (3 .2 2 ). The roots of this equation have negative real parts if and only if A > 0, A2 4 B, B > 0 which is assured by condition (i) and (ii) of Theorem 3*2. The condition A 4 B assures that the corre­ sponding eigenvectors of G have no zero components and Theorem 3*2 is proved. EXAMPLE.

Consider the system of equations

x" + x 1 = e(l - x2 -x2 )xj

+ e P cos t = e f 1(x1, x2, xj,x^, e,

t)

,

(3.23) x2 +

ff2x 2 = e ^1-

- xg)x^ = s f2 (xt, x2, x»,

x^, e, t)

,

where e > 0, p 0, a 2 4 m, m an integer. Applying the method of successive approximations [6], it is easy to see that (3 *2 3 ) has a periodic solution Xj(e, t) = x j(£* ^ + 2*)> j = 1, 2 , analytic in e for 0 < e £ eQ, eQ > 0, (3.24)

X.,(o, t) = a sin t,

where a > 2 Furthermore,

X2 (0, t) = 0

is the positive solution of the equation

, 1 - (S)2 + E = 0 .

2

a

2 Jt

j = 1 , 2 , and the numbers A, B of Theorem 3 . 2 are given by 4a = (2 -a2 ) 1 6 B = (2 - a2 )2 + 1 6 a2 - 9a2/4 and A 4 B, A2 4 B, B > 0 for every real a 4 0 . Therefore, from Theorem 3 .2 , the solution (3*24) of (3 .2 3 ) is asymptotically stable to the right for 0 < e < eQ.

STABILITY OF PERIODIC SOLUTIONS

103

THEOREM 3*3* Consider the system (3 . 1 ) and suppose that (3 .4 ) is a periodic solution of (3 * 1 )• Suppose e > 0, c*i

a2=

=

...

a.

0,

* of^ =

> 0, j

°, j = ji+ 1 ,..., n, for

Pj >

P2j - i ( ° ) = 1 kj £D/Tnj = pj ( ° )

^

= q +

1,

...,

all e, 0 < e < eQ, and J’ = 1' 2'

a ),i^

j

=

2,

...,

v; v £ q ; 2v;

k = 2 v + 1, ..., n + (3.25)

n,

|i ;

(°) J* P2j(°) (mod •••* T2 r are obtained from Lemma 3 *2 . The matrix M of the theorem is precisely the matrix G of Lemma 3«2 if the equations (3*9) are reordered so that A = diag(p2v+1, ..., &2V’ P1 * p2 v* P2 r+1 * ***' pn+ii^# Therefore, each of the eigenvalues ^j_v ° ? the matrix M determines two character­ istic exponents ^j-i' T2j = ^2 j-l' with T2 j- 1 = j( 0 ^ + e xj_v + 0(e), j = v + 1 , ..., r.The real parts of these characteristic exponents are also negative for 0 < e < eQ, and the theorem follows from Theorem 2 .1 . REMARK 3*3* For v = l in Theorem 3*3, the condition (ii) re­ ferred to can be replaced by condition (ii) of Theorem 3 .2 . This theorem should clarify the remark preceding Theorem 3 •1• In fact, this theorem is mentioned only to illustrate the procedure for cal­ culating the characteristic exponents of (3«9) which are two by two complex conjugate, r - v which are equal for 6 = 0 and none of which are close to the origin in the complex plane. §k.

AUTONOMOUS DIFFERENTIAL SYSTEMS

Consider the autonomous system of equations Xj + a 3Xl + ajXJ = e fj(x, x 1, e), j = 1, 2, ..., n , (*.1) J = ^ + 1> •••' n; to^ t < + 00 '

Xj + PjXj = £ * V X'

where e> 0 is a real parameter x = (x^ ..., xn ), x f = (xj, ..., x^), each fj is real and analytic in a neighborhood U of the origin of the (x, x T, e) space. The parameters of., p., 0 1 are real analytic functions J

J

y

Of B, 0 < £ < £q , eo ^j^£^ = ^^j "" ^j ^ ^0, j = 1, 2, «•«, M>, 0 < e £ eQ. Furthermore, we define numbers p., j = 1, 2, ..., n + \i, by the relations (4.2)

P2j-1 =

~ aj +

j ) = p2j* 3 =

•••>

j=n+l,

Pfi+j = ~ ^j* ..., n.

STABILITY OF PERIODIC SOLUTIONS

The system of autonomous differential equations (4.1 ) satisfying all of the above conditions shall be referred to as system (4.1 )» Notice that the vectors x, x* used above do not have the same dimension. Suppose there exists a real periodic solution, X(e, t) = (X-j, ..., Xn ), of system (4.1 ), analytic in e for 0 < s < sQ, sQ > 0 with X.Q s Xj(0 , t) = aj cos(rj o>t + cpj ), j = 1 , 2 , ..., v; v < m- , (4.3) Xjo 5 Xj^°' t) = °> 3 = v + 1* •••* n > where a -, cp • are real numbers, rn * = k./m., k., m. relatively prime J J J J J J J positive integers j = 1 , 2 , ..., v, and X(e, t + mT) = X(e, t), T = 2 jr/co, m = m 1 ...mv. Many methods have been given for obtaining such periodic solutions (for example, see [4a, Section 8 .5 ] and [4b]). As in the previous section, the linear variational equation associated with the solution X(e, t) of (4.1 ) is given by *

(4.4)

z! = A(b )z + e C(t, s)z

where A = diag(pi; ..., Pn+li)> the pj are defined by (4.2) and the matrix C = (Cj^)* j, k = 1 , 2 , ..., n + n, is given by (3 .1 0 ) with (4.5)

f]

= fj

(X, X«, e), fjx£ = fjx^ (X, X-, e)

and X = (X1, ..., Xn ), X 1 = (X], X^). The matrix fore, satisfies the relation C(t + mT, e) = C(t, s).

C(t, e),

there­

We wish to apply Lemmas 3 . 1 and 3* 2 to obtain sufficient con­ ditions for the asymptotic orbital stability of the periodic solution X(s, t) of(4.1 ). Before proceeding to the formal discussion, observe that one expects each of the given numbers a^ to satisfy a relation of the form cr. = **jaj + °(e), j = 2 , 3 , ..., v, to obtain a periodic solution of (4.1 ) of the form (4.3 )• The number co (also the numbers r., cp ., j = 1 , 2 , ..., v) is then determined as a function of e and the numbers a. so that (4.3) satisfies (4.1 ). In particular o> = a1 + o(e). As we know, one could just as well obtain as a function of od and e simply by making a convenient change of scale in the variable t. In the present context, itis convenient to assume that the latter alternative is chosen, namely, is independent of e and the numbers crj , j = 1 , 2 , ..., v, are convenient functions of e so that (4.3) satisfies (4.1).

HALE

Another observation to be made is that the linear variational equation (4.4) always has a periodic solution of period mT [5, p. 322] and, therefore, one of the characteristic exponents (since they are only determined up to a multiple of coi/m) must be ko>i/m, where k is an integer or zero. THEOREM 4.1. Consider the system (4 . 1 ) and suppose that (4.3) is a periodic solution of (4.1). Suppose e > 0, cr1 = a2 = ... = arq = 0, a . > 0, j = q + 1, ..., [i; > 0 , j = n + 1 , ..., n; for all e,

0 £ e < e0> eo > °'

811(1

p2 j_1 (0 ) = i ICjco/mj = p2 j(0 ), j = 1 , 2 , ..., v, v £

(4.6)

,

pj(°) ? pk(°) (mod

J = 1> 2> •••* 2v; k = 2v + 1, ..., n + n ;

pj(°) 4 Pk^0 ^ (mod a)IjL)^

iJ

j, k = 2v+ 1,

..., n

+n

,

where cd! = ai/m, m = m 1 ...mv, and the are defined by (4.2). The periodic solution (4 .3 ) of(4.1 ) is asymptotically orbitally stable to the right, 0 < e 0 , if the following conditions are satisfied:

(i)

f

mT f j x , (XQ, X£, 0) dt < 0, j = v + 1,

o

where XQ = (XlQ, ..., XnQ), X£ defined by (4.3); (ii)

q

,

^

= (X*Q, ..., X»Q ) are

2v - 1 of the eigenvalues of the 2 v x 2 v matrix G = H - D, D = diag(d1 ...d2y), H = (hjk), j, k =

1, 2, d 2j._1 = i

•••, 2v, 1 1 m e ~ 1 [Oj(e) -

PT

1 h 2 j - l ’2 k-l

“ iSr

a j ( o ) ],d 2j._1 = - d 2 j,

-1

J

j =

1,

2,v

)cu't f jxk + f j x £ ]e

dt

’

o

1 h2 j - l , 2 k = imT / o

pT

[(1°k)

-1

1 (mk i +mii f jxk ~ f j x ' l e

h2 j , 2 k - 1 = ® 2 j - 1 , 2 k ’ h2 j , 2 k = ^ 2 j - 1 , 2 k - 1 5

) ®' t dt ’

k = 1, 2,

...,

v

,

;

STABILITY OF PERIODIC SOLUTIONS

where the functions under the integrals are evaluated at (XQ, X£, 0), P2j_i(°) ~ p2k-1 ^ = “jV0'1' mjk ^ integer or zero, j, k = 1, 2, v, are distinct, have negative real parts and the corresponding eigen­ vectors have no zero components. PROOF. The proof of this theorem is exactly the same as the proof of Theorem 3*1 if we observe that one of the characteristic ex­ ponents of (4.4) being a multiple of coi/m is equivalent to one of the roots p of the equation |G - pl| = 0, being zero. One then applies Theorem2.2 rather than Theorem 2.1 to complete the proof. THEOREM 4.2. Consider the system (4.1 ) and suppose that (4.3) with v = 1 is a periodic solution of (4.1 ). If 6 > 0 and the a., pj, tr. satisfy the conditions of Theorem 4.1 with v = 1, i.e., P 1 (o )

(4.8)

= ico =

p 2 (o),

t Pk^0) (raod 0)1

j=

pj(°) i Pfc(0) (mod °>1), j i

1, 2;k = 3} 4, ..., k,k = 3, 4, ...,

n +

n ;

n + n ;

the periodic solution (4.3) with v = 1 of (4.1) is asymptotically orbitally stable to the right, o < s < sQ, 60 > °>

if T

(4.9)

J*

o

fjxt (Xq* Xq, 0 )dt < 0 , j = 1, 2 , ••., q ^

j

where XQ = (XlQ, XnQ), X* - (XjQ, ..., X«Q ), XlQ = a cos (cd t + q>), XkQ = 0, k = 2 , 3, •• n; a, cp real numbers. REMARK 4.1. In his thesis at Purdue University, E. W. Thompson [12] has obtained this same result by another method. The method used by E. Thompson was to first eliminate the zero root of the variational equa­ tions and then apply some known results of H. R. Bailey and L. Cesari [2]. Also, A. Andronov and A. Witt [1], (see also [11, p. 153]) have ob­ tained this result for the case v = 1 , q = n = n = 2; i.e., for a system of two second order equations with no "large" damping terms. The method used by A. Andronov and A. Witt was to evaluate the characteristic poly­ nomial of a matrix of order four. As we shall see in the proof of this

HALE

theorem, no determinants are required. PROOF. Exactly as In the proof of Theorem 4.1, we apply Corollary 3«1 to obtain the real parts of the characteristic exponents, T2j -1 = T2j* 3 =

•••> ^

^ = ^+

, •••> n

as

T

fe IO

R(T2j-i)= - r 1 + 2T

v * (x°' xi > o)dt + o(e2)' j - ^ 2> •••> »* *

(4.10) T R(V j ) = ' Pj +

We may assume that

fI t1

fjXj(Xo' Xo' °)dt +

= o,

t2

= 0(e)

j = U +

3=1

’

and we know [5, p. 81 ] that

n+ |i T X

n

T|!n TJ = v/ tr (A(e) + e C(t, e))dt = °

3=1

T ^ ji_ + e

c*j -

^ j=n+1

ri

/O ( 1i-1

J

J-1

From (4.10) and the fact that

* i ^J > • J=n+1

t1

= o,

we have

T R(t2)T = e J

f1x, (XQ, X£, 0)dt + 0(e2) ,

and Theorem 4.2 now follows immediately. One could prove this theorem in another way by evaluating directly the eigenvalues of the 2 x 2 matrix G in Theorem 4.1. EXAMPLE.

Consider the system of equations

x" + cr2x1 - e(l - x2 - x2)x.| = e f(x^ x Q, x£, s) (4.11) x2 + a2X2 " £(1 - X2 - x2)x^ = 8 g(Xl, x», x2, e), e > 0 , a a 2 are analytic functions of e at e = 0 with cr2(o) = */2a1(o), f(-x1, x2, x2, e) = - f (x«|, x2, x£, e), g(x^ x], -x2, e) = - g(x^ x», x2, are any analytic functions in a neighborhood U of the origin in

STABILITY OF PERIODIC SOLUTIONS

(x1, x2, xj, x^, s) space. It is known [7, P« 3 0 0 ] that there are two periodic solutions of this equation of the form (i) (ii)

x 1

= a1 sin(t +

cp)

+ o(e),

x2

= 0, a1 = 2 + o(e), a = 1 + o(e)

x1 = 0, x2 = a2 s i n ^ t + cp) + o(e), a2 = 2 + 0(e), a2 = J

2

+ 0 (e)

of periods 2k and */2*, respectively (we are choosing o2 as func­ tions of € rather than choose the period). For (i) conditions (4.9) for j = 1, 2, are equivalent to

o

o

It follows from Theorem 4.2 that (i) Is asymptotically orbitally stable to the right, 0 < s < eQ, eQ > 0. Similarly, one shows that (ii) Is also asymptotically orbitally stable to the right for e sufficiently small. THEOREM 4.3* Consider the system (4.1) and suppose that (4.3) is a periodic solution of (4.1 ). Suppose e > 0 , o?1 == ... = a = 0 , cfj > 0, j = q + 1 , ..., n; s.^ 0 . j = ja + n, for all e, 0 < e £ e Q

p2j_l (°) = i kjto/mj = p Pj(o)

f* Pk^0) (mod

(0 ), j = 1, 2,

v; v < q. ;

J = 1> 2>•••>

2v>

P2j-1^0^ ^ p2 j(0 ) (mod ^ fi),p2j_-| (0 ) “ P2k-1^0^ = rajkCDi , m-k

an integer or zero,

pj(°) f P ^ 0) (mod

j, k = v + 1, ..., r; v < r < q

;

j = 2v + 1, ..., 2r; k=2r+i,

..., n + n

;

pj(°) f P]j(0) (mod a)1i), j / k, j, k = 2r+ 1 , ..., n

+ \x ,

where 0 if condition (ii) of Theorem 4.1 is satisfied, and, In addition,

HALE

mT

f f*jxl

(^.13)

(X , X', 0)dt < 0, j = r + 1, ..., q

,

where XQ = (X1Q, X^), X' = (X'0, X«Q) a^e defined by (4.3); and the eigenvalues x.1 , >• of the (r-v) x (r-v) matrix M = N - P, P = diag(p1, ..., Pr_v), N = (njk ), j, k = 1, 2, ..., r - v ;

Pj = i

11m

e_1

CTv+ j (e) " ffv+ j (0)]>

U.UO

mT

P

___I -v ,k -v "i j-------------------

J’ = 1' 2 ’

r " v

'

Km,.. - m ) a i ' t H1.

- mT/

°> e

k '

J1

dt'

o j, k = v + 1, where the functions cjk are defined in (3 •10 ) with the f given by (4.5) are distinct, have negative real parts and the corresponding eigenvectors have no zero components. PROOF. The proof is exactly the same as the proof of Theorem 3»3, replacing everywhere the words Theorem 3«1, Theorem 2.1 by Theorem 4.1 and Theorem 2.2, respectively. REMARK 4.2. For v = 1 in Theorem 4 .3 , the condition (ii) re­ ferred to can be replaced by mT /

EXAMPLE.

Consider the system of equations

x11 + a^x, = X1 (4 .1 5 ) x" 2 + 2X2 = x"

3

+

f i *t (xo’ x a + ±p ^ 0

CESARI

REMARK. Though not needed in the sequel the following considera­ tions may be of some interest. Given a function f(t) e C^ with w [f] = 0, it is of interest to knowwhether its unique primitive F(t) e 0cD with m [F] = o is a definite integral of the form

J

t f(u)du

i with 0 < | < T. For functions f(t) = e^cp(t) e C^, w [f] = w [cp] = 0, this is true, for instance, if the function $(t) above happens to be real, and then we may take for £ any zero of $(t).In particular this is true for a + ip = o, f = cp real. For A + ip = 0, f= cp = cp1 + icp2, cp^ cp2 real, w[f] = w [2 ] = 0, there are two points 0 < i2 < T, such that t F(t) = /

t du *

/

h

92 du

,

^2

and the pointsi 2 may be distinct, as for f(t) = e1^ = cos t + isin t, F(t) = - ie^ = sin t - i cos t, ^ =0, t2 = n/2, ^ 4 i2,(mod *). If f(t) = e^'^^cpft) g C^, "))i [f] = w [cp] = o, then there is always a de­ composition f = f1 + f2 and two points o < i2 < T such that f . € cffl, Stf [fj] = o, t / fj(t) dt = f fj du ,

j = 1, 2 [J. K. Hale, 9a, proved this by the use of faltung integrals]. w [cp] 4 0 then not even this is true as the example shows: f(t) = e a^ w(f) = 0, »t(a) 4 °> t F(t) = u-1eat = a-1 J

e011 du

,

+00

+ according as w (or) < 0, or »j (cr) > 0. eatcp(t) e C^, a £ 0 (mod a>i), we have

F(t) = (eaT - 1)"1

J

t+T

t For

a = 0 (mod aii) we have

Note that for f(t) =

f(u) du

.

If

NONLINEAR LIPSCHITZIAN SYSTEMS

F(t) = T~1

J

t+T uf(u) du

t [J. Moser]. (I.ii).

If

f(t) = eatcp(t) s Cm, M[f] = 0,

and

F(t) e

is

its unique primitive with M [F] = 0, if V > T is any constant, there is a constant N, depending only on a, T, V,such that

(1.3)

|F(t)| < N

J

then

T |q>(u) | du,

0< t < V

.

o If a = 0 (mod o)i) we may take N = 2, if \o +ima>| > 8 for all m = 0, + 1, + 2, and some 6 > o, wemay take N depending only on 8, T, V. If F(t) is periodic then (1 .3 ) holds for all t [J. K. Hale, 9a; see also H. R. Bailey and L. Cesari, 2]. Given a = ot + ip, a, p real, with a + imo> 4 m = 0, + 1 , + 2, ..., let 8 = min [|a + imco|, m = 0 , + 1, ..., ]. For every crT = a1 + ip1, or1, p1 real, with |a1 - a | < 8/2 we have |a’ + ima>| > 8/2 for the same m. (l.iii). Let o = oe + ip, a, p real, be any complex number with m = 0,+ 1 , ..., and let 0 < 8 < min [10 + imco|, m = 0 , + 1 9 •••]> ° < let cr1 be any other number with |cr! - or | < 8 /2 , let cp(t) be any complex valued function periodic of period T = 2*/a>, L-integrable in [0, T],‘let f(t) = eatcp(t), f.,(t) = eaftcp(t), hence [f] = W[f1] =0, and let V > T be any given constant. Then the unique primitives F, F1 of f, f1 of class and mean values zero verify the relation a + ±ma> 4

(1.5)

|F(t) - F,(t)| < |ff - o' | N'

J

T | 0, all 3 = 1 , ..., n, and m = °> ±^> ± 2> •••> then the differential system y 1 = Ay + f(t), y = (y^,..., yR ), has exactly one solution y(t) = (y1# ..., yn ) whose components are periodic of period T. Also, there is a constant N de­ pending only on A and T and not on f(t) such that n

T

f If^t)!dt,j = 1 , ..., n,

(1.6)

|y j(t)| < N h=1

0

PROOF. It is not restrictive to suppose that A = ls Siven in triangular form, i.e., a.^ = o, j > h, ajj = p y j = 1, •••, n. Let p. = ofj + lPj, oi y Pj real, 3 = 1 * •••, n, y = max |orj |, M = erT, Q the sum in (1.6), a = max la^l* The last equation of the system, ^n " pn^n + ^n^^

*

has a unique periodic solution of period T = 2jr/co, which, by using the notation of this Section, can be written as p t 7n (t)

= e n

p —p u

J

e

n fn ^

du

>

where the integrand is of class C^ and has mean value zero, the Integral is the unique primitive of class C^ and mean value zero, has the form

NONLINEAR LIPSCHITZIAN SYSTEMS

121

n e $(t) considered in (1 .2 ), and hence yn = ®(t) is periodic of period T. Let N be the constant of (i.ii) depending only on 6, T, and V = T. Then by (i.ii) for all ° < t < T, we have

MNQ

The last but one equation of the system is yA-1 ." pn-iyn-i + [an-i,ny (t) +

>

where the expression in brackets Is periodic of period equation has a unique periodic solution given by

yn_!(t) = ePn"1 f e and we have, for

T.

Hence this

Pn'1 [an_ ^ nyn (u) + ^ ( u ) ] du

,

0 < t < T, T

|yn_l Ct)| < MN

du

[|an_ ^ n | |yn(u) | +

< MNQ(aMNT + 1) . By repeating this procedure

n times wefind, for

|yj (t)| < MNQ(aJVLNT + 1 and these relations hold for all

t

j=1,

since 7j(t)

0 < t < T,

n, is periodic.

(b). A form of Brouwerfs fixed point theorem. For the use In §5 we mention here the following statement which has been shown to be equivalent to the Brouwer fixed point theorem for a cell in the Euclidean space ER [C. Miranda, Un^sservazione su un teorema di Brouwer, Boll. Unione Mat. Ital. 3, 19^1* 5-7]• (l.v). If K C En Is a cube whose 2n opposite (closed) faces are Ky Kj, j = 1, ..., n, if f(x) = (f.,, ..., fR ), x e K, x = (x^ ..., xn ), is a real vector function, continuous in K, if ^as opposite constant signs on the two faces K*., K., (j = 1, ..., n), then / \ there is atleast one point xQ € K o, such that f*(x0 ) = °> i.e., f j ( x 0 ) ==

n# An Immediate corollary of (l.v) Is

122

CESARI

(l.vi). If K ( E is a cube whose 2n opposite (closed) faces are K yi K ”j = 1, ..., n, if M is a compact topological space, and f(x, m) = (f1, ..., fn ), x e K, m € M, is a real vector function, continuous in K x M, if for some m € M, f.-(x, m ) has opposite • n ° J ° signs on the two faces K^, K . , (j = 1 , ..., n), then there is a neighbor­ hood N of mQ in M and, for each m e N, at least one xQ = x0(m), m e N, x° e K°, such that f [xQ(m), m] = 0 for every m e N. Indeed, by continuity, there is a neighborhood N of mQ such that, for every m €N, fj(x, m) has the same constant sign on Kl [K!j] as f-;(x, m ), and thus f(x, m) hasoppositeconstantsigns on Kj, K y (j = 1, ..., n )• A slightly different form of (l.vi) is the following one: (l.vii). If K C ER is a cube whose 2n opposite (closed) faces are KI, Kj, j = i, ..., n, if M2 are compact topological spaces and f(x, m.,, m2 ) = (f^ ..., fn ), x €K, m1 e M1, m2 € M2, is a real vector function, continuous in K x M1x M2,if for some m2Q e M2, fj(x, nii, m2o^ has °PP0Site constant signs on the two faces K y Kj, (j = • • • > n), independently from m1 e , then there is a neighbor­ hood N2 of m2o in M2 and for each m1e M1, and m2 €N2, at least one xQ = xQ (m^ m2 ), m1 € M1, m2 € N2, x°e KO, such that ffx^m^ m2 ), m ^ m2] = o for every m1 g M1, m2 € N2« (c). Schauder!s fixed point theorem* We shall need in §3 following form of Schauder*s fixed point theorem:

the

(l.viii). Any continuous mapping f : K -- > K, from a con­ vex, closed, compact subset K of a linear space M has at least one fixed point y e K, fy = y. [See J. Schauder, Der Fixpunkt in Funktionalraumen, Studia Math. 2, 1930, 171-180. Also, S. Lefschetz, Topics in Topology, Annals of Math. Studies, No. 10, 1 9 4 2 .] We do not repeat all definitions, but only the following ones: K convex means that x, y € K implies tx + (1 - t )y e K for all 0 < t < 1; K closed means that M K is open, and hence xn -- > x, xR € K, x €M, implies x € K; Kcompact means that every sequence [xn ]> xn e K, possesses a convergent subsequence. We shall need in §3 also the following statement: (1 .ix). Any continuous mapping f : subspace K of a metric space M, which is a contraction in K, exactly one fixed point in K (Banach*s fixed point theorem).

K -----> K, has

If d(x, y) Is the metric in M, then f a contraction means that d(fx, fy) < md(x, y) for all x, y e K, and a constant m < 1. That K is complete means that every Cauchy sequence in K has its limit

from a comp

NONLINEAR LIPSCHITZIAN SYSTEMS

123

in K. Thus K is closed [H. Hahn, Reelle Fimktionen I, Chelsea 1 9 ^8 , p. 1 1 8 ]• The uniqueness follows by the obvious remark that fx = x, fy = y, x, y e K, x / y, implies 0 < d(x, y) = d(fx, fy) < md(x, y) with m < 1, a contradiction. The existence follows by proving that any sequence [xn] with xn+1 = f(xn ), n = o, 1, ..., x e K arbitrary, is a Cauchy sequence [Cf. E. A. Coddington and N. Levinson, Ordinary differ­ ential equations, McGraw-Hill, 1955, p. ^1]• §2.

A TRANSFORMATION

3

IN FUNCTIONAL SPACES

We shall consider first, for the sake of simplicity, a differ­ ential system of the form (2 . 1 )

dy/dt = Ay + eq(y, t, e), - « > < t < + «

>

where y = (y1, ..., yR ), A = A(e) is a constant real or complex n x n matrix, e > o is a real parameter which will be supposed to be sufficiently small, q =(q1, ..., qR ),and each q .(y, t, e) is a real or complex function of t, y, e. We shall make on A and q the follow­ ing assumptions: (a). There are numbers o> > o, 5 > 0, e > o, and integers 0 < v < n, a. > 0, b - > 0, j = 1, . . . , v , such that the n character/ \ j =1,..., n, of istic roots pj(e), A are continuous functions of e in ° < e < eQ, and verify the relations Pj(o)

=

i

a.aj/bj>

Ip j (0)

-

inM>/t>0 | > 8 > o ,

j

=

1,

V

,

(2 .2 ) j=

v

+ l, ..., n, bQ = b., ... by

,

m = o, + 1 , + 2 , ... In addition we assume A = diag(A^ Ag), where A1, (n-v) x (n-v) matrices, A = diag[p.j(e), ..., p A2 = diag[pv+1(e), ..., pr (e )].

are

v x v and (e)], and (*)

Actually, the last requirement (*)is unnecessarily restrictive, and all of the present results are valid under a much weaker assumption replacing (*); namely (**) B2 is any matrix whose coefficients are continuous functions of £ in 0 < s < s . It is only for the sake of simplicity that we use condition (*) [See remarks at the end of §2]. Also, bQ in (2 .2 ) need only be any common multiple of b^, ..., by. Finally, A(e ) could actually be any n x n matrix whose ele­ ments are continuous functions of £ in 0 < £ < eQ, having a matrix Aq

CESARI

as above for its canonical form, i.e., provided there exists a matrix P(e) whose coefficients are continuous functions of e in 0 < s < eQ, with detP(e) 4 0 and PAP-1 = AQ. (K) There exists a number R > 0 and a function t(t) > 0, - oo< t < oo, L-integrable in every finite interval such that |y^| < R, i = 1, ..., n, - oo < t < oo, implies |q.(y, t, e)| < \|r(t), j = 1, ..., n. Given (; > 0 there exists | > 0 such that |yj |, |y2 1 < R, 0 < e1, s |yi | K iy & - 1> • • • > n, |e - e J < £, implies kj(y1/ t,e1)

-

qj(y2, t, e2)|
0. Sometimes we shall replace (K) by the stronger assumption: (L). Condition (K) holds, and, in addition, i = 1, ..., n, 0 < e < £q > implies n kj-(y\ t, e) - qj(y2> t, e)| < +(t) ^

y^ ,

y2 < R,

|y] - y2 1, j = 1 , . . n, - °° < t < °°

£=1

Condition (p) will be replaced by a condition of quasi periodicity in some theorems. Also, it may well occur that we can take R = oo, or that we may take R any arbitrary constant and \|r(t)depends on R, all with obvious simplifications in the statements of the present paper. REMARK 1. Let us give here in a few words the actual meaning of condition (a). In most cases (2 .1 ) will be the canonical form of a real n p system containing at least one equation of the form x1 + o x^ = s f1, and then p1 = ia, p2 = - ia. Then the first condition (2 .2 ) is satisfied by simply taking v = 2 , = a, a1 = 1, a2 = -1, b1 = b2 = 1, while the second condition (2 .2 ) requires that the remaining characteristic roots are not "close" to any one of the numbers im or, m = 0 , + 1 , + 2 , ... . If the real system contains, besides the equation above, a first order equa­ tion of the form x^ = £ f y then we have = 0 , and the first con­ dition (2 .2 ) is satisfied by simply taking v = 3 , 00 = a, a1 = 1, a2 = - 1, a^ = 0, b1 = b2 = b^ = 1, while the second condition (2 .2 ) requires again that the remaining characteristic roots are not "close" to any one of the same numbers above. In §5 we shall actually consider these and more general situations. Integers a., bj 4 + 1 certainly occur in connection with subharmonic and ultraharmonic solutions [cf. 8]. As a further informal

NONLINEAR LIPSCHITZIAN SYSTEMS

125

comment on the general process discussed rigorously below we wish to mention that periodic solutions of system (2.1 ) can be expected to be of iT.t

the form y. = Cj e where

J + 0(e), j = 1, ..., v, y. = 0(e), j = v + 1, ..., n,

I t . = Pj(°) + °(£ )

in the autonomous case, and where the numbers

Cj = cj(e ' do n°t necessarily approach zero as e ---> 0. This is actually the case in innumerable known examples [see, e.g., 8, 9c for references].

Thus it is natural to look for periodic solutions of (2 .1 )

in the "neighborhood" of solutions of (2 .1 ) for e = 0 of the form P.jTo )t yj = CJ 9 = v' yj = °' = v + 1' n* Note that, by force of the continuity requirement in condition (or), we may assume eQ > 0 and 8 > 0 sufficiently small so as |Pj ( e )

-

la ,j® /b j|


6 ,

5,

j

=

1,

v,

0
v, Let n = n(c.|,

periodic vector functions

any

tja.

•••# 0 ) = e C

, •••,

0 ).

z(t)

of the form

,

are constants, C = col (c1, ..., cv, 0 , cv, b Q, a>) he the class of all continuous

cp(t) = (cp.,, •••, (u), u, e]J

J qj[q>(u), u,e ] du

,

,

j = v + 1,

n ,

j = 1, ..., v ,

- P -u

,

and we shall analyze these relations below.

j = v + 1,

...,

n,

NONLINEAR LIPSCHITZIAN SYSTEMS

127

Obviously the products e J q.j[cp(u), u, £], j = 1 , ..., v, are periodic of period T = 2 *b0 /a>, and are L-integrable in [0 , T]. c . 4 0, d . is Thus -V* in (2 .1 1 ) Is the usual mean value, and since defined, j = 1 , ..., v. By (2 .6 ) we have - I t

e (2 #1 3)

.u J {q.j fcp(u), u, e] - djq>j(u)} =

-I t .u

= {e and, by (2 .1 1 ), class C^, and primitive, say and finally, by

* J qj[qp(u), u, e] - Ojdj) - djCPj(u)

these functions are periodic of period T = 2 itbQ/o), of mean value zero. Hence, by §1 , there is one and only one tyj(t), of (2 . 1 3 )> also periodic and of mean value zero, (2 .9 ), we have *

(2.14)

,

*j(t)

“

,

+ eT|rj(t)]e J

I t •t ,

j

=

1,

v

,

y,«(t) satisfy (2 .5 ). Also, the same functions J -p.u tj(t) are absolutely continuous. The functions e J q*[q>(u), u, e], j = v + 1, ..., n, are obviouslyof class C^, with |- pj + ima)/b0 | > 5 > 0 for all 3 = v + • • • > n> m=s °> + + 2 > •••> hence have mean value zero according to §1 . Thus (2 .1 2 ) is 3ustified. Also, there is one and only one primitive of class C^, and of mean value zero, and —p »t of the form e J tjQ(t), Vj0(t) periodic of period T (^jQ of mean value not necessarily zero), and then ij(t) = e^jQ(t), 3 = v + 1 , ..., n. Also, the same functions ^j0(t) are periodic and absolutely continuous. Thus (a), (b), (c) are completely proved, and 3 nR C ft. that Is, the functions

Me shall take in ft the uniform topology (§3 ) and we shall prove that, for s sufficiently small, 3 transforms a closed sphere ftQ around z(t), ftQ C ft-^, into itself under the hypotheses of continuity (K). Also, ( § 3 ) , there is a compact convex set ftQ CC ftR which is transformed into itself by 3 . This implies (§3 ) that 3 has in ft* at least one fixed element y(t), y = 3 y, by Schauder!s fixed point theorem. Under the Lipschitz condition (L), ; | ftQ is a contraction (§3 ) and thus the fixed element y = 3 y is unique in ftQ. In any case, y(t) satisfies the integral equation

y (t) = z (t) + eeBt f

e"^

(q[y(u), u, e] - D[y]y(u)J du

(2 .1 5 ) DC « D[y]C = w (e“Buq[y(u), u, s])

.

,

128

CESARI

Since both y(t) and D depend upon the integers a^ ^ o, bj > 0, and the complex constants c • 4 o, j = 1, ..., v, besidesa) and e, we J shall write y(t, a, b, c, a>, e), D(a, b, c, o>, e), where a = (a^, ..., ay), b = (b1, ..., by), c = (c1, ..., cy). By (2 .1 5 ), y(t) is abso­ lutely continuous, has first derivative y !(t) a.e., and, by differ­ entiation, we obtain (2 .1 6 )

y 1(t) = (B - eD)y(t) + eq[y(t), t, e]

;

in other words, y(t) satisfies a differential equation analogous to (2 .1 ). The vector function y(t) will be a solution of (2 .1 ) provided the equation B - eD = A is satisfied, or, in component form, (2.17)

iajoj/bj - sdj (a, b, c, a>, e) =

pj(e),j

= 1, ...,v.

These equations are called the determining equations of system (2 .1 ). Under the Lipschitz condition (L), - I a contraction, and the actual determination of the unique fixed element y(t) e and of the numbers d^ ..., dy can be obtained by a method of successive approximations, namely (n \

y

It .t It t (t) = z(t) — (c^e , •••} e , 0, ..., 0 ) ,

y ^ ( t ) = z (t ) + eeB^ D (m“l)C =

J

e'^Cqfy

^(u), u, e] - D^m"1\ ^) du ,

te^qfyfa"1}(u), u, e]},

C = col (c-j,•••,0 ^, 0 ,•••, 0),

^ = diag(d j

m = 1, 2, ...,

^,...,dy

^,o,.#.,o),

where each integrand is of class and mean value zero, and each in­ tegral is the unique primitive of class and mean value zero. The uniform convergence of the process, i.e., y(t) = 11m y (m)(t) ,

D[y] = lira D[y(m)] ,

as m -- > 00, is a consequence of the fact that £ is a contraction in This is the method of successive approximations mentioned in the Introduction and used in most papers listed in the references. REMARK 2. A few informal words on the definition (2 .7 ) of the transformation 3: may be of interest. First, the term D[q>] h, bjj = pj(e). Note that the least n - v equations (2.1 ), written In inverse order, are 3n = Pn(e)yn + S(ln » ^n-i = Pn-l(e)yn-1 + V l , i / n + e(ln-l

'

yn-2 = pn-2^e^yn-2 + bn-2,n-lyn-l + bn-2,nyn + eq-n-2

'

ii j

'v + 1

=

p v + 1 ( £ ) y v+l

+

Y

V

i

,

^

+

e% + i

•

i= V+2

The transformation y = n cp can now be defined by the same re­ lations (2 .9 ), (2 .1 1 ) for j = 1, ..., v, and, for j = v + 1, ..., n,

+n(t) = eepnt Jf e~pnuqn[q>(u), u, e] du

*n-l(t) = ePn"lt/ e Pn_1

,

{e(ln-l[cp(u)' u’ e] + V i , A (u)} du

V - 2 (t) = ePn'2 / e Pn"2 {Bqn_2[q>(u),u,e] + V

’

2, n - l V l (u) + V 2 ,iA i(u)} du'

+v+1(t) = ePy+1 / e PV+1 {e m = °9 ± ± 2 > ••• • For the sake of simplicity we shall refer in the following mainly to systems of the form (2 .1 ). All considerations of §3 extend with obvious changes to the more general situations mentioned above. (e). The elementary case

v = 0.

CESARI (3•i i i )•

System

(3 *2 0 )

y« = A(e)y + eq(y, t, e)

,

y = (y^ •••> yn )> q = (q^ •••> qn ), satisfying (a) and(K) with v = 0, has always a periodic solution for all 0 < e < and some e1 > 0 sufficiently small. Indeed, for v = o, there are no aj, b *, c a n d (a) requires only that Pj(°) ^ imw, j = 1 , ..., n, m = o, + 1 , ..., where 2 it/a> is a period of q. Then, for 0 < s < s Q and some e Q > 0 , 8 > 0 , we have also |pj(e) - imo)| > 5 > o for the same jM s and m fs.The trans­ formation ^ , or | = ^ 0 . Finally ^ has a fixed element y(t) e nQ.The determiningequations (2 .1 6 ) are nonexistent (thus certainly satisfied) and y(t) is a periodic solution of (3 .2 0 ). Under condition (L) the corresponding method of successive approximations is the usual one y^0^ = 0, y ^ = 3:y ^m~ 1 \ m = 1, 2 , ... [See A. Lyapunov, Probleme general de la stabilite du mouvement. Ann. of Math. Studies, 1 7 , 1 949 ]• (3•iv). System (3 .2 1 ) where y, q, v = 0 , and [0 , 2 jt/o)], for all 0 “
0 sufficiently small. Finally, we may replace A(s) in (3 .2 2 ) by a n x n matrix A(t, e) whose elements are continuous functions of t and e t - 00 < t < + 00, 0 < e < e , periodic in t of period 2jt/o> (the same cd as for q and g), and whose characteristic exponents, say x..(b), j = 1, ..., n, (all defined mod coi) verify the relations Xj(o) 4 0 (mod coi), j =l, ..., n. The last extension contains, as particular cases, results of Antosiewicz and Diliberto, since the former assumed R(x.) < 0 , and the latter R(x-) 4 °> j = •••> n* Furthermore we assume that the funcJ tions q and g are periodic and L-integrable in the period (in the sense of condition K) instead of periodic and continuous (S. P. Diliberto and M. D. Marcus, On systems of ordinary differentiable equations. These Contributions, Vol. 3, 1956, 237-241 ). An extension in a different direction can be obtained as follows. Suppose that system (3-21 ) satisfies conditions (a) and (L) with v = 0 and |R(p.)| > 5 > °, j = 1, •••, n, suppose that F(t) € QP, where QP is the class of all vector functions whose components F.(t) are J L2 -quasi periodic, i.e., r—i Fj(t>

ILX t -

CESARI

We shall suppose also that qj(y, t, e) e QJ> and qj[y(t), t, e] e Q? This condition replaces the periodicity condition (pj of (K) as mentioned in §2 . Note that we have now 7j(t)

z(t) s o and

(pj + lXm r1cjmelXmt'

Z |(pm + V S m ' 2
< co < o)q2. Take V1 = 2jtb0/cD02, V2 = 27rb0/o^)1,Vl< V2 < V. Since o>01 < < 03q2 and T = 2*b0/o), we have V1 < T = co1, e = e1, and 2 2 J J J J c = c,(D = a),8 = s2. We shall omit to indicate all the parameters which are not essential. Note that V1 < T2 < T1 < V2 < V. For

j = 1, ..., v,

we have by (2 .1 1 )

t2

I

t1

®

- °]>7'

o

/

at

o

where qj3[yS(t)] denotes qj[y(t, c8, a>s, es), t, es], s = 1, 2 . By manipulation, by introducing the new variable of integration u = o^t/o^ m2 in the second integral, and changing u into t we have T2 dj2 - dji = (cj2 - cj!)T21 I

T2

T]t“1

+ cj

J

elTj2\ j2[y2(t)] dt +

-ix.^t e

J'2

-jqj g [y2 (t )] - q ^ [y1(a>2t/2, and thus the

145

NONLINEAR LIPSCHITZIAN SYSTEMS (^•4)

q.j [ y ( t ,

while (^•5)

C2 ,

CD,

e 2 ),

e2 ] -

t,

qj[y(t,

c 1,

a>, e 1 ),

t,

e 1 ]

,

in the autonomous case it is q.j[y(t, c2,

cd2,

We shall denote by A 0 < t < V, j = 1, ..., p""" 1 oo V/o) and hence |t in the autonomous case

e2 ), e2 ] - qj [y((u2t/cu1, c 1,cu1,e1 ), s 1]

•

the number A = max |y-P(t) -y..(t)| for all ^ 2/1 n. For every 0 < t we have

|yj (t, C2, O)2,62) - yj(o32t/o31, o1, 031, £1)| <
1, £1)| +

+

|y j ( t , O 1 ,

O)1 , £ 1 ) - y j f ^ t / c o 1 ,

1 |,

C 1 , CD1,

£ 1 )|


^ J n + y

C P -jy ^ j — “f + Pj2*^"2 J ^

—

^9

M-j t,

n

;

S

•

System (5.4) isof the type considered in §2, where the first equations and the last n - r equations replace the first v equations of §2; hence 2v + (n - r) replaces v, N = n + m. replaces n, and n + n-[2v + (n-r)] = fi + r - 2 v replaces n - v. The numbers pj(o) =Ity j = 1, . v, of §2 are now replacedby the following 2v + (n-r) numbers It.,,

- ixr

1t v ,

- i T y , o,

o

'

..., yN; t, e,

J* = M + 1' •••>

n

,

(the zero repeated n - r times), where tj = ajOi/bj, j= l , ..., v. In other words, for the numbers a^, bj of §2, say a b l , we have now the

2v

NONLINEAR LIPSCHITZIAJM SYSTEMS

j = •••> v > aj = °> v numbers c of §2 the

integers agJ._1 = aj, a2j = - a^, b2,j-i = b2j = b« = 1 , j = r + 1 , ..., n. Let us take for the J following 2 v + (n - r) numbers c2j-i

c. J

Then the vector

= cj ,

c2 j

= "

°y

cj

^

°*

153

COTI1P l e x > j

=

1»

•••> v

4 ° f o■ ? real, j = n + r + 1 , ..., i-i + n

.

J

z(t)

>

of §2 is now the N-vector iT.t z(t) = (cje

-iT.t

J , - Cje

J

,

j = 1 , •••, vj 0 , •••, 0 j 0 , •••, 0 ; ^j*J = |i + r +

1 , ..•,

n + n),

where the zero is repeated (2\± - 2 v) + (r - n)times. In the present situation we may consider the space ft, analogous to the one of §2 , of continuous periodic N-vector functions cp(t) = (cp^, ..., cp^) of period T = 2 *b0 /o>, bQ = b 1 ..-by, with cp^.^ = - 92j-,j = 1 , ..., n, 2j._1(t)j- = Cj, wje

tn£cpj(t)} = Cj, Then

z(t) y ft,

j |fj= e J J e J q.j[q>(ii), u, e ] du,

(5.8)

tj = cj + £ f ^qj[q>(u), u, e] - djcpj(u)| du , j = n + r + 1, ..., n +

j = 2*i+l, ...,

where D = D[cp] = (d^ ..., d2y, o, ..., o, d^+p+1, ..., d^+n), f " i T nu

°jd2j-1 = W l e (5.9)

cjdj

'

1

J q_2j [q>(u), u, £ ]|,

= w q. [cp(u), u, s]

,

j = 1, ..., v ,

j= |i + r + l, ..., n + n.

(5.1). LEMMA. For every cp € ftR we have and d2j._1 = d2 ., j = 1, ..., v, d. real, j = n + r + 1, n+n. PROOF.

and

"I

e]J

f i T iu

- Cjdgj = w |e

n+ r,

t € ft,

We have

^2j-1 = ^j* ^ j = ~ **j real> J = •••> v> a11^ I t .u - I t .u hence Q.2j_i = “ ^ y while e J ,e J are complex conjugate. By (5-9) we conclude that d2 ._.j = dgj, and then the integrands in (5*5) are complex conjugate and of opposite signs. Thus, also their unique primitives of mean value zero have the same property, and (5.5) implies ^2j-i = “ ^2j> j = 1, ..., v. Since p^2 = p ^ , j = v + 1, ..., n, we can repeat on relations (5.6) the same reasoning above and thus t2j_i = “ ^2j* ^ = v + ^ 120 Since Pj is real, q^ is real, j= 2ia + 1, ..., y. + r, by (5*7) we conclude that \|r. is real, j= 2|i + 1, n + r. Finally, for j = + r + 1, ..., ii + n, relations (5*9) and (5.8) assure that dj and are real, j = p. + r + 1, ..., n + n. Thereby, (5-i) is proved. By (5*i) we conclude that s ftR C ft and, as in §§2, 3 , there is a sphere ftQ about z(t) in ftR with s ftQ C ftQ> for e > 0 sufficiently small. A fixed element y(t) = (y1, y^) exists in ft , £ y = y, and

NONLINEAR LIPSCHITZIAN SYSTEMS D[y]

verifies the same relations of (5«i)«

We shall consider now the determining equations of §§1, 2, 3, which are in number of 2v + (n-r), namely Pj1 = lTj ‘ Ed2j-1» pj2(e) = - ±Tj “ Ed2j'

i = '*

*•*» V ,

J = r + 1,

P-j = e^+j >

By the remark above we conclude that for every j = 1, ••., v, the two equations above are equivalent, and thus we have actually only v + (n determining equations

(5.10) Pj =

,

j = r + 1,

n

.

1 9 J, 1 j• = 1, We shall write c- = v, nj = > j = 1, ..., n - r, Xj 4 °> n* 4 °> 8j all real, 9j(mod 2 « ) , X = (x.j, ^v), e = (0i> ev)> H = •••> 1n-r^ d2j-i = pj + 1Ci^ pj> Qj real»

j = 1> •••> v ,

dM.+r+j. = Rj. ', Rj. real >,

1=1, .... n - r ,

where Pj = Pj(a, t>, *■> e)> Q.j = Qj (•••), Rj = Rj(...). Then equations (5.1o) reduce to the following 2v + (n-r) real equations ePj = Qfj(e) , (5.11)

sQj = ajbj1u> -

j = 1, ..., v , >

eRj = Pr+j(e) >

j = 1^ •••> v , j = 1 , ..., n - r,

where QTj(o) = 0 , ajbj1co - 7j(o) = 0 , j = 1 , v, Pp+j(o) = 0 , j = 1, ..., n - r. Equations (5*11 ) are the 2v + (n - r) determining equations in real form. By (5.9) we have - 1 "l e 1 f!e"l T J iu Pj + iQj = Xj e Ja>,'-

Rj = nj1 w f ^ +r+j} ,

1 >

j = 1 , ..., v , j = 1 , ..., n - r

156

CESARI

and, by obvious manipulations, also P- = J

J

[cos e .w{f . cos a -b^asu] - sin e.w{f. sin a -b^cou)] J

J

J J

J

J

,

J J

Q. = xT1 [- sin e.wCf. cos a.b^1 ( 0 j rj) ect, has topological index one with respect to the origin. By force of continuity only, this holds true also for the(per­ turbed) continuous mapping Uj = Pj(^> 9> n, e) -£”1Qfj(e), Vj = Q. + s”1 (7 j(e) - ajbj1a)), Wj = Rj - £“1Pr+j(E), 0 ,r\) € a, provided £ > 0 is sufficiently small. We shall discuss this point In more detail in the papers mentioned in the Introduction. For the present argument, see, e.g., J. Leray and J. Schauder, Topologie et equations functionnelles, Ann. Ec. Norm. (3 ) 51, 45-78, 1934. Let us consider now real systems (5*1 ) where the functions do not depend on t, i.e., autonomous systems, " i p Xj + 2ajXj + CTjXj = £f .(x, x !, 1 , xj + PjXj = £ f y x > x!'

e ),

fj

j = 1,n,

. j = M + 1, ..o, n

,

NONLINEAR LIPSCHITZIAN SYSTEMS

159

satisfying the hypotheses listed at the beginning of §5 , where now 03 in condition (K) is actually arbitrary. As mentioned in §2 and §4 we shall suppose u) e U, where U is a sufficiently small neighborhood of any number for which CTj(°) = ^ = 1' •**> v ’ Pj1 pj2 ^ ^ ima)/b0, j + v + 1 , ..., n, m = 0 , + 1 , ..., and we may expect to satisfy (5*11) by a convenient function to = 03(e) with cd( o ) = a) . Thus (5.iv). The same as (5«ii) for system (5 .2 0 ) with to the list of parameters.

03(e)

added

For every solution x(t) of (5 .2 0 ) also x(t + 0 ) is a solution forevery arbitrary constant phase 0. Thus we may expect that (at least) one of the phases Q y j = 1 , ..., v, considered above, say 0 ^ remains arbitrary and we put e1 = 0. Note that now the 2v + (n - r) expressions P, Q, R are still functions of 2v + (n - r) arbitrary parameters, namely X y j = 1, ..., v, 0 j = 2, ..., v, r\y j = 1, ..., n - r, and a) e U. We may now denote by 0 the (v-1 )-vector (©2, ..., 0 ).Again, as above, let us suppose that the functions f .(x, x !, e) have continuous J ! first partial derivatives with respect to the variables x1, ..., xn, x1, ..., x^. If equations (5.17) hold for some \ Q, 0Q, r\Q and 03 = 03q, and the Jacobian (5.21)

c^(P, Q, R ) / ^ ( X , 0,

for the same \Q, 0 , solution x(e), 0(e), 03 e U for all e > 0 now the Poincare type

ji,

03) ^

0

t]0, oiQ, and 6 = 0, then equations (5*11 ) have a T](e), 03(e), in some neighborhood of 0Q, t\q and sufficiently small. In analogy with (5.iii) we have theorem for autonomous systems:

(5*v). THEOREM. If real \ Q, 0Q, |i , 03Q exist such that equa­ tions (5.17), (5-21) hold, then for e > 0 sufficiently small system (5*20) has a real periodic solution of period T = 2*b0/o3 of the type (5.19) with0 .j = 0 (and then also all other solutions with t replaced by t + 0 , 0 arbitrary). We proceed to prove a theorem of a more general type (others will be given in §6). For the sake of simplicity we shall consider a system (5*2 0) with v = 1, r = n, = 0 , i.e., an autonomous system of the form " + al 2 x1 x1 = ef1(x, x 1, e) , (5 .2 2 )

" 2 2 Xj + 2dfjX. + cTjXj = sfj(x, x 1, e),

j = 2,

Xj +

j = n + 1, ..., n ,

xt'

(-L ,

CESAR I

where 1= v < n < n, x = (x1, ..., xR ), x 1 = (xj, ..., x^), °> tfj(e),r©al functions of e, Pj(£) 4 °> (or constants;, o < e < eQ. Also, we assume that a = b = 1, and hence (o) = cdq, (o), Pj2(°) 4 imco, j =2, ..., |i, m = o, +1, +2, ... . Thus= 03q = a^ (0),and there will be only one c. The functions f . areassumed to satisfy a condition (L). There is only one equation (5.10), namely, ico - ed1 (c, co, e ia1(e), or, in real form, two equations (5*11 ), P1 = 0, co - eQ = cr^ (e)• Since there is only one phase e1 it is not restrictive to take it equal to zero, i.e., c = X, 0 < r 1 < ^ < real and positive. Thus the two determining equations are actually

5 23)

( .

P., (

X,

co,

e)

=

0,

03 -

eQ-j U ,

03,

e) = ^ ( e )

.

Note that P^ (X, 03, 0) = (T\)~1

T f1 [\co~1sin 03t,0, ..., 0, X cost o>t,0, ..., 0; 0]cos cot dt o

J

(5.24) Q1(A,, 03, 0)

= -

T (T^')""1JT o

sin

cot,

0, .. ., 0, X cost

cot,

0 ,. •

0; 0]sin

where T = 2a/co. ¥e shall suppose that for some XQ we have P1(X, and that for two X*, x " , r1< X 1 < X0 < x" < r2, the two numbers P^X*, coQ , 0 ), ? ^ ( x " , coQ , 0 ) have-opposite signs.

cot d t

coQ ,

0)

(5.vi). Consider system (5*22) where the functions f. are in­ dependent of t, Lipschitzian in x, x 1 and continuous in e, for |xj| < R, j = 1, •.., n, |x11 < R,j = 1, ••., \i, 0 < e < eQ. Suppose that ay a y aPe contimio' us factions of e [or constants] and that crj (0 ) > 0, j = 1 , ..., [I, Pj(°) J °> j = n + •••> n> and either a - ( o ) / or Qfj(o) = 0, aj(°) f °> o,1 (0), j = 2, ..., (i. Let coQ = 0 ^ ( 0 ), 0 6 ^01' “02^ X € [IV P2]' Wlth ° < ^01 < ^O < “o2' ° < P1 < r2 < R * If there exists X ' , XQ, x " , r1< X' < X0 < x" < r2 such that P1 (XQ, 03q, 0 ) = 0 , and ? ^ ( x l , coq , 0 ), P1 ( x " , coo, 0 ) have opposite signs, then there

is an e1 > 0 such that, for every e, least one periodic solution of the form x1(t,

e)

= X(e

0 < e < e1,

)co” 1 (e )sin o)(e)t

+ o(e)

system (5*22) has at

,

(5.25) xj(t, e) = 0(e),

j = 2, ..., n

,

for conveniently chosen co(e) e ^02 ^ ^(£) € [*•*> Xu]» System (5 .2 2 ) has also any other solution we obtain by replacing t by t + e, 0 an

NONLINEAR LIPSCHITZIAN SYSTEMS

arbitrary constant. PROOF. Let us use statement (l.vi). Take F1(x, e) * P (x, cd, e), where cd € U,say U = [cd!, cd"], cd* < cdq < cd", X e V = [X!, X,r], £ € [0, eQ].Thus F,, (X1, cdq, 0), F.j (x m, cdq, o ) have opposite signs, and, if we suppose U = [cd*, cd"] sufficiently small, also F1(xf,cd, 0), F^ (xn, cd, o), cd e [cd1, cdm], have opposite constant signs. Take F2(X, cd, e) = cd - a., (b ) -efy (X, cd, e), and note that F2(cdq, X, 0) = CD0 “ (0) = 0, F2 (cd1, X,0 ) < o, F2(cd,!, X, 0) > o for all X e [XI, x«]. By (l.vi), with M = [0, e2], K = [X1, X,f] x [cd*, cd"], we conclude that there are a neighborhood [0, e ] of 6 = 0, and x(e) e [X*, x,f], cd(6) e [cd^, cd"] for every 0 < e < e2, such that P1 = F2 = 0. Then (5.vi) Is a consequence of (5.1v). REMARK 1. Theorem (5.vi) is more general than the usual theo­ rems which are proved by Poincare periodicity condition, since no differ­ entiability property is involved (other theorems will be given in §6). Note that the condition of (5.vi) Is certainly satisfied if (X, cdq,o ) happens to have derivative with respect to X at X = XQ and &P.J /dx 4 0 at that point. Finally, under usual conditions where the functions f . have J t continuous first partial derivatives with respect to x.,, ..., xn, x1, ..., x^, then P1, F2 have also continuous first partial derivatives with respect to X and cd, and M F ^ F2 )/d(x, cd) = x, F2cd “ F1cdF2X' w^ere F1X = P1X, F2cd = 1 + o( e), F1cd = 0 (1 ), F2X = 0(b). Thus we have a(P1, P2)/&(X, cd) = SP^ax + 0 (b)

.

The condition just mentioned assures that this Jacobian is / o for X = XQ, cd = cdq, b = o, and finally assures the uniqueness and continuity of cd(e ), x (b ) for o< b < , and some e1 > o sufficiently small. Note that, under the same conditions, the Inequality T ^

fxi [Xcd sin cDt, o, ••., o, x cos cDt, o, ..., o^ 0 ] dt < o , o 1

or the equivalent Krylov-Bogolyubov condition SP1/dx < o [in both X = XQ, cd = cdq, e = 0], together with < o, j = 2, ..., n, p. < o, j = n + 1, ••., n, assure that the same periodic solution Is asymptotically orbitally stable. [For these and more general stability conditions see J. K. Hale, 91 , and E. W. Thompson, 11]. REMARK 2.

The following considerations may help to understand the

CESARI

generality of (%vi). The real numbers (or functions of e) p^(e), j = ii + 1 , ..., n, of.(s), cTj(e), j = 2 , ..., n, have no bearing on P1 (x, a), o). We require for them only p. ^ o, j = n + 1 , ..., n, aj(°) > °> J = •••> n> and either ofj(o) ^ o, or a.(o) = o, aj(0) ? 0(mod xj) =(X,, 0, ..., 0, xj, 0,

0; 0) ,

g1(x, x', e) = f1(x, x', e) - Z1(x1, xj) , then f1 = Z1 + g.,, g^x.,, o, ..., 0, xj, 0, no bearing on P1(X, xi ) + zi (“ xi» xj ) " z! (xt>~ x j ) - Z., (-

g1 has

x1, x j )]

,

and we define Z12, Z^, Z1^ analogously by changing the signs (+, -, -) into (+, +, + ), (-, -, +) , (-, +,-), then Z . . [ Z 1 0 ] is even in xand odd [even] in x1, Z^tZ^] is odd in x1 and odd [even] in x1. We have f1 = Z1 + g1 = Z11 + Z12 + Z13 + Z1^ , and Z12, Z13, Zlif, g1 have no bearing on P., (x, 03, 0 ). We have

Pt(\, to, 0)= (TX)-1

T J Z11 (Xa3~1sin 031, cos 03t) dt , o

where T = 2*/o3, and 1 03^ In the particular case where u = 0, k ^ 1 ,

then \2h+2k-2 P (x, (D , 0 ) = 2 -2 y a ^ ' x L, 1 0 h!kl(h+k)l ;(h+k)l v 2a1

where a1 = cr1(0)• If, for instance, aQl and the coefficient of the maximal power of X are 0 and) of opposite signs, then certainly (5 .2 2 ) has a periodic solution [cf. J. K. Hale, 9c, for the analytic case]. EXAMPLE 1. The real system (5 .2 2 ), without restricting its generality, can be written as follows:

1 63

NONLINEAR LIPSCHITZIAN SYSTEMS

M + al 2x1 = 6[Z11(x1, / 1N x1 x1 )+

/

1Nx1) + / 1X Z^3 ( x } , x1)

+ Z^k ( x v (5.24)

!1 | 2 Xj + 2ajXj + CTjXj = Xj

+ p .X.

= e fj(x ,

e f j( x ,

x ',

x,' ) + g, (x, x', e)] , j = 2, ..., n ,

x ' , e ) ,

j

e),

=

n +

1,

...,

n

,

where ..., f2, ..., fn are Lipschitzian in x, x 1, e where Zn [Z12] is even in x^ and odd [even] in x^, Z^fZ^] is odd in x1 and odd [even] in xj, where g ^ x ^ o, ..., o, xj, o, ..., 0; o) = o, where a .(e), aj(e), are continuous functions of e [or constants] with aj(0) >0, j = 1, ..., ii, Pj(°) 4 °> j = ^ + •••> and either Qfj(o) 4 0> or ofj(0) = 0, aj(0) 4 0, mod a1(0), (o) = coQ, j = 2, ..., n. If we take Z ^ = (1 - x2 )x.j, then P1 = (i/2)(l - X,2/ W 2). Thus for every e, 0 < e < and some e1 > o, system (5*24) has a real periodic solution of the form (5 .2 5 )

x1 = XoT1sin (cot + e) + 0(e), x. = 0(e),

j = 2, ..., n ,

with X = 2a1(0) + 0(e). For Z^2 = Z13 = Z ^ = 0, g1 = 0, n = n = 1,we have, as a particular case, the well known Van der Pol equation tf 2 2 1 x1 + cr1x1 = e(i - x1)x1. By (9*i) we can see that thesolution (5 .2 5 )is asymptotically orbitally stable. If we take Z11 = (1 - x2 - x.J2 )xj, then P1 = (1 /2)(1 and (5*24) has a real periodic solution of the form above with X = a1(0 ) + 0 ( e ) , co = a ^ ( 0 ) + 0 ( e ) . As in Example 1with Z ^ = (1

EXAMPLE 2.

- |x1 |)xj.

- X2 / a 2 )

Then

T P, = (T\)_1

J

(1 - Xad 1 |sin

cot

|) cos2

cot d t

,

o and P1(x, coQ , 0 )= (l/2)(l - X/no^ ).Hence, system (%24) hasaperiodic solution of the form (5*25) with X = *a1 + 0 ( e ) , co = a1 + 0 ( e ) . EXAMPLE 3* The real system x"+ x -e(1 -

y' )

x2 - y2)x' = ef,(x, y,

+ eg,(x, x', y, y' )y ,

y" + 2y - e(1 - x2 - y2)y' = efp(x, x 1, y) + eg (x, x 1, where

f,(- x, y, y 1) = - f,(x,

j, y'),

y,

y 1)x ,

f2(x, x 1, - y) = - f2(x, x', y).

1 6k

CESARI

There are two periodic solutions of this system given by x = X sin (cot + 0 ) + 0(e), y = 0(e), co = 1 + 0(e), X = 2 + 0(e) ; x = 0(e), y = X sin (cot + 0 ), co = 21/2 + 0(e),

x= 2 + 0(e)

[See J. K. Hale, 9c, for the analytic case. In 9i it is proved that both solutions are asymptotically orbitally stable.] The following further examples have been studied among others, in the lines above, by R. A. Gambill and J. K. Hale [8] with emphasis on the existence of harmonic and subharmonic solutions. In [1 ], [3], and [91] the stability of some of these solutions has been discussed. In all examples e Is a small parameter. x" + cr2x = B cos 2 cot +

4.

5.

1

Ly + a2y = e6^ + B cos

^

#y +

2

+ £vx y *>

r x,f + kx = cos t + e[cos t • x + ky3 + by] ]

^ yM + y=e[cos

t • y + cx°]

1

^yM+

2

,

;

r xM + a2x- = e(a - y j 2 )x' + P cos t

9.

;

+ eyxy 2

eax + eA cos t • x +epx^

2

ii

;

;

e(l - x2n)xf + epco cos (cot + a)

x" + x= r xM + cr2x=

8.

• x + ebx^

x" + cr2y = B cos cot + ex^

6.

^*

soccos 2cot

,

2

= e(5 “ px )y! + r 003 2t i

Example k is a nonlinear Mathieu equation with a large forcing term, 5 is a Duffing equation, 6 Is a generalization of the Van der Pol equation, 7 is a system of nonlinear Mathieu equations, 8 is a system of nonlinear Mathieu equations of which one has a large forcing term, 9 is a system of Van der Pol equations. §6.

REAL SYSTEMS PRESENTING SYMMETRIES

We shall consider now real systems (5.1 ) presenting symmetries as defined below. In addition to the hypotheses listed at the beginning of §5, we shall suppose

NONLINEAR LIPSTCHITZIAN SYSTEMS QCj - 0,

j - 1, .••,p. ,

o < V < H =r < n, pj = °,

j =n + 1,..., n

Then we have 7 ^ = CTj, p^ tions (5 «2 ), (5 • 3 ) become

=i c r j ,

pj2

y2j-i =lffjxj + xj’y2j = lajxj

= -

l a y

-Xy

(6.2)

xj = (2iCTj)_1

= 1 , ..., n,

j

j

if u =(x,, ..., Xu,),v = (xm+1,

fj(^.-v,w,-u*,vS-t,e) = f (6.4) fjtu^-v^w^-uSvS-t^s)

and equa­

- y2j} ' =

1,

..., x^),

hence x = (u, v, w), x r = (uf, v f), for some

.

j = 1> — » ^ V j = x j j =(i + 1, ..., n ,

+ y2j^ xj- = 2"1(y2 M

^6 ' 3 ^

Also,

165

...,

n.

w = (*M+1> ..., xn ),

0 < m < v,

.(u,v,v,u*,v',t,s),

we have

j = 1, ..., m

= - fj(u,v,w,u!,vSt,B),

fj(u,-v,w,-uf,v?,-t,6 ) = - fj(u,v,w,ur,v!,t,e),

,

j * m + 1, ...,

n,

j = \i + 1, ..., n .

For m = 0 allfj are odd withrespect to (v, u 1, t).For m > o the first m of the f . are even and the remaining ones are odd withrespect to (v, U 1, t). We shall take 0 . = n/ 2 , j = 1 , ..., m, 0 . = 0, j = m + 1, ..., v, J J and hence c- = ix., j = 1, ..., m, c- = X., j = m + 1, ..., v, X. real; j j J J J we shall take Cj =j = n + r + 1, . \± + n. Then we have T pj = -

o

T

fj sln Tjt dt’

= "

o

(6.5)

fj 003 Tjt dt * =

T

Pj = (TXj)"1 J o

m

»

T

f. cos Tjt dt,

Qj = - (T^j)-1/ fj sin t^t dt , o. j = tn + 1, ...,

V

,

RJ where the arguments of the f !s, besides t and e, are given by the (5 .1 2 ) of §5 , thus, in the present case, by relations analogous to the (6 .3 ).

CESARI

1 66

We shall restrict the space n to all cp = (cp.,, ..., 9p+n) as in §5 which besides verifying 92j_-,(t) = j = 1> ..., n, cp. real, j = 2m- + 1^ ..., ji + n, have the following properties: cp2j_1(-t) = -

0 sufficiently small. We may take, for instance, f = x + x + x? , or f = |x | + |x1 |, or f = x |xf|.Also we may takef = (2 + x) (|xf| + x). EXAMPLE 2 . The system x" + x = b (1 - |y| )|x!|x, yn + 2 y = e (1 - |x|)|yf|y, has two families of periodic solutions of the forms X

=

Xco

c o s (cot

x

=

0 (b),

y

=

+

0)

+

Xco_ 1 c o s

0 ( e ), (cot +

y

= 0) +

0 ( e ), co

= c o (X , e ) = 1+

0 (e),

= c o (x ,

co

e)

=

0(e ) ,

2 1^2

0 (e).

+

EXAMPLE 3 . The system xM + x = sf1 (x, y, z, x*), yf = z’

ef2 (x, either

z > x ')'

z* = efV x>

with

z, 0 ) = 0

fi

and

fj(—x, y, z, x *) — — f .(x, y, z, x !), j = 1 , 2 , 3 , or f1(x, y, z, -xf) = f1(x, y, z, x 1), fj(x, y, = - fj(x, y, z, x !),

z, -xf) =

j =2 , 3 ,

has a four parameter family of periodic solutions, for every small, of the form x = Xco^sin (cot + 0 ) + 0(e ), or X

w ith

&

=

=

Xco"*1 c o s (cot

c o (x ,

t)-j ^

t)2 ,

+

0)

+

b ), 0 ,

0(e ), x.,

t)^ ,

y

=

t)2

T] 1 +

o(s),

a r b itr a r y .

z

=

T)2 +

e sufficiently

0(b )

,

NONLINEAR LIPSCHITZIAN SYSTEMS

EXAMPLE 4. A third order equation x MI + a.,xM + a2 x* + a^x = ef(x, x r, xM, t, e), with f periodic could be studied in the lines of §§5, 6 . For Instance, if f(x, -x!, xM, -t, e) = - f(x, x 1, xM, t, e), then equation x ,!I + a2 x* = ef, a> 0 , has periodic solutions of the form x(-t, s) = x(t, e), x(t, o) = - c1 ao>cr~2 b~1 cos ab'^t + c2, a, b integers, c1 , c2 4 o real, provided ab“1co - bH = cr, where H = H(a, b, c.,, c2, oo, cr, e) and

H = (c1 T ) " 1

T f [x(t,o),x!(t,0 ),xr,(t,0 ),t,0 ]sin ab~1 a>t dt + 0 (b) . o

J

If a = aa)/b, and there Is a c l 0 = ci0 ^c2 ^ such that H = 0 and dH/dc.j 4 0 for c., = c1o, then the equation has a non-zero solution for every c2; that is, a one-parameter family of periodic solutions [J, K. Hale, 9f ]• BIBLIOGRAPHY [1 ] Bailey, H. R., Harmonics and subharmonics for weakly nonlinear

Mathieu type differential equations.

To appear.

[2 ] Bailey, H. R., and Cesari, L., Boundedness of solutions of linear differential systems with periodic coefficients. Archive Rat. Mech. Anal. 1^, 1958, 246-271 . [3 ] Bailey, H. R., and Gambill, R. A., On stability of periodic solutions of weakly nonlinear differential systems. J. Math. Mech. 6 , 1 957 6 5 5 -6 6 8 .

[4] Cesari, L., (a) Un nuovo criterior di stabilita per le soluzioni delle equazioni differenziali lineari. Annali scuola Norm. Sup. Pisa (2 ) 9 , 1940, 1 6 3 -1 8 6 . (b) Sulla stabilita delle soluzioni dei sistemi di equazioni differenziali lineari a coefficienti periodici. Mem. Accad. Italia (6 ) y\_, 1 941 , 633-695[5]

Cesari, L., Asymptotic Behavior and Stability Problems in Ordinary Differential Equations. Ergbn. d. Mathematik und ih. Grenzgebiete, Heft, 1 6 , 1 959, vii, 2 7 1 .

[6]

Cesari, L., and Hale, J. K., (a) Second order linear differential systems with periodic L-integrable coefficients. Riv. Mat. Univ. Parma, 5* 1954, 55-6 1 ; 6, 1955> p. 159* (b) A new sufficient con­ dition Tor periodic solutions of weakly nonlinear differential systems. Proc. Amer. Math. Soc. 8, 1957, 757-764.

[7] Gambill, R. A., (a) Stability criteria for linear differential systems with periodic coefficients. Riv. Mat. Univ. Parma, 5, 1954, 1 6 9 -1 8 1 . (b) Criteria for parametric instability for linear Sifferential sys­ tems with periodic coefficients. Ibid. 6, 1955* 37-43. (c) A funda­ mental system of real solutions for linear differential systems with periodic coefficients. Ibid. 7, 1956, 311-319*

CESARI

[8] Gambill, R. A., and Hale, J. K., Subharmonic and ultraharmonic solutions for weakly nonlinear systems. J. Rat. Mech. Anal. 5, 1956, 353-394. [9]

Hale, J. K., (a) Evaluations concerning products of exponential and periodic functions. Riv. Mat. Univ. Parma, 5, 1954, 6 3 -8 1 . (b) On boundedness of the solutions of linear differential systems with periodic coefficients. Ibid. 5 , 1954, 1 3 7 -1 6 7 . (c) Periodic solutions of nonlinear systems of differential equations. Ibid. 5, 1954, 2 8 1 -3 1 1 . (d) On a class of linear differential equations with periodic coefficients. Illinois J. Math. 1 , 1957, 98-104. (e) Linear systems of first and second order differential equations with periodic coefficients. Ibid. 2 , 1 9 5 8 , 5 8 6 -5 9 2 . (f) Sufficient conditions for the existence of periodic solutions of systems of weakly nonlinear first and second order differential equations. Journ. Math. Mech. 7, 1958, 1 6 3 -1 7 2 . (g) A short proof of a bounded­ ness theorem for linear differential systems with periodic co­ efficients. Archive Rat. Mech. Anal. 2, 1959, 429-434. (h) On the behavior of the solutions of linear periodic differential systems near resonance points. These Contributions, 5 , 1959, (i) On the stability of periodic solutions ofweakly nonlinear periodic and autonomous differential systems. These contributions, 5, 1959.

[1 0 ] Puller, W. R., Existence theorems for periodic solutions of systems of differential and differential-difference equations. Ph.D. Thesis, Purdue University, Lafayette, Indiana, 1957• [11] Thompson, E. W., On stability of periodic solutions of autonomous differential systems. (To appear;.

VII.

THE APPLICATIONS OF A FIXED POINT THEOREM TO A VARIETY OF NON-LINEAR STABILITY PROBLEMS Arnold Stokes §1.

INTRODUCTION

We use a fixed point theorem for locally convex linear spaces, due to Tychonoff1 to reduce the study of the boundedness and stability of certain n-dimensional vector differential equations to the study of the corresponding properties of related first-order equations. In this manner, we very simply obtain a variety of known results, and in certain cases, we are able to clarify some of these results. Fixed point theorems have been used to study the stability of p systems of differential equations before, notably by Hukuwara , and o Bellman , but the present approach, by using Tychonoff*s theorem rather than Schauder!s result^ gives many more results in a greatly simplified fashion. §2.

DEFINITIONS

We will let E be the real vector space of all continuous func­ tions from the non-negative reals into Rn, the n-dimensional vector space The results presented here are based on material contained In the author!s thesis, written as partial fulfillment of the requirements for the Ph.D. degree at the University of Notre Dame. The author wishes to thank Dr. J. P. LaSalle for his valuable suggestions and constant encouragement during the development of this study. The author also is grateful to the Office of Naval Research for their assistance while at the University of Notre Dame. 1 A. Tychonoff, Ein Fixpunktsatz, Math. Ann. 1 1 1 , 767-776, 1935* 2 M. Hukuwara, Sur les Points Singuliers des Equations Differentlelles Lineaires, J. Faculty of SciV, Ifckkaldo Imp. Univ. Ser. 1/ Math. 2/ 13-88, 1934-36. “ 3 R. Bellman, On the Boundedness of Solutions of Non-Linear Difference and Differential 'Equations, flrans. Araer. Math. £>oc., 6 2 , 357-386, 1 9 ^7 • J. Schauder, Der Fixpunktsatz in Funktionalraum, Studia Math. 2 , 1 7 1 -1 8 1 , 1930. 173

STOKES

over the real field. The topology on E shall be that induced by the family of pseudo-norms ^Pn^=i^ where for x € E, p (x) = sup ||x(t )Ij n o< t E be continuous, and let A be a closed convex subset of E. T(A) C A and T(A) is compact, then there exists a fixed point of T in A.^

If

If T Is a compact (completely continuous) operator and if A is bounded, then T(A) is always compact. We then obtain: COROLLARY 1. Let E be as in Theorem 1. Let T : E -- > E be continuous and compact, and let A be a closed, convex, bounded subset of E. If T(A) C A, then there exists a fixed point of T In A. §3.

REDUCTION TO A FIRST-ORDER EQUATION

We shall consider theee types of n-dimensional systems here: (1 )

x = f(t, x)

(2)

x = A(t)x + f(t,

x), where

||X(t)||
0 (3 )

x = A(t)x + f(t, x), where

||X(t)|| < Ke_crt ||X(t)X'1(s)|| < Ke"a(t_3), K, a > 0

■=

This theorem, while not explicitly stated by Tychonoff, follows from a theorem given in No. 1 in the same manner that the second fixed point theorem of Schauder's in No. 4 follows from his first theorem.

175

APPLICATION OF A FIXED POINT THEOREM

where by X(t) is meant the principal matrix solution of the system x = A(t)x, where A(t) Is a continuous function of t for t > 0. A(t) is a constant matrix, or periodic in t, examples of systems satisfying the Inequalities in (2 ) or(3 ) arewell-known* into

Rn,

(4)

In addition we shall assume that and that ||f(t, x )|| < G(t, ||x||), for t >

f

If

is continuous on Rn+1

0, x e

D( Rn

where G(t, r) is piecewise continuous on R , positive for t, r > 0 , and nondecreasing in r for fixed t; and D is some subset of Rn. With (1 ) above, we shall associate the integral operator

(5)

t Tb (x)(t) = b + J f(s, x(s))ds % o

,

and with (2 ) and (3 ) above we shall associate the operator t (6)

T-b(x)(t) = X(t)b + J

X(t)X“1(s)f(s, x(s))ds

,

o where b

Is a vector

in Rn.

Clearly fixed points of these operators correspond to solutions of the associated equations, the solution passing through b at t = 0. Also it is evident that both operators are compact in the topology of our function space E. To apply Corollary 1, we shall proceeds as follows: Let B be a bounded subset of Rn, and let A be a subset of E defined by a positive real-valued function g(t), continuous for t > 0 , that is, (7)

A = {x e E | ||x(t)|| < g(t)} .

For such ag, A isclosed, convex, and bounded in the topology given above on E. We will further assume that for x e A, x(t) e D for all t > 0, so that (4) maybe used. Now takeb e B, x € A. To apply Corollary 1 to either of operators in (5 ) or (6), we must show T^(A) C A, or, by (7 ), ||T^(x)(t)|| < g(t). Using the operator in (5 ), we obtain

the

176

STOKES

||Tb (x)(t)|| < ||b|| +

J

t ||f(s, x(s))||ds

o

< Hb|| +

J

t G(s, ||x(s)||)ds,

by (4)

,

o

J G(s, g(s))ds, t

< libII +

o Thus, for (5 ), we have

(8)

T^(A) C A

if g

by the definition of A, as G is non-decreasing.

satisfies

t ||b|| + jT G(s, g(s))ds < g(t) o

for b

e

B, t > o

.

Now consider (6), and let A(t) be such that X(t) satisfies the inequalities in (2 ). In precisely the same fashion as before, we have ^(A) C A if g satisfies t

(9)

K ||b|| +

J

KG(s, g(s))ds < g(t),

for b e B, t >

0

.

o

if g

If X(t) satisfies

satisfies the inequalities in (3 ), we have

^(A) C A

t (10)

K ||b|| e ~ a t + J K e~ a('t ~s '>G ( a , g(s))ds £

for

g(t),

b € B, t > o .

o Equations (8) and (9 ) may be combined into

(11)

t K1 ||b|| + J ' K^G(s, g(s))ds < g(t), o

for b £ B, t > 0 .

where K1 = 1 If equation (1) is beingconsidered, being discussed.

or

K, if

(2 ) is

Now g will clearly satisfy .(1 1 ) if it satisfies the differ­ ential inequality:

APPLICATION OF A FIXED POINT THEOREM

(12) and (13)

177

g(t) > K.,G(t, g(t)), g(0) > K1 ||1d||, for b e B, t > 0 , g will satisfy (1 0 ) if it satisfies the differential inequality: g(t) > - crg(t) + KG(t, g(t)), g(0) > K ||b||, for b € B, t > 0 ,

where by a solution is meant a differentiable function g or (1 3 ) wherever G Is continuous.

satisfying (1 2 )

Thus, to demonstrate the existence of a solution to any of the equations (1 ), (2 ), or (3 ) which satisfies ||x(t)|| < g(t),we must choose B C Rn and a g satisfying (1 2 ) or (1 3 );(and, of course, g must possess the property that ||x(t)|| < g(t) implies x(t) e D). §4.

STATEMENT OF THE THEOREMS

(i) On existence in the large. We will consider equation (1 ) and assume that f Is dominated by G everywhere, that is, D = Rn. THEOREM 2. If the equation r = G(t, r) possesses the property that for any rQ > 0 there exists a solution defined on [0, 00), passing through rQ at t = 0, then for an arbitrary vector b e Rn, there exists a solution of (1 ) defined on [0, 00) which passes through b at t = 0 . PROOF. Takeb e Rn, let B = {b}. Choose g to be a solution of r = G(t, r) with g(o) > ||b|| which is defined on[0 , «). Clearly such a gsatisfies (1 2 ), so the operator in (5 ) maps A into A, where A is defined as in (7 ). So by Corollary 1, T^ has a fixed point in A; that is, there exists a solution of (1 ) defined on [0, 0 0 ) passing through b . at t = 0. COROLLARY 2. Assume ||f(t, x)|| < M(t )L( ||x||), for t > 0 , x € Rn, where M, L are piecewise continuous, positive, and L is non-decreasing. If

ds = 00 L(s)

for positive rQ, then for any b € Rn, there ex­ ists a solution of (1 ) passing through b at t = 0 .

1 78

STOKES

PROOF. Observe that the equation be solved for any rQ > o, for if we write r

r = M(t)L(r), r(o) = r Q

may

t

I -J— ds = f M(s)ds 0 l (s )

i

,

note that the function of r on the left is strictly increasing, so the inverse function exists, and by our assumption concerning L, the domain [0, co), of the inverse functions is [0, oo), so the solution through rQ at co) for all r Q > o. The result follows from t = 00 is defined on [[ 00,, «) Theorem 2. Corollary 2 is a theorem due to Wintner^, which was later improved 77 b y him by removing the requirement that L be non-decreasing . Later gg Conti generalized this result still further, b y replacing the norm function b y a more general function. The above proof is considerably simpler than W i n t n e r !s original proof, however. For an example of dominated as as in of a function dominated in Corollary Corollary 2, 2,con­ con­ sider any anyfunction function f(t, f(t, x) x)such such that that ||f(t, ||f(t, x)|| x)|| == o(||x||) o(||x||) as as ||x||x|| ----->> for fixed t . (ii) On Boundedness. We will consider Equations (1) or (2 ), and again assume assume that that ff is is dominated dominated bbyy G G everywhere, everywhere, oror DD == Rn Rn .. THEOREM 3* If the the equation equation r r= =G(t, G(t, r)r) possesses possesses the property that for any rQ > 0 there exists a solution defined and bounded on [[0, 0 , 00), passing through rQ at t = 0 , then for' an arbitrary vector b e Rn there exists a bounded solution of (1 ) or or ((22 )) defined defined onon [[00,, K 11 ||b||.

A. Wintner, The Non-Local Existence Problem of Ordinary Differential Equations, Amer.’J . of Math./ 6 7 / 2 77-28V, 1945. 7 A. Wintner, The Infinities in the Non-Local Existence Problem of Ordinary Differential 'Equations, Amer. J. of Mat h . , '66, 173-178, ’ 19 ^6 . R. Conti, Limitazioni !in A m p i e z z a f della Soluzioni di un Sistema di Equazioni Differenziali e Applicazioni, Bolletino della Unione Mat. I t a l .. "Ser."' 3, ’ l‘ l"/ "3W-350"/ 1956*

APPLICATION OP A FIXED POINT THEOREM

179

COROLLARY 3* Assume ||f(t, x)|| < M(t)L(||x||) for t > 0, x e Rn, where M, L are piecewise continuous, positive, and L is non-decreasing. If IX)

for positive

rQ, and

J

00 M(s )ds 0 , there exists a solution to r = M(t)L(r) defined on [0, 00), passing through rQ at t = 0 , and further that

J

00

M(s)ds < 00

o implies that this solution is bounded. With reference to the following theorem, observe that the above boundsin general depend upon the initial conditions. For an example of afunction dominated as in Corollary 3 , let f(t, x) = B(t)x, where B(t) is an n x n matrix, continuous in t, such that 00 J

||B(t)||dt < =»

.

o Q

Such a theorem appears in Bellman , where the proof uses Gronwall!s in­ equality. The above corollary shows that the linearity of L is not essential, but rather that it suffices that r ~— J- ds r L(s) o ^ R. Bellman, Stability Theory of Differential Equations, (McGraw-Hill Inc., New York, 1953)•

180

STOKES

diverges as

r --->

(iii)

On UltimateOnBoundedness» Ultimate Boundedness» Here we useHere ultimate we useboundedness ultimate boundedness

in the sense of Yoshizawa1°; that is, through every point there passes a solution which is not only bounded but as

t ---> °o,

enters a bounded

region, which is independent of the initial conditions• sider Equation (3 ), and again we assume THEOREM b.

If

G

rQ

at

for any

rQ,

G

everywhere.

there exists a solution to

r > - err + KG(t, r)

t = 0

We will now con­

is dominated by

is such that for every positive

initial condition the inequality

f

passing through

which is ultimately bounded, then there exists a solution of (3 )

b e Rn ,

passing through

b

at

t = 0

which is ultimately

boundedo PROOF. where

B = (b),

The proof is the same as for the above two theorems, for

bounded solution to

b € Rn ,

and

g

r > - or + KG(t, r),

COROLLARY k. R(e) > 0

and here

For any

such that

e > 0,

is taken as any ultimately with

g(o) > K ||b||.

assume there exists

||f(t, x)|| < N, ||x|| < R, t > 0,

||f(t, x)|| < e||x||, ||x|| > R, t > 0.

Then all

solutions to (3 ) are ultimately bounded. PROOFo

Here let N,

r < R

G(t, r) = | er, By Theorem k, all we must show is that for ist ultimately bounded solutions to at

t = 0 , where

0
> K |||b||,

Now choose

ee > 0

r > R e

sufficiently small, there ex­

r > - or + KG(t, r)

for

passing through

ban anarbitrary arbitraryvector vectorinin

such such that that e e NK/a asas t t---------> > °°, and and so so ggis is ultimately ultimately bounded. bounded. And And g(o) > K ||b||, so all that remains is to show that g satisfies the differ­ ential inequality: N,

g(t) < R

g (t) > - ag(t) + K / eg(t), g ( t ) > R T. Yoshizawa, Note on the Boundedness and the Ultimate Boundedness of Solutions of x = F(t, x), Mem. of the Coll. of Sci./ Univ. of Kyoto, Ser. A 29, Math. No. 3, 275-291, 1955-

181

APPLICATION OP A FIXED POINT THEOREM Now

3^3 an^

g(t) = p(eK - cr)K ||b | e^(€K“a ^ ^^

( O ( HHO

g(t) + crg(t) lbll ep(€KH,)t[P }t[3eK 4- 0(1 0 (1 --p)] crg(t) = K l llbll 6K 4p)] ++ NK NK . .

By our choice of p, this is clearly > NK for for all all t >> 0, 0 , and and so so half half K2 the inequality is satisfied. Now Kg(t) = K 2 ||b|| eP(€K ”a )t ++ NK2 nk2/a, /^, and and 1 *0 , we see that the inequality upon comparison with ( (1*0,

g ( t ) > - crg(t ) + eKg(t) reduces to the two inequalities p e K + cr (1

-

P) >

e K

and eK NK/a

NK >

.

(1 -- P), P), But the first of these is equivalent to to a(l a(l -- p) p) >> €e K K(1

€ < a/K,

and the second reduces to

choosing

e < a/K,

eK/a < 0 . The corollary follows.

rQ < 8,

A result of this type, where L(r) = r, is given by Bellman^, where again Gronwall^ inequality is applied. As before, Corollary 5 demonstrates that the linearity of L is not important. For an example of a function of this type, consider

f(t,

x)

^ R. Bellman, Stability Theory of Differential Equations, (McGraw-Hill Inc., New York, 1953).

APPLICATION OF A FIXED POINT THEOREM

183

where the i-th component is the product of a polynomial in the components of x, without a constant term, with a function cp^(t), where 00

o (v). On Asymptotic Stability of a Critical Point. Here we shall consider (3 ), and assume that (3 ) has a critical point at the origin, that Is, f(t, 0 ) = 0 . By asymptotic stability we mean, of course, that theorigin is stable, and in addition, there exists a 5> 0 such that through any b, ||b|| < 8, there passes a solution which approaches 0 as t -- > oo. As this is again a local problem, we will assume f to be dominated in G in some neighborhood of the origin, that is, D = (x e R| ||x|| < tjJ• For out purposes, here we wish to take r\ as a function of g , where g is a small parameter, and we will also assume that G(t, r) = G(t, r, g ) is also a function of this parameterTHEOREM 6. If the inequality r > - ar + KG(t, r, e) possesses an asymptotically stable point at the origin for g sufficiently small, then (3 ) possesses an asymptotically stable critical point at the origin. PROOF. Take € > 0 so small that the differential inequality above is asymptotically stable. Then there exists a 5 > 0 such that for 0 < rQ < 5, there exists a solution to the inequality which passes and for all t > 0 , through r at t = 0 , tends to 0 as t -- > is bounded by rj(€ ). Then, to apply Corollary 1, let B = {b e Rn | ||b|| < 5], and choose g to be a solution to the inequality having the above properties, with g(o) = 5. The theorem follows. COROLLARY 6. Assume that for any e > 0, we have ||f(t, x )11 < e ||x||, for t > 0,11x || < t j (g ). Then in (3 ), the origin is asymptotically stable. PROOF• By Theorem 6, we must show only that for sufficiently small e a solution g(t) can be found to the differential inequality g(t) > - crg(t) + eKg(t) which is bounded by tj(g ) and tends to 0 as t -- > 00. But for g < cr/K, we may choose g(t) = tj(e )e ^ and th corollary is proved. As an example, we may take any o(||x|j) as ]|x11 -- > 0 uniformly in t. is originally due to Perron.

f(t, x) for which !|f(t, x )|| = This result, by now classical,

STOKES

184

COROLLARY' 7* Assume that for any T(e) > 0 such that

|f(t, x)|| < |

* k„ ||x|| +. Jtu b ||x|| + tu

e > 0, we have

n ^x i|i| 1+a || ITa, 0 < t < T(e) i| i| „x i| i| 1+a ,+cl, t > T(e) ,

for ||x|| o, a > o. (3), the origin is asymptotically stable.

ly small

Then in

PROOF. Again by Theorem 6, we must show only that for sufficient e, a solution g(t) can be found to the inequality

r kg(t) + tbg(t)1+a, 0 £ t < T g(t) > - crg(t) + K 4 h 1+a I eg(t) + tbg(t)1+a, t > T1 ,

where

T1 > T(e),

which is bounded by

T](e) and tends to

,

0 as

t -- > »

Now, for 0 ^ r < 1, 0 Ti]• g0 = g(T1). We now wish to define g for t > T1. Take e > 0 so that e < a/K, and choose p so that 0 T(g ), and let g(t) = g0e^^€K”cr^ t"T1 Clearly, by our choice of €, g -- > 0 as t -- > °°, and as gQ < r\(e), g(t) < tj(0 for all t > 0. We now wish to show that for T1 sufficiently large, and gQ sufficiently small, g satisfies the differential inequality g(t) > - ag(t) + eKg(t) + Kt^gCt)1"1*8, for t > T1. Now g ( t ) - (eK - a ) g ( t ) = (p - 1 )(eK - a )goeP (€K_CT} ( t _T 1 } = a g ( t )

,

where a = (p - 1)(eK - a) > o by our choice of p and e. So we must show that ag(t) > Ktbg(t)1+a, or, as g is positive, oc > Ktbg(t)a. But Ktbg(t)a = g g K e - ^ ^ ^ ^ l

• tbeaP(€K-a)t

•

Now choose T1 so large that t^apfeK-a for t > T^ and choose g0 so small that g^Ke“a^ €K“a ^1 < See . Clearly both these choices can be made, and the corollary is proved. Corollary 7 appears in Coddington and Levinson 11 . There we also have an example of a system to which Corollary 7 applies. 11 E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equa­ tions, (McGraw-Hill Inc., New York, 1950 J.

VIII.

QUADRATIC DIFFERENTIAL EQUATIONS AND NON-ASSOCIATIVE ALGEBRAS Lawrence Markus

§1. STATEMENT OF THE PROBLEM AND THE RESULTS In order to obtain a thorough knowledge of the qualitative be­ havior of the solutions for a class of non-linear differential equations, we classify and analyse differential systems which have quadratic poly­ nomials as coefficients. We find all such differential systems in the plane, and an interesting collection of such systems In higher dimensional spaces. The method of classification is based on an algebraic technique and thus differs from the geometric methods which are customarily used in the qualitative theory of differential equations. While the algebraic method is somewhat more intricate than the geometric analysis in dimension two, It is available in higher dimensions where the geometry is more diffi­ cult to manage. To each quadratic differential system we attach a certain nonassociative, but commutative, real linear algebra. The problem of affine equivalence of differential systems Is shown to be the same as the iso­ morphism problem for the corresponding algebras. This is analogous to the classification of linear differential systems by canonical forms for the coefficient matrix, that Is, by means of certain linear endomorphisms of a vector space. Thus we first find all real, commutative, two-dimensional, linear algebras, see Theorems 6, 7, and 8. This yields all quadratic differential systems, up to affine equivalence, in the affine plane. It is easy to show, Theorem 9> that there are just six geometrical types of quadratic differential systems with isolated critical points in the plane. Repre­ sentation for these six geometric types are pictured in Figures I through VI. Finally there is a brief discussion of quadratic differential systems In the real projective plane and in higher dimensional spaces. The standard summation notation is used throughout. 185

MARKUS

186

§2.

THE ALGEBRA CORRESPONDING TO A QUADRATIC DIFFERENTIAL SYSTEM

DEFINITION. )

A quadratic differential system is 9

dtT" =

where the n

q

real constants

i = 1, 2, ..., n

i a^

,

± * a ^ = a^..

are normalized so that

DEFINITION. The related real linear algebra vi of the quadratic differential system is defined by the multiplication table for a basis u1, u2, ..., uR as Uj • u^ = aj^u^. Clearly the n-dimensional algebra ?i is commutative but it may not be associative. THEOREM 1.

Two quadratic differential systems

('/)

x1 = and

y)

y1 =

k = i, 2,

are equivalent under a non-singular linear trans­ formation x if andonly if their related algebras respectively, are isomorphic.

Write

?i

and

y

,

PROOF. Suppose x1 = h^y^ carries thesystem cf)into B^b^ = h^B^ = 5^, the Kronecker symbol,so y^ =B^x^". f * = Bfx1 «

B^bJb^yV =

.

Thus ajk ~ Bibjbkars

or ajkbi ~ bjbkars

Now consider the related algebras with multiplication tables

$

Then

187

QUADRATIC DIFFERENTIAL EQUATIONS

The linear transformation of ''i onto' .>( defined by is now shown to be an isomorphism. For u . • uk = ajku± -- > Sjfcbju, and ■1 r.

.s„ _ .r-. s i jr * k 3 " j k r s i

But *1 ,r, s J jk i " j k rs Conversely, if there which is expressed in terms of ii the linear transformation x Here we say that equivalent.

(p

*

is an isomorphism between ?i and S , the given bases by u, -- > b ^ . , then i = bjy carries cp into* cp, as required. and

(p are affinely (or linearly)

It is of interest to relate the algebraic properties of ?i to the behavior of the solution curves of cp • The next three theorems give certain general results in this direction. THEOREM 2. c? )

x1 =

The quadratic differential system 1, j, k = 1 , 2,

n

has an isolated critical point at the origin if and only if the related algebra ?i has no nilpotent ele­ ment e 4 0 for which e • e = o. PROOF. If cp has another critical point P than the origin 0, then the line OP consists of critical points since cp is homo­ geneous. Take OP to be the x1-axis and then the coefficients of cp (in the new coordinates) satisfy a^ = 0 for I = 1, 2, ..., n. But then the first basis vector u1, for the related algebra ?i, satisfies u., • u1 = 0 . Conversely if e • e = 0 , take a linear transformation of the x-space so that e = u1, the first basis vector of?i . Then a^ = 0 and the x1-axis consists of critical points of Q . Q.E.D* THEOREM 3*

(?)

X1

= a ^ x ^

The quadratic differential system i,

j,

k

=

1,

2,

n

has a solution which is a ray to or from the origin

188

MARKUS

If and only if the related algebra ?i has a non-zero idempotent (i.e., an element e ^ 0 for which e • e = e ). 1 i PROOF. If (}> has a ray solution, saythe x -axis, then a1 1 = for i = 2 , 3 , ..., n. Thus the basis vector u 1 in?( satisfies u 1 •u 1 = xu1for some X ^ o. Take e = 1/x u 1 . Conversely if ?i has an idempotent, say u 1 • u 1 = u1, then a^ = 0 for I = 2 , 3 , ..., n and a^1 = 1 so thex1-axis Is a ray solution of q). Q.E.D. COROLLARY. If n = 1, 3> 5* 7* ••• is odd, and if has an isolated critical point at the origin ( has no non-zero element e • e = 0 ), then has a solution ray( ?i has an idempotent element). PROOF. This is an Immediate consequence of the topological resuit that the sphere Sn“ 1 cannot support a continuous, non-vanishing, c tangent vector field. Q.E.D. THEOREM k. (?)

X1

The quadratic differential system

= a ^ x ^

i,

j,

k

=

1,

2,

..

has an invariant r-plane, 1 < r < n, through the origin if and only If the related algebra ?i has an r-dimensional subalgebra. PROOF. Say the r-plane spanned by the coordinate axes of x 1 , x2 , ..., xr Is invariant under the flow of . Then a"I ^ = 0 for r + 1 < i < n and j, k < r. Thus the basis vectors u1, u2, ..., u^ of span an r-dimensional algebra. The converse also follows from the same observation. Q.E.D. If has an r-dimensional ideal then, in a certain sense, ()} has a projection on an (n-r)-dimensional quadratic differential system. However if si splits as a direct product, then so does , as is In­ dicated in the next theorem. THEOREM 5 . Let c? be a quadratic differential sys­ tem with related algebra ?i . If there exist Ideals ^ and a , intersecting only in zero and spanning ?i , then is affinely equivalent to a product differ­ ential system of the form

QUADRATIC DIFFERENTIAL EQUATIONS

189

i, j, k = 1 , 2, ..., r

x1 = and X1

=

aj^x*

i, j,

k

=

r +

l,

n

.

PROOF. We require that a^. = 0 for i < r with j > r or k > r; and for i > r with j < r or k < r. Choose a basis u2, ..., up, •••> ^ 'l such that u 1 , ..., Up is a basis for the ideal ^ and ur+1> •••> is a ^as:i-s ^or the ideal a . Then the multiplication tensor ajk has the desired properties. Q.E.D. §3.

EXAMPLES AND APPLICATION OP QUADRATIC DIFFERENTIAL SYSTEMS

In the study of the potential flow from a doublet we are lead to the differential equation M

dx

=T

2

2

-* 2xy

whose solution curves describe the streamlines. The equi-potential lines" are the orthogonal trajectories which thereby satisfy

The quadratic differential system x = 2xy,

• 2 2 y =y -x

corresponds to the algebra 4) of Theorem 8, which is the complex numbers. Next consider the quadratic differential system x = yz - x2 y = xz - y2 z = xy - z2 which occurs in a solution of the Einstein gravitational field equations, cf. [9]. The corresponding algebra has a basis of idempotents e^ e2, e^ with

MARKUS ©1

* ®-j

®i

y

~~

e1 ’ e2 = - i e3' 61 •

®2' e2 • e3 = - \ ei

e3 = " £

•

There is just one (independent)nilpotent of index two, namely, e1 + e2 + e^. Next consider the Jacobi elliptic functions1 u = sn t, v = cn t, w = dn t for a fixed parameter value of k2 on o < k < 1. Then du = vw , — dv = - uw , dw ,2 — — = - k uv dt dt dt This corresponds to an algebra with basis that e1

• e1= o ,

e1

* e2 = " \ kS

e.,, eg, e^

e2 •

e2 = o ,

’ e1 '

e3 = " i

.

of nilpotents such

e3 # e3 = 0 ®2 ' e2 * e3 = | el

•

Let Ei ■ i v

Es = i e 2>

e3 = e3

and then E1 • E1 = 0, Eg • Eg = 0,

E3 • E3 = 0

and E, •Eg = -

1 E 3,

E, •E 3 = -

1 Eg,

Eg •E 3 =

1 E,

Thus for all k, the corresponding quadratic differential systems for the elliptic functions are linearly equivalent. One of the major applications of quadratic differential systems occurs in interaction processes which depend on collisions of entities. For example, second order chemical processes, or biological interactions are often of this nature. Following Volterra [l8], we consider n > 2 species c^, a2, ..., an of creatures with populations N1, N2, ..., NR respectively. Say that the members of cr^ each have mass and also assume that each species The author wishes to thank Professor G. Birkhoff for calling attention to this example and for discussing with him the topic of quadratic differ­ ential equations.

QUADRATIC DIFFERENTIAL EQUATIONS

191

has a natural net birth-death rate s^. Assume that when individuals of species and o • meet, there is a certain probability that the member of will destroy and eat the member of ex.. Then the fluctuations of the populations with time t can be studied by the differential system

dN± , -rr= = ( dt

V

^ i

L

P. a, .N. ) N, p±

iJ ] /

\

i

>1 for i = '1, 2 , ..., n. Here the a^. are real biological constants and the conservation of mass requires that

pi One computes easily plN1 + P2N2 + ... + PnNn = + ... + enPnNn> which leads to a first integral in certain important cases. For example, if e1 = e2 = .•. = en = 0 we have a quadratic differential system with an integral P1N1 + ... + PnNn = const, which is a hyperplane. To bemoredefinite consider the

( '1 *

”2 )

*2 - ( '2 * ^

case of two Interacting species,

",

», ) "2

•

Then an integral is £p r ^1 ^2 1 ^ 1 1 r ”*^2 ^ 1 2 ^ 2 1 N12 exp | 2--1'| N2 1 exp j ^ j =const.

In case

s1 = e2 = 0 we have the quadratic differential system N, = P27N1N2,

N2 = -

for 3

> 0,

9*1o > 0, 7 = ~ > 0 P1

.

MARKUS

This corresponds to the algebra 3 ) of Theorem 6. As a second example of Volterra’s theory consider the problem of the three fish. There are three species o^, o2 , a 3 with populations N1, N2, N3, respectively. We assume that the net birth-death rates £1 = 62 = e3 = that is, each species is in equilibrium with the en­ vironment whenever the other two species are removed. Assume that the first type of fish eats the second type, which eats the third, which eats the first. Then a12 > 0 , a23 > 0, a^ > 0 . The resulting quadratic differential system corresponds to an algebra with a basis of nilpotents e.j, e2, e^ such that e1 • e1 = 0 , e1 • e2 = Ae1 Here the positive

e3 # e3 = 0

- Ce2, e1. e3 = - Be1 + Ee3, e2 • e3 = De2 - Fe3

.

constants A, B, C, D, E, F are given by

2A = P2ai2

2BB - 2

P3a21 Po

2E =

2D

0,

e2• e2 =

_ 3

- 2C

P,a 131

Pla21 P2

2P = P2a32 ■■■JP3

Volterra’s theory can be modified to apply to military situations, where it is termed Lanchester’s law [5], but here the conservation of mass is no longer required since civilized warriors do not (as yet) eat one another. Finally we point out that by increasing the dimension of the space of dependent variables we can make a differential system with nonhomogeneous quadratic polynomial coefficients into a quadratic differential system. For example, consider the damped harmonic oscillator with the 2 equations of motion x + hx + a> x = 0, for positive constants h and o>* In the affine phase space this is written •

x = y,

y = - o)2x - hy

.

Consider the differential system in the real projective plane, with homo­ geneous coordinates (x, y, z) and write the differential equations in an affine plane "at infinity" where x 4- 0 . In this affine plane use the coordinates v -Z , X

z -i X

QUADRATIC DIFFERENTIAL EQUATIONS

193

and then Z =

-

ZV,

V

Now introduce a new coordinate w system in R , •

= -

Z

The solutions are solutions

ZV,

*

v

=

-

=

-

O)

2

-

,

2hv

-

V

2

and consider the quadratic differential 2 2

a>

w

2hvw

-

-

2

V

•

y w

=

of this enlarged differential system, for the harmonic oscillator.

§4.

0

in the plane w = 1,

REAL COMMUTATIVE ALGEBRAS OF DIMENSION

TWO

LEMMA. Let vi be a real 2-dimensional commutative linear algebra. If ?i doesnot contain a (non-zero) nilpotent element with squarezero, then ?i con­ tains an idempotent e withe • e = e 4 °« PROOF. U1 • U1 = If

a ^

=

0,

For some basisu1, u2 of

?f

we have

+ a ^ U g , u 1 • u g = a]gu 1 + a ^ U g , u g • u g = a^u., + aggUg. th e n

a^ 1

40

and

ta k e

e - - r ui ai1 which is idempotent.

1 = 0, If a22 e

2 4 0 and one could take then a22 = 4

-

u2

22

which is idempotent.

2 1 Thus suppose a11 4 0 and a22 4 °* Consider the real cubic polynomial a22x3 + (2a12 - a22)x2 + (a]1 - 2a22)x - a2.,. This has a real non-zero root xQ. Thus a 22X 0 + 2 a i2X 0 + a i1

x 0 fa22X 0 + 2 a i2X 0 + a i1 1

*

If a | 2 x 2 + 2afgX0 + a 2 , = a ^ x 2 + 2 a ] 2 x Q + a 1 1 = o, then E = u, + x QU g is nilpotent which is impossible. Then E • E = XE where \ = a11 + 2a12xQ + a22x2 4 0. Define the idempotent e = 1/ \ E. Q.E.D.

MARKUS

We first classify the quadratic differential systems in the plane where the origin is not an isolated critical point. Here the related alge­ bra contains a non-zero nilpotent element with zero square, that is, a nilpotent of index two.

ro

CVJ

CVJ

10)

0

e2 = 0 , e1 • e2 = e2 e2 = 0 , e1 • e2 = ei e2 • e2 - e2 e2 = 0 , e2 • e2 = e! e2

5) el * e = 0 , 01 6) el * e = 0 , ei 7) el * e = 0 , ei 8) el * e = 0 , ei 9)

0 , e1 •

11

0

1) el * e = 0 , 2) e1 * e = 0 , e2 3) e1 * e = 0 , e2 4) el # e = 0 , ei

0

CD

THEOREM 6. There are ten (types of) real commutative 2-dimensional algebras which contain a nilpotent of index two.

e2

e l, e 2

e2

= e 2> e2

* e2 = ke; = e2 or ei * e2 = ei' or ei * e2 = e2" Exac'tly one °f these alternatives must hold. For only in the first case does e1(which is the unique nilpotent up to a scalar multiple) annihilate the algebra upon left multiplication; in the second case e^ generates a 1-dimensional ideal and this Is not so in the third alternative. Now take a basis in the algebra ?( with e1 •e^=0, e1 * e2 = 0# Write eg • eg = fe1 + ge2 for real constants f, g. By a change to a new basis, still preserving the properties e1 * e1 = o, e1 • e2 = 0 we can always obtain e2 • e2 = e2 or e2 • e2 = e1. These yield the algebras 4) and 5) above. They differ in that k) contains an idempotent whereas 5) does not. Now take a basis in the algebra e1

*e 2

=

el •

Talce a new b a s i s E 1 = ae-j,

so thatE1 • E1 = 0 then

?i with e.j a

4 °,

and E1 • E2 = E1. Write

E2 ‘ E2 =

and

•e1 = o, Eg = X e 1 + e 2

e2 • e2 = fe1 + ge2 and

+'f E, + gEg

.

If g = 0, choose a = 1, \ = 1 / 2 f which makes E2 nilpotent and this is not allowed here. Thus g 4 °. If g 4 we can obtain E2 *Eg = gE2* If g = 2, one obtains either Eg • E2 = E1 + 2E2 or E2 •E2 = 2Eg. These yield the algebras 6) and 7) above. The algebra 7) has no 1-dimensional subalgebra, other than that generated by a nilpotent element, and so 7) is different from every algebra in 6). If k = 2 in 6) then every element generates a 1-dimensional algebra whereas if k 4 2 there is only one (up to scalar multiples ) non-nilpotent element which generates a 1 -dimensional subalgebra. Now consider the Case 6) with k 4 2 . Here the 1 -dimensional space generated by e2 is dis­ tinguished. But then e1 • eg = e1 normalizes e2 so that k is an invariant of the algebra and distinct values of k determine non-iso­ morphic algebras.

MARKUS

Finally take a basis in

?i

corresponding to the third alterna­

tive: e1 • e1 = o, ei

* e 2 =e 2

*

Write e2 • e2 = fe1 + ge2* Replace e2 by E2 = |ie2, \x 4 Then E2 • E2 = ii2fe1 + ngE2. If g = 0 , we can obtain algebras 8) or 9). If g 4 °> we obtain algebra 1 o). In this last case we exclude k = 0 since this yields an algebra with a basis of nilpotents. In the algebras 8 ), 9 ) and 1 0 ) the idempotents ae1 each de­ fine a linear transformation upon multiplication of ?i . Moreover only e1 produces an eigenvalue of + 1 and that for the eigenvectors be2, b 4 o* Thus if there is an isomorphism of algebra 8 ) onto 9 ) or 1 0 ) it must carry e1 to e., and e2 tobe2. This distinguishes between algebras 8 ), 9 ), and 1 0 ). Also for distinct values of k in the alge­ bras 1 0 ) we obtain non-isomorphic algebras. Q.E.D. The only algebras occurring in this Theorem 6 which are associ­ ative are 1 ), 4), 5 ), and6 ), for k = 1 . The algebra 6 ) for k = 2 is power associative and allthe remaining algebras of Theorem 6 are not power associative. The necessary and sufficient condition for the power associativity of real commutative algebras is the identity x2 • x 2= p (x • x) • x, cf. [1 ], and a direct computation yields the desired result.

CD

0

0 ,

2 ,

2

-
so as t o have a u n iq u e form f o r ea ch p o s s ib le

a lg e b r a . Thus c o n s id e r cc ^ 0 ,

oc ^

id e a l.

cc ^ 2 .

e 1 • e 1 = ae^

Here

e2

•

oc

e2

fo r

U sing - e

and

e2 • e2 = e2, e 1 • e2 = e2

i s th e u n iq u e idem p oten t w h ich g e n e r a te s an

+ (1 - - ) e r oc

^

a s a b a se f o r th e a lg e b r a we f i n d t h a t th e a lg e b r a s w ith v a lu e s

a

and

a r e iso m o rp h ic and t h a t no o th e r v a lu e o f th e p a ra m eter y i e l d s an iso m o rp h ic a lg e b r a .

Thus we p i c k o u t a s i n g l e r e p r e s e n t a t i v e f o r ea ch a lg e b r a o f

Case 2 c) b y demanding oc

>

a

-

1

Thus 0 < or < 1 and oc > 2 e n fo r c e s a o n e -to -o n e co rre sp o n d en ce betw een p a ra m eter v a lu e s oc and isom orphism c l a s s e s o f a lg e b r a s i n 2 c ) . C o n s id e r Case 3a) w ith

oc =

2

(so

p ^ 2 ).

id em p o ten ts i ei

80,1

i

e2

•

There a re j u s t two

QUADRATIC DIFFERENTIAL EQUATIONS

The product of these idempotents is

i

(i ei ) + i ^

•

Thus isomorphic algebras in 3a) must yield the same value for i +1

.

Thus p is different for different algebras. The case is isomorphic to an algebra with a = 2 and p 4 2.

p = 2

Now turn to Case 3b): e1 • e1 = ore1, e2 * e2 = Pe2 ' ei ’ e 2 = ei + e 2 with cc 4 0, a / 2, p

4P

4 2, ap

4 4, a + p / ap .

There are three idempotents E2 = p e2 »

E! = a 01' and

The table E, • E2 ' ? E, *

i E2

E, • E, = - v--P' 1 3 if _ ap \

2

■

E, + - E 1 a 3 9

. E = I E + SL.±£ .r . Sfi. E 3 p 3 4-ap 2

shows that the algebras of 3b) with parameter values or (p, oc), or 4 - ap \ a, -------- J, a+p-ap/

or

, (a, p) = (a, P),

/4-ap ( -------- , a \ a + p - ap

or Qf+P-QfP

, P ), /

- ap or ( p, \ Qf+p-Qfp

MARKUS

200

are isomorphic.

Also these are the only Isomorphic parameter sets.

To obtain a unique parametrization of the algebras in 3b), we must find a fundamental domain, in the (a,p)-plane with the deletions a 4 °> & 4 2, P 4 0, p 4 2, ap 4 a + p 4 ( - t£ t ? S )

T : (p, oc)

.

We compute T2 = R2 = I, the identity. Also RTR = TRT. Call Z = RT (first T then R) and 7? = I, RZ = Z2R. Thus the group con­ sists of six elements I, Z, Z2, R, ZR, and Z2R. Use the fact that

ap oc

+ p - ap

is a monotonic decreasing function of p, for each fixed a. A careful study of the geometry of the six transformations of the group I, Z, Z2, R, ZR, Z2R shows that every allowable point of the (a,p)-plane is equivalent to one and only one point of the fundamental domain OC -

with a > 2,


0, g2 < 4(1 - 2d) • e1= e1 , e1 • e2 = ^ e2, e2 • e2 = - e1

c)

d) e1 • e1= , e1 • e2 = de2, e2 • e2 = - e1 + ge2, with d 4 °> d 4 g2 4 ^d2, g2 < 4(2d - 1), g > 0. 4) e1 • e1 = e.,, e., • e2 = e2, e2 • eg = -

.

Distinct values of the parameters yield non-isomorphic algebras. PROOF. Let e1 be the idempotent element of % . Then there is a basis e^ eg in ?i such that exactly one of the following three cases holds: CASE 1:

e1•e2

=

CASE 2:

e1•e2

= e1 +

CASE 3:

ei*e2

= de2> for d ^ 0.

To see this Case 1. If c / 0, d Case 1. If c = o, d d 4 0. Herereplace

0 e2

write e1 • e2 = ce1 + de2. If c = d = o we have = o replace e2 by E2 = - ce1 + e2 to obtain 4 0 we haveCase 3* Finally suppose c 4 and e2 by =2 - r h

(if d 4 1) to obtain Case 3 again.

ei - e2 If d = 1, c 4 0 use

E2 = \ e2 to obtain Case 2.

Thus every

Also an algebra For in ?i the element The linear transformation a null space in just Case vector, e1. In Case 3

Case 1, 2, or 3*

?ican fall Injust one of these three cases. e1 is distinguished as the unique idempotent. T1 : ?i -> ?iof multiplication by e1has 1.In Case 2 T1 has just one independent eigen­ T1 has two independent eigenvectors.

Now consider an algebra cases designated by: 1A)

lies in

y

of Case 1. We consider three sub­

e2 • e2 = ge2 for g 4 0

MARKUS

2 02

IB) e2 # e2 IC) e2 • e2

= fei ^OP f > 0 = + e1 + ge2 for g 4 0.

Suppose e1 • e1 = e^ e1 • eg = 0, * e2 = fel + ge2* Sub” case 1A) is inadmissible since it allows a basisofidempotents in . If g = 0, f 4 0 thenthere are nilpotents unless f > o. If g 4 f 4 o ^ have Case 1C), upon replacing e2 by E2 = |f |~1//2e2. -.1 / 2 In Case 1 B ) replace e2 by E2 = f“ 1 e2 to obtain the result e1 • e1 = e^ e1 • Eg = 0, Eg • E2 = e1 which Is listed in the theorem. In Case 1C) the possibility e2 • e2 = - e1 + ge2 is not allowed since It admits a basis of idempotents.Also in case e2 • e2 = e1 + ge2 we must have g2 < 4 to prevent a basis of idempotents. But the values + g and - g yield isomorphic algebras under the automorphism e1 -- > e^ e2 -- > - eg. Therefore every algebra ?i of Case 1 is listed under 1) in the theorem. It is easy to check that each algebra of 1 ) has exactly one idempotent and no nilpotent of index two. Also in an algebra ?i of 1 ) the element e^ is distinguished by e., • e1 = e1. Also the subspace Xe2 is the null space of T^ multiplication by e.,. The condition e 2 • e2 = ei + ge2 fixes the pair {e2, - e2}. But then the restriction g > 0 shows that no algebra in 1 ) corresponds to two distinct admissible values of g. Now consider Case e2 * e2 = f0l + ge2*

^

2 ),

e1 • e1

g ^ 0 we replace

e1 • e2 = e1 + e 2, e2 by

to obtain e1 • e1 = e^ e1 • Eg = e1 +Eg,E2 • Eg = he1. Therefore we can always take g = 0 in Case 2. If f > - 1 there are two independent idempotents in ?i and so we are lead to the Case 2 ) listed in the theorem. Each algebra of Case 2 ) has no nilpotent of index two and only one Idempotent. In an algebra y of 2 ) the element e1 is dis­ tinguished by e1 • e1 = e1. The affine space {e2 + ore1) is distinguished by the relation e1 • e2 = e1 + e2* But the only member of {e2 + ae1) whose square is a scalar multiple of e1is e2, which is thereby dis­ tinguished in ' . ' i. Thus distinct values of f yield non-isomorphic alge­ bras In Case 2 ). Now consider Case 3, e1 • e1 = e.,, e1 ' e2 = de2 > e2 * e 2 = **ei + &e 2 f with d 4 0* If f = 0there is either a nilpotent of index two or two idempotents in vi so we must have f / 0. Replace ep by E2 = If |_1//2e2 and thereby we can assume that f = + 1 or

203

QUADRATIC DIFFERENTIAL EQUATIONS

f = - 1. Further we separate thecase where theoperator T1, of multi­ plication by e1#is the identity, that is d = 1. We are then lead to three subcases of 3: I)

e1• e1 = e ^ e1 • e2 =

e2, e2 • e2 = + e1 + ge2

II)

e1• e1 » e ^ e1 • eg » d 4 o, d 4 1

de2, e2 * e2 = ei + Se2'

III)

e1• e1 « e ^ e1 • e2 = de2, e2 • e2 = - e1 + ge2, d 4 °> d 4 1•

An algebra ?r of Case 3 has a distinguished element e1 with e1 • e1= , e ^ . Thus the operator T1 is an algebraic invariant and in 31) it is the identity. Case 3 II) is distinguished from 3 III) in that the eigenvector e2 has a square with e2 • e2 = e1 + ge2, instead of

e2 * e2 = " 81 + ge2# In 3 I) the possibility e2 * e2 = ei+ Se2 is eliminated since such an algebra always has a basis of idempotents. Also the possibility e2 # e2 = ~ el + &e2 I’es‘tricted since if whereas if g > 4 there are two

o

g

= k there are nilpotents idempotents.

Thus take g2 < k and define new basis vectors E2 = Xe1 + ne2 where \x > 0 and \ are defined by

E1 =

Then one computes directly E 1 • E 1 = E 1, E 1 • E2 = E2, E2 • E2 = which is algebra 4) In the theorem. This algebra is the complex numbers. We next show that in 3 II) the admissible values of the para2 1 2 meters are g < 4(1 - 2d) and g = 0, d = -g . Use (ae1 + pe2 ) = (a2 + p2 )e1 + (2apd + p2g)e2 to show that there are neither nilpotents

nor idempotents, with

p 4 °>

f°r g2 < Ml - 2d).

But if g2 > 4(1 - 2d)

then there are two idempotents and so this is not admissible. If g2 = 4(1 - 2d) there are two Idempotents in unless g = o, d = ~ . Now the interchange of e2 and - e2 replaces g by - g and so we can take g > 0. Thus algebras of 3 II) are those of Cases 3a) and 3b) in the theorem. In Case 3b) the arithmetical condition g2 < 4(1 - 2d) shows that d 4 d 4 ~ . Thus 3a) is different from any algebra in 3b). In an algebra of 3b) we can distinguish the idempotent e^, then the pair {e2, - e2) by e2 • e2 = e1 + ge2* Since g ;> 0 there Is just one algebra for each admissible parameter value. Next consider the cases 3 III).

We shall show that the admissible

MARKUS

values are exactly those where d 4 °> d / 1> g 4 ± 2d, g2 < 4 (2d - 1) and also g = 0, d = | . Use (are., + pe2 )2 = (a2 - p2)e1 + (2apd + p2g)e2* Since g 4 + 2d, there are no nilpotents of index two. Since g2 < h (2d - 1), or g = o, d = ^1 , there are no idempotents other than O O p e1. Now suppose d 4 o, d ^ 1 but g = 4d or g > 4(2d - 1) (ex1 2 2 of index two. ” cepting g = 0, d = ^). If g = 4d there are nilpotents If g “ > 4(2d - 1) then there are idempotents other than ep (provided we exempt g = o, d = ^). Therefore we have the algebras of Cases 3c) and 3d) In the theorem. ”

-

Finally we must non-isomorphic algebras. idempotent e^, then the Interchanging e2 and there is a unique algebra

note that two parameter values in 3d) define For such an algebra ?ifirst distinguish the eigenspace of e2, then the pair {e2, - e2). e2 replaces g by - g and since g > o foreach admissible parameter value. Q. E. D.

Only the algebra 4) in Theorem 8 Is associative. Moreover none of the other of these algebras in Theorem 8 is even power associative. In Theorems 6, 7, and 8 we have found all real commutative alge­ bras of dimension two. An Interesting by-product of our classification Is the following result. COROLLARY. The only real commutative two-dimensional algebra which is power associative, yet not associ­ ative is: e1 • e1 = 0, e1 * e2 = ei> e2 * e2 = 2e2

#

This is a Jordan algebra, that is, it satisfies the identity (uv)u2 = u(vu2 ). §5.

QUALITATIVE BEHAVIOR OF QUADRATIC DIFFERENTIAL EQUATIONS IN THE AFFINE PLANE

We shall study the topological behavior of the solution curve familites for quadratic differential systems in the plane, with an isolated critical point at the origin. Thus we shall utilize the algebras listed in Theorems 7 and 8. Consider the algebra 1) of Theorem 7: e1 • e1 = e1# e2 • e2 = e2, e1 • e2 = o . This corresponds to the quadratic differential system

QUADRATIC DIFFERENTIAL EQUATIONS

x = x2 ,

2

y• = y2

Corresponding to the three idempotents e^ e1 + e2, and e2 there are three lines through the origin which define solution curves. The compu­ tation

where e = arctan y/x, displays these ray solutions along the lines x = 0, y = o, and y - x = 0. We designate a ray solution by + if the radial velocity is positive and - if the radial velocity is negative. For each sector between the ray solutions we write an arrow to the right -- > if the angular velocity is positive and an arrow to the left o, and 3b) for a < - l , o < p < i , in Theorem 7* •

III. x = 2x2 + 2xy, y = - y2 + 2xy, which represents Cases 2b), 3a) for p < o, and 3a) for P > 2, in Theorem 7« IV. x• = 2x 2 + 2xy, Cases 3a) for

y* =2y + 2xy, which represents 0 < p < 2 in Theorem 7*

All of the above have four or six directions of approach to the origin, and next there are systems with just two directions of approach. V.

x = x2 + y2, y = y2 which represents Cases 1 ), 3a), and 3b) of Theorem 8.

VI. x = x2 - 2y2 + 2xy, y = 2xywhich represents Cases 2), 3c), 3d) and 4) of Theorem 8.

QUADRATIC DIFFERENTIAL EQUATIONS

PROOF.

2

By examination of e =

x2 + y2

we sketch the solution curves and find the schemes for the quadratic differential systems as follows. In Theorem 7, we find 1)

---> +

2a) + +

-- > + + + - for all a 3a) -- > + for p < o

2C )

/\

i

+

V

- + -- > + 2 3b)

+ 2, 0 < (3 < 2, b - ap > 0, and for a < -- 1I , 0 < 0 < 1 +— >

-> - +

3b)

-- > + for all d and g

3c)

+ 00 as n — > + oo. -

(iii) Pn > P e ft as n ------ > + 00. (iv) rp (t ) --- > Q e ft as n -— > + 00. n (v) {Tp (t )} has no limit point in ft.

LAGRANGE STABLE MOTIONS

223

We assume, without loss of generality, that 0 is at the origin. Let o 2 1 /2 * p = {x^ + ... + xn)' .The condition E =E means that every point of egress from o> is a pointof strict egress; in other words, p = p(t) along any orbit which intersects cd cannot have a relative maximum on the surface p = X. This, in turn, is certainly the case if for every P e H(cd) such that: (a) P is not a critical point; and (b) 2a dt d2§ >0. In fact, this last statement is actually stronger we would have — dtd than the statement E = E and could be weakened somewhat if necessary. In terms of the notation employed for system (1.1 ) we have: xi

(^.1 1 )

p2 =

X 1=1

n (4 .1 2 :

z^

dt

i=1 n (1U13)

^

n

: (! *£♦ 1^

=f3 = p ( ^ ^ + i=1

j=1

Thus the sufficient condition under consideration obtains the following form: COROLLARY 1. Let X > 0 be arbitrary. If for every regular point PCx^ ..., xn ) = P(x) such that n 2

X

i=1 and at which n

pp =

X xjf^x) i=1

= 0 ,

224

MENDELSON

there holds the Inequality n

n

(4.14) i-1 then there exists a semi-orbit wholly contained in the open sphere of radius X and center 0. (4.2) Next let co be an open box with sides parallel to the coordinate axes; i.e., the open set defined by the inequalities - x^ < x^ < > 0, > o, (1=1, ..., n). Following the same line of reasoning as in (4.1 ) we obtain COROLLARY 2. Let X± > 0, > 0, (1=1, ..., n), be arbitrary. If for every regular point P(x.j, ..., xn ) = P(x) lying on the Ith face of the boundary of cd (i.e., such that x^ = - x^ or x^ = n^) at which fj_(x^ •••> xn ) = 0, there holds the inequality (4.21

then there exists a semi-orbit wholly contained with­ in the box - X^ < x^ < up (I = 1 , ..., n). Other topologically equivalent choices of cd will result In still further analytical expressions for the above sufficient condition. Columbia University BIBLIOGRAPHY [1] MENDELSON, P., "On dynamical systems without improper saddle points," to appear. [2] FLIS, A., M0n a topological method of the study of the behavior of the integrals of ordinary differential equations," Bull. Acad. Polon. des Sciences Cl. Ill, Vol. II, No. 9 (1954), pp. 415—41B. [3] WA^EWSKI, T., "Sur un principe topologique de l!examen de l fallure asymptotique des Integrales des equations differentielles," Ann. Soc. Polon. de Math., Vol. XX (1947), pp* 279-313* [4] WAfcEWSKI, T., "Sur une methode topologique de l ’examen de 1 Tallure asymptotique des integrales des equations differentielles," Pro­ ceedings of the International Congress of Mathematicians (1954 J, VoiY "37 PP« 1 3 2 -1 3 9 .

XI.

THE LOCAL THEORY OF PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS I.

IDEAL SYSTEMS

J. Andre and P. Seibert INTRODUCTION The theory of automatic control, which Is an important field of modern engineering, gave rise to the investigation of a class of differ­ ential equations which, until very recently, have not been studied by mathematicians; these equations involve piecewise continuous functions. Among the previous contributions to the subject we mention a paper of Solncev [10] which gives a stability theory for two-dimensional piecewise continuous systems, the monograph of Flugge-Lotz [5] who investigates various piecewise linear systems and stresses mainly the engineering point of view, two notes by the authors [la], [lb] concerned with n-dimensional piecewise linear systems, a number of recent papers on the problem of optimal control (e.g., those of Bellman, Glicksberg, Gross [2], Boltyanskii, Gamkrelidze, Pontryagin [3], [6], [9], Krasovskil [8], Bushaw [4]), and a paper of Krasovskil [7] on stability in the large of piecewise continuous systems. The present paper gives a generalization of the two above-mentioned notes by the authors. The systems under consideration are piecewise of p type C and the set of discontinuity Is assumed to consist of certain hypersurfaces of class C^. The latter are called the "switching spaces’1 of the system, their points "switching points11.* The precise hypotheses and the definition of a solution will be given in §1. In §2 we study the 1 It will be noted that in most of the papers on optimal control ([2], [3], [6]> [8], [9]) the right hand sides of the differential equations considered depend discontinuously on t and not, as in the other papers (including the present one), on x. However, since the problem treated in these papers is to find optimal switching times for a given initial point, the solution of this problem for all initial points is actually equivalent to that of finding certain switching spaces in the sense considered here (provided, the uncontrolled system Is autonomous). The systems studied by Krasovskil In [7] (which are generalizations of those occurring in the control problems) have right hand sides which also depend discontinuously on t. 225

226

AJMDRE AND SEIBERT

behavior of the solutions near the switching spaces and give a classifica­ tion of the switching points. The latter consist of three principal types: transition points (i.e., points at which a solution traverses the switching space), end-points (at which two solutions "end", (i.e., are not continuable beyond the point), and starting points (at which two solutions "start", i.e., are not continuable Into the past). At switching points of some of the other classes (which are less frequent) solutions may fuse or fiburcate. In §5 we describe qualitatively the entire sets of solutions passing through given closed subsets of a switching space. For this purpose we introduce "local flows" (In §4) which are defined on compact subsets of the phase space and can be con­ sidered as "local dynamical systems" in analogy to the concept of local groups. The system of solutions around the switching spaces can then be conceived as a complex of local flows, connected with each other by certain neighborhood relations. The second part of the paper will be devoted to systems with switching delay. These formally belong to a class of difference-differential equations. They are distinguished from the systems considered in the present part'of the paper (also called "ideal systems") by the prop­ erty that the discontinuous function changes sign shortly after the tra­ jectory has reached the switching space, rather than at the exact moment of transition. They usually represent a better approximation to physical reality than the Ideal systems and, in contrast with the latter, their solutions are continuable indefinitely Into the future, i.e., they have no end-points. §1 . THE CONCEPT OF PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS 1. The system (S). Consider m real single-valued functions s^(x), defined in E31 (n-dimensional euclidean space) and satisfying the following conditions. (a) The functions ^(x) are of class C3 (I.e., they possess continuous partial derivatives of first, second and third order). (b) At no point of E11 does any function sji(x) vanish simultaneously with its gradient ( S£V

5£V N

1 Strictly speaking, this is true only for systems with "constant time lag", not for those with "threshold".

PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS

227

Under these conditions the sets (1.1)

S^ : s^(x) = 0

(ii = 1, ..., m) ing notations:

are smooth hypersurfaces of En. We introduce the follow­ m

s- U ^ ■ H=1

s(x) = (31(x),

Sm (x))

.

We now consider the domains into which the by the hypersurfaces S^. Denoting by sgn(cr1, ..., (sgn , ..., sgn am )= (o^/larj, ..., o^/laj) (o^ associate to every point x € Sn - § a vector e(x) (1.2 )

space is decomposed of^) the vector real, 4 o), we can bysetting

e(x) = sgn s(x)

Furthermore, we associate to every vector e with coordinates (open) domain1 (1.3)

De = {x e E11 - § : e(x) = e}

+ 1 an

.

Apparently, every such domain is bounded by hypersurfaces

S^.

Finally, we associate to every vector e (for which De is 4 0) an n-dimensional vector function f(x, e) which is defined and of class C2 throughout E11 2 and which does not vanish on the boundary of De. After these preparations we consider the differential equation (1.4)

x = f(x)

(x = (x.j, ..., xn ),

• = d/dt)

where the vector function f (x) is given by (1*5)

f(x) = f(x, e)

for

x e D@

1 which may be empty or disconnected. 2 As long as we restrict ourselves to ideal systems, it is sufficient that f(x, e) is defined in an open set containing the closure 5e of V

ANDRE AND SEIBERT

228

and undefined on §• form (S)

Using (1 .2 ) and (1.5)> we can write (1.4) in the

x = f(x, sgn s(x))

.

2. The Concept of Solution. Given a point p of a domain D0, there exists exactly one solution of the differential equation (S, e)

x = f(x, e)

which passes through p at the time t = 0. We denote it by x(t, p). Since the system (S) coincides with (S, e) for all x e D@, the func­ tion x(t, p) is also a solution of (S) if it isrestricted to a t-interval I for which1 x(I, p) C De holds. We now extend the concept of a solution of (S) by the following definition: DEFINITION 1.1. We call a (single valued) vector function x(t), defined in an interval I, a solution of the system (S) if it satisfies the following conditions: 1°

It is continuous throughout

I.

2° For every t e l , to e(x(t)) =sgn s(x(t)) = e], satisfies equation (S, e).

such that x(t) e D0 [which is equivalent the function x(t) is differentiable and

3° The set of values no cluster point in I.

tel

for which x(t) € S holds, has

A solution according to this definition satisfies equation (S) in the ordinary sense for almost all values t e l . In the second part of the paper we will extend the notion of solution in order to include also certain paths contained entirely in § which are related to a phenomenon in the theory of discontinuous control (after end-point motions). §2.

SWITCHING POINTS AND THEIR CLASSIFICATION

3. Topological classification of switching points. The hyper­ surfaces S^ are called the switching spaces of the system (S). Points in a switching space, or switching points, will usually he denoted by u. x(I, p)

denotes, as usual, the set

{x(t, P)^el *

PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS Unless the contrary is stated, it will always be assumed that to exactly one switching space index

n.

S^,

Under this assumption

two domains

D0

corresponding vectors

e+ + 1

u

belongs

and we will therefore drop the

u

belongs to the boundaries of exactly

which we denote by

which we assume to be

229

De+

and

De~,

respectively.

The

and

e~

differ only in their ji-th coordinate

in

e+

and

- 1

in

e” .

The solutions of

the equations

+ x = f(x, e“ )

+ with the initial value

u

will be denoted by

x~(t, u).

We first classify the switching points of the cruves definitions:

x+ (t, u)

and

u

+

77 =

+

+

7“ =

lim e (x“ (t, u)), t —» —o

both exist, the point

d

11m eM(x"(t, u)) t -» +0

u will be called

+ an

with respect to each

by introducing the following

If the limits

+ (3.1)

x “ (t, u)

A-point (with respect to

+

x"(t, u))if

+

7“ = + 1, 7“

= + 1,

+

+

an E-point

if 7" = + 1, 7” = + 1 ,

an L-point

if 7” = 7” = + 1 ,

+

t t 71 = 72 =

an R-point if In for which

+

e

the case

+

satisfies

x e D q I*

the function

solution of

of an

(xi(t, u)) = +1,

x-(t, u)

(S).

+

(S, e-)

+ -

A-pointthere exists a t-intervalB = (0, p) and

consequently

x-(B, u) C

Since

and this equation coincides with (S)

x^(t, u),

restricted to the Interval

Analogously we find that for some interval

the same function does not satisfy +

1

B,

for is a

(-a,

0)

(S).

In the case of an E-point the situation is vice versa, i.e.,

x~(t, u)

satisfies (S) for small negative values of

t,

but not for

small positive values.

+ If

u

isanR-point,x ” (t, u)

certain open neighborhood of

t = 0,

exists an interval around

in which

is a solution of (S) in a while in the case of an L-point there

+ 0

x ” (t, u)

does not satisfy (S).

Consider now the case where some of the limits (3 . 1 ) fail to exist.

This situation occurs whenever the Intersections of one or both

+ of the curves

x “ (t, u)

with

S

cluster at

u.

We call

u

230

MERE AND SEIBERT an A -point if

7”

*

i

an E -point if

*

an L -point

71

exists and is equal to + 1 , exists and is equal to + 1 ,

in all other cases.

* * # A -, E -, L -points have the above stated

It is easy to see that

properties of A-, E-, L-polnts, respectively, except that

x “ (t, u)

usually satisfies equation (S) in certain intervals clustering at

t = 0

and contained (in the case of A - [E ]-points) in the semi-neighborhoods which are void of solutions In the case of A- [E-] points. We now extend the classes

A, E, L

respectively. The set of all X-points + + to x~(t, u)will be denoted by X~. classification

u

#

by the sets

#

A , E , L ,

(X = A, E, L, R)

with respect

Then the complete topological

of the switching points is obtained by considering

intersections of the sets

X+

X+ n Y~

points of the sets

and and

Y “ (X, Y = A, E, L, R).

X” n Y+

the

Since the

are of the same topological

character with respect to the solutions passing through them, the sets of topologically equivalent switching points are given by the expressions

Y~)

(3.2) xy = (x+ n (X, Y = A, E, L, R). of these classes to

The symmetry relation

AE,

XY = YX

Behavior of solutions at switching points.

there exists a unique solution with

which is defined in an open interval around which we denote by

x(t, u),

r

u

If

u

as Its initial value = 0.

for

t > 0

for

t o [t < 0]. We call the AA- and ALpoints starting points, the EE- and EL- points end-points. If u is an AR - [ER-] point, both x+ (t, u) and x”(t, u) are solutions of (S), one being defined only for t < o [t > 0], the other for positive and negative t-values. This implies the existence of two solutions actually passing through u. Assume, e.g., u € A+ n R” [u g E+ n R” ]. (In the case u e A” n R+ [u e E~ n R+ ] the situation is analogous.) Then x1(t, u) = x"(t, u) and r x“(t, u) xp(t, u) = J L x (t, u)

for

t < o [t > 0]

,

for

t > o [t < 0]

,

are both solutions of (S) passing through u. Since for t < o [t > 0] these two solutions are identical, the situation can be described as a bifurcation [fusion] of trajectories.2 Through an LL- point there apparently exists no solution of (S). - If u is an LR- point, there is exactly one solution [either x+(t, u) or x“(t, u)] with u as its initial value. It is defined for positive and negative values of t and does not cross the switching space. Apart from the transition (AE-) points, the LR- points are the only switching points at which the existence and uniqueness theorem holds in the strict sense. - If, finally, u belongs to the set RR, both x+ (t, u) and x~(t, u) are solutions of (S) around t = 0. This implies the existence of altogether four solutions passing through u, the latter therefore being a point of fusion and of bifurcation simultaneously. Normal and exceptional switching points. Beside the topological qualities of a switching point which we analysed in the preceding setion, it isofsignificance whether the curves x~(t, u) associated to the point u aretangent to S ornot. We introduce the following notions:

1

+ i.e., the restrictions of

2

x“(t, u)

to

t ^ o [t £ 0].

The concept of solutions introduced in the paper Andr£-Seibert [la] was slightly less general than the one we use here. According to the former, there exist fusions of trajectories, but no bifurcations.

232

ANDRE AND SEIBERT

Fig. 1 .

PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS A point

u

e

233

S will be called a normal switching point if

+ neither of the curves x“(t, u) are tangent to the switching space S at u, otherwise an exceptional point. In particular, all points of the types A*, E*, L * are apparently exceptional. The same is true for all points of the types LX, RX (X = A, E, L , R ) , since for all of these at least one of the curvesx"(t, u) fails to traverse S at the point u. Denoting the set of all normal points by S°, we therefore have the in­ clusion S° C. AA U AE U EE

(5.1 )

.

The sets of normal starting, end-, and transition points will be denoted by A, E, T, respectively: A = AA n S°,

(5.2)

E = EE n S°,

We call an exceptional point

T = AE n s°

.

u

of first order, if only one of 1 the curves x”(t, u) is tangent to S at u and the contact is of 1 order; in all other cases we say u Is of higher order* By S' we de­ note the set of all exceptional points, by S,T that of all exceptional points of higher order, and by S1 = S 1 - S" the set of exceptional points of first order. The Intersections of any subset Q of S with S°, S', S", S1 will be denoted by Q °, Q', Q", Q1. 1 Since at points of

+

+ types L L, LR, RR relation

both curves

( 5 .3 )

x~(t, u)

are tangent to

LL U LR U RR C S " D e t e r m in a t io n

o f th e

ty p e

of a

S,

we have the

.

s w it c h in g

p o in t .

In o rd e r

to

state the criteria for the type of a given switching point u e S = S it 2 1 ^ Is sufficient to assume s € C , f € C instead of the stronger conditions

^

+

formulated in §1.1.

Then the functions

tions x = f(x, e~) Taylor expansion:

are of type

3 (x_(t, u)) = — m-

^

ax

C2

3 (x "(t, U)) m-

x”(t, u)

which satisfy the equa­

and we can apply the following

t

+

t =o

= c“(u)t + 1/2 c~(0t, u)t2

1 /2

2 S^( x " ( t , U) dx

T=0t

(o < 9 < 1 )

.

i— -------------In the case of the sets A, E, T [vid. (5.2)] we omit the superscript 0. — It should be noted that the sets S°, S*, S", S1 comprise only that part of S which is not contained in any other switching space.

ANDRE AND SEIBERT

234

Here + jl + C7(t, U) = -r 3 (x“(t, U)) 1 dt1 ^ and

+

+

c£(u) = c£(o, u) (i = 1, 2)

.

For the present purpose it is sufficient to examine the signs of the co+ + efficients c“(u) and c“(u). An obvious calculation yields the follow­ ing formulas: 1 (6.2)

c"(u) = f(u, e-) 1

c~(u) = f(u, e") ^

dx

3 (u) ,

3 ^(u) ) f(u, e-)'

(6.3) + f(u, e") ( f(u, e~) ^ s (u) V dx / dx H-

;

Here ~ denotes the gradient when applied to a scalar and the Jacobian matrix when applied to a vector:

d2 The operator — ^ , applied to a scalar, denotes the Jacobian matrix of dx the gradient. From the formula (6 •i ) we immediately obtain the following theorem: THEOREM 6 .1 . If u is a point of the switching space S (and of no other switching space), the following p implications hold: 1 By * we denote the operation of transposition. 2 The upper [lower] signs +, - in the superscripts corresponds to the upper [lower] signs .

PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS

c~(u) ^ 0

implies

u

€ A”

+ < c”(u) ^ o implies

u

i € E~

,

+ c~(u) = o and

+ . + c“(u) ' o together imply u e R“

+ c”(u) = 0

+ s c”(u) £ o together imply u eL-

and

+ + The coefficients c~(u), c“(u) formulas (6 .2 ) and (6 .3 ). In particular,

u

235

,

are given bythe

is

a normal transition point if c|(u) and c”(u) are either both positive or both negative, a normal starting point if c”(u) < 0 , a normal endpoint if

c|(u) > 0

c|(u) < 0

and

and c~(u) > 0 .

Unconsidered in this theorem remain only those exceptional points of higher order which involve contacts of higher than first order [i.e., at least 3 -point contact]* The three last implications require only the hypotheses f € C°, s e C1. If these are satisfied, the functions c^ are continuous and we obtain the following corollary: 1 COROLLARY 6 .1 . The sets A, E, T of normal switching points are open relative to S. Since every point of S1 involves a conclude from the third and fourth implication + 1 continuity of c”(u) (assuming again f e C ,

contact of first order, we in Theorem 6 . 1 and from the 2

s e C ):

COROLLARY 6 .2 . 2 The set S1 of exceptional points of first order consists of the sets (AL)1, (EL)1, (AR)1, (ER)1, all of which are open relative to S1 . The boundaries of the sets A, E, T may contain parts of intersections of S with other switching spaces. The latter, however, being closed sets, do not affect the validity of Corollary 6 .1 . 2 All sets with superscript 1 consist of exceptional points of first order.

236

ANDRE AND SEIBERT

§3-

TOPOLOGICAL PROPERTIES OF THE CLASSES OF SWITCHING POINTS

7- The exceptional classes. Let u he an exceptional point of S ( - S ) [vid. §2.5]. Then the conditions s^(u) = o

(7.1)

< c7(u) = f(n, e“) — s (u) = o » H y M dx ^-

hold, the second either for e+ or e“ [vid. (6 . 1 ), (6 .2 )]. The func+ •] tions s^(u) and c“(u) are of class C . Therefore, if the Jacobian Mmatrix J7(u) = ~ (s (u), c7(u)) 1 dx t1 1

~ (s (x), c7(x)) dx M*

'J

of the system (7- 1 ) is of rank 2 , it follows from the implicit function theorem that the set of all exceptional points in a certain neighborhood of u constitute an (n- 2 )-dimensional differentiable submanifold of S + [or, if c”(u) both vanish, the union of two such manifolds]. We in­ troduce the following terminology: + If the matrices J^(u) are of rank 2 at all points of S [or of a part Q of S] satisfying the corresponding conditions (7 . 1 ), we say, the system (S) satisfies condition A on the switching space S [or on the subset Q]. We can then state the following theorem: THEOREM 7.1. If condition A is satisfied on a subset Q1 of the switching space S, the set Q 1 of ex­ ceptional points in Q consists of (n-2 )-dimensional differentiable submanifolds of S If it is not empty. Every point u e Q 1 possesses a neighborhood N such that Q! n N is contained In the union of at most two of the connected manifolds constituting Q* (namely, one (connected) component of each of 1

S.

It is assumed that Q

contains no points of switching spaces other than

PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS

237

the sets {u € Qf : c|(u) = 0 } and {u e Qf : c”(u) = 0 )). If, in particular, U is an exceptional point of first order (vid. §2.5), the set Q! n N is contained in a single (connected) manifold. We now establish a similar theorem for the set SM of exceptional points of higher order [vid. §2.5]* Every point u € Sn satisfies one of the two following sets of equations: (7 .2 )

3 ^ (u) = c|(u) = c"(u) =

(7 .3 )

sn(u ) = o"(u) = CgCu) =

+

(the latter either for c£ or c£). Jacoblan matrices of these systems:

0,

+

Denote by

0

J2 (u),

Jp(u) = — (s (u), c*(u), c"(u)) ^ dx ^ 1 1

,

J“(u) =

.

3

(s (u), c"(u), c"(u)) dx ^ 1 d

(u)

the

We say, condition B is satisfied on

S [or QC S], if the matrices + J2 (u), ^(u) are of rank 3 at every point of S [or Q] satisfying the corresponding set of conditions (7 -2 ), (7 - 3 )• Then, in analogy to Theorem 7*1, we obtain THEOREM 7 -2 . If condition B is satisfied on a sub­ set1 Q of the switching space S, the set Qff of exceptional points of higher order In Q consists of (n- 3 )-dimensional differentiable submanifolds of Q f if it is not empty. Every point u e Q" possesses a neighborhod N such that Q,f O N is contained in the union of at most three of the connected manifolds constituting Qn.

REMARK 7 .1 . By comparing the number of (scalar) equations enter­ ing into the conditions A and B with the number of independent vari­ ables, it is easy to see that, in general, both conditions hold in the entire switching space. Vid. Footnote on p. 236

ANDRE A ND SEIBERT

238

REMARK 7*2. Since S" is ofoflower lowerdimension dimensionthan than S S1 1 (pro­ (pro­ vided that that condition condition BB is is satisfied satisfied on on SSTT and and the the latter latter is notis not empty), S 1 [= S f - S M ] is apparently dense in S f: (7 . 0

S 1 = S'

.

8. The classes of normal switching p o i n t s . For the following considerations we assume conditions A and B to be satisfied through­ out the switching space S. Then the following lemma holds: +

L EMM A 8.1.

+

If If c~(u) c~(u)= =0 0

and and

c~(u) c~(u) 4 0 4 hold, 0 hold,

the function c” assumes positive and negative values in every neighborhood of u. PROOF. Under the assumptions c|(u) = o, c*(u) > o the curve x + (t, u) is is contained contained in in the the set set D@+ D@+ for for |t| |t| o

for

v e N

,

c*(v) > o

for

v e N

.

so small that

(8.2)

Then, due to the continuous dependence of x + on the initial values, there ifI Idenotes denotes inter­ exists a neighborhood neighborhood N^ CC NN of of uu such such that, that, if thethe inter­ and 55 is is chosen chosen small small enough, enough, the the set set xx++(I, (I, val (0, 55)) and N^) ) contains contains outside N. N. Then Then (8.1 (8.1 )) and and (8.2) (8.2) imply imply no point of S outside

x+ (I, N1 ) C De+

.

x I and Since x + (l, N^ ) is a homeomorphic image of the product set S, the the set set xx++(I, (I, N 1 )apparently apparently N 1 is an open set set on on the the hypersurface hypersurface S, 77 nn D0 + of of uu ( (ccHH-- :: neighborhood neighborhood of u). u). contains a semi-neighborhood oc oc D 0+ On the other hand, it follows from the hypotheses (and (6.1 )) that 0 ++ tends --->> -- 0, 0, which which leads leads to to aa contradiction. contradiction. x + (t,u) C DD0 tends to to uu for for tt --If c|c|< < 0 0 isis assumed assumed inin N N instead instead ofof c|c|> > 0,0, the the preceding preceding argument needs only to be modified b y considering the interval (- 5 , 0) instead of the same.

(0, 6). -

For the coefficients

c~, c“ the proof is exactly

PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS

E, T

We now consider the boundaries relative to and denote them by dA, dE, dT.

239

S of the sets A,

From Theorem 7*1 and Lemma 8 . 1 we easily obtain the following pair of inclusions: ( 8 .3 )

(A L )1 U (A R )1 C SA n ST

(8.4)

,

(EL)1 U (ER)1 C. SE n ST

.

Consider, e.g., the case u e (A+ n L-)1. [The other cases are treated in obvious analogy.] Here c| > o, c~ = 0, c” > 0 [Theorem 6.1]. According to the lemma there exist points v, w e S arbitrarily near u with c|(v) > 0 , c!j"(v) > 0 ; c|(w) > 0 , c~(w) < 0 . This Implies v € T, w e A [Theorem 6 .1 ]. After these preparations we can prove the following theorem: THEOREM 8 .1 . If the conditions A and B are satisfied on the switching space S, one of the following statements Is true, provided that S Intersects no other switching space: (a) (b)

S consists entirely of normal switching points of one type, i.e., S = A, E,

or T.

The sets T and A U E are both non-empty and their boundaries dT and d(A U E) coincide. Thus the closure T of T separates A from E and the components of A and E from each other and, con­ versely, the components of T are separated from each other by A U E.

If S intersects other switching spaces S^, the preceding statements hold for every component of S - U^(SX n S). PROOF. We first show that the set of exceptional points S 1 is empty if all normal points are of the same type. (The converse is obvious because dA, dE, dT C. S*.) Corollary 6 .2 yields (8.5)

S1

(AL) 1 U (AR) 1 U (EL) 1 U (ER) 1

.

From (8.3), (8.4) it follows that all sets on the right-hand side of (8.5) are empty, hence S1 = 0 . Finally, (7*4) Implies S! = 0 , so that we have

ANDRE AND SEIBERT

240

the case (a). Suppose now that S! and therefore also S1 are non-empty. Then it follows immediately from (8.3), (8.4), and (8 .5 ) that both T and A U E are non-empty. The same relations imply (dA)1 £ dT, (dE)1 C. dT, (ST)1 C. d(A U E).

(8.6)

Observing that dA, dE, dT are closed subsets of S and (8.6), we obtain

and using (7*4)

dA = dA n S' C. dA n S1 = dA n S1 = (dA)1 C. ST = dT

and the analogous relations dE Q bT

and

dT C. d(A U E)

.

Hence,

d(AU E)

(8 .7 )

= dT

,

which yields the separation property of case (b). The extension to the case where spaces is immediate.

S intersects other switching

§4. LOCAL PLOWS 9.

Definitions and elementary properties. As we have seen in §2.4, the qualitative behavior of the solution of a system (S) differs in many respects from that of a dynamical system in the usual sense, defined, hy the solutions of differential equations satisfying a LIpschitz condition. In order to study families of solutions of (S) qualitatively, we Introduce the notion of "local flows". DEFINITION 9 .1 . Let Y be a compact connected topological space. Denote by tQ, t1 two continuous maps of Y into the negative and positive halves of the real line (both including 0 ) respectively, and by I(y) the Interval [tQ(y), t1 (y)]. Then a continuous mapping cp of the set ft = ((y, t ) : y e Y, t € I(y)} onto Y defines a local flow SF = (Y, I, cp) if cp has the following properties: (a) y g Y, t € I(y) implies [tQ(y) - t, t1(y) - t]. (b)

cp(o, y) = y

I(q>(t, y)) = I(y) - t =

holds for all y e Y.

PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS

2^1

(c) If y e Y, t €I(y), t + t1 € I(y), therelation cp(t + t1, y) = q>(tS y ), and define a pair of sets YQ, Y 1

/(q) = 7 (p)

y, = which y 1 of the connected set Y, are connected. PROPOSITION 9 .2 . The mapping q>(t, y) (as a function of t for every point

are continuous,so are are images under yQ and

is univalent y cYQ) . 2

I.e., a Hausdorff space with the property that each of its points possesses a neighborhood which is homeomorphic either to an open k-sphere or to a semi-k-sphere x^2 + ... + 2 1 ^ > °* It should be noted that Definition 9 . 1 excludes the case where I(y) is the whole real line. Therefore, in particular, critical points [ 0 such that $ defines a local flow on the set Y = $([0 , T], C). PROOF. Choosing T so small that Y borhood H of the kind mentioned in footnote 2 , point y € Y possesses a unique representation t € [0 , T]). Then, if we define I(y) = [- t, T of Definition 9»1 are obviously satisfied.

is contained in a neigh­ it is clear that every y = o(t, yQ ) (yQ € C, - t], the conditions

So far only Interior properties of local flows have been con­ sidered, i.e., properties which do not depend on an imbedding of Y into a larger space. Now we are going to study properties which depend sub­ stantially on such an imbedding. We therefore assume Y to be a subspace of a topological space X. Closure, Interior, and boundaryof Y will always be understood relative to X. DEFINITION 9 .2 . A local flow & 1 = (Y!, I1,cp1 ) is called a subflow of the local flow & = (Y, I, cp) if Y>( Y( X,I»(y) = (t e I(y) : 9 (t, y) e Y 1}for all y € Y 1 and cp1 is the restriction of cp to the set ft1 = {t, y : y € Y 1, t e 1 1(y)]. If

is a subflow of

we write C

PROPOSITION 9*4.

SF.

Consider a local k-flow & = (Y, I, 9 )

1 I.e., a continuous mapping of the product space X x I (I: real line) onto X which satisfies the conditions (b), (c), (d) of Definition 9 . 1 in which Y and I(Y) are replaced by X and I. 2 In this context we mean by a "hypersurface without contact" a hyper­ surface H of X with a neighborhood N such that H does not contain more than one point of any component of the intersection of any path with N.

PIECEWISE CONTINUOUS DIFFERENTIAL EQUATIONS

243

and denote by BY the boundary of Y. If a path 7 of SF contains a point p g BY not lying in YQ or Y1, the entire path 7 is contained in BY. PROOF. To every point p €Y we associate the set1 = cp(— tQ(p), Yq) [which is a (k- 1 )-dimensional manifold with boundary]. Since tQ and t^ are continuous, given two numbers oc g (tQ(p), 0 ), 0 e (0 , t.,(p)), there exists a neighborhood of p relative to such that cp(t, p T) is defined for all t € I = [or, p] and all points p! e If p is an interior point of (i.e., possesses a neighbor­ hood relative to which is homeomorphic to a (k-1 )-sphere), it is also an interior point of the set (tQ(pn ) - tQ(p), pn ) ultimately belong to Thus pn g cp(I, Q^). Therefore, the intersections of Y and cp(I, Q^)with a sufficiently small neighborhood of p coincide. This implies the equiva­ lence of p g BY and p g BQp. Now consider two points p g By and q = cp(t^, p) (t g I(p)). Then p g Bq^ and, choosing as before (with t^ g I), 9 (t^, y) maps homeomorphically onto = ^(t^, Q^), so that q g BQ^. This Implies q g BY which proves the proposition. DEFINITION 9.3. A subflow = (Y1,I*, 9 1 ) of a local k-flow SF - (Y, I, cp) is called a face of $F if Y 1 is contained in the boundary of Y. We also say, SFX is incident with SF and use the 2 notations C

B

SF

and C B^

n B o) such that the functions x±(t, u) satisfy (S) in the Intervals I+ = [0, p] and l“ = [a, 0], respectively, + /+ x / x and define local flows on the sets x(I“, U). Moreover, x(t, u) defines a local flow & = & + U SF~ on x(I, U ) (I = I" U I+) [vid. Proposition 9•7]• In the case U C A [E] the set U is the common entrance [exit] set of two local flows defined by solutions of (S) on the sets + + x (I", U) [x“(I~, U) ], respectively, where I+ = [o, p], I” = [a, 0]; - a, p > 0 and sufficiently small. The ambiguity occurring in connection with starting and end points can be eliminated by a modification of the phase space E11, namely by "cutting" it along the sets A, E and replacing every point u e A, E by a pair of points u“, situated on either "edge" of the cut. More precisely, this procedure can be described as follows: Denote by V the set V = A UE and consider an open set X( E11 satisfying the condition X n S = V. Then complete the set X - V (with respect to the euclidean metric), and finally replace E11 by the space D* = (E11 - X) U (X - V)

,

where the closure is to be understood in the sense of the new topology, de­ fined by the Cauchy sequences in X - V. Obviously, the construction of the space D does not depend on the choice of X. Denote by cp the continuous mapping of D onto En which leaves every point of E11 - V fixed. Each point v e V possesses two Images under t = cp"1 which we denote by ^±(v) = v“ and which are distinguished from each other by the property + v”

g

D + e“

ANDRE AND SEIBERT Apparently, through every point p e D solution of (S). This we denote by x(t, p).

there exists exactly one

11. Local flows around the sets (AL)1 and (EL)1. If U is a subset of (AL) , it is easy to see that it Is contained in one of the sets A+ n L “ and A“ n L+. Indeed, in the opposite casethe connectedness of U would imply the existence of a point u € u n A+ n L" n A“ n l + which we could, e.g., assume cj > 0 [vid. Theorem 6.1]. would contain points of A” tinuous. Therefore, without

,

to belong to A+ n L~. This would imply Since, however, every neighborhood of u n L+, on which cj = o, cj could not be con­ loss of generality, we may assume uc A+ n L"

.

Relation (8.3) implies U dA n dT. Moreover, U Is in the intersection of the boundaries of two components A1, T1 and T:

(11.1)

contained of A

u £ sa, n ar1 .

This follows Immediately from the connectedness of TJ and from Theorem 7.1,1 according to which every point u e U [C. S1 ] lies on the boundary of exactly two components of S°. For every point (11.2)

u €U

x~(t, u) € De+

holds for sufficiently small (11.3)

the relation

t* [Theorem 6.1].

x“(t, v) € De-

(0 If (0

< |t I ^ 2 3 ' & 3k* The re1 ^ 10113 0 1»0 (11.7) obviously hold also in the case under consideration. Since (1 1 . 1 ) implies U C 0 (and, o f co u rse, another on th e hemisphere y^ < 0 ) . This so lu tio n i s near the

“1/2

c ir c le y3 = 5 on th e two sphere. m ately - ny2 (y2 / 3 - 1 / 5 ) + 1/ 5 a + by2 enough, a < 0. Hence,

J

But tnen (y, f ( y )) 13 ap p roxi­ which i s n egative f o r |a| la rg e

T

(Y> f(y))dT < 0

o where T i s th e period o f the p e rio d ic s o lu tio n . p e rio d ic s o lu tio n on th e hemisphere y^ < o«

S im ila rly f o r th e

The theorem then a p p lie s , and the s o lu tio n

x = 0

of (5 ) i s

ASYMPTOTIC STABILITY IN 3 -SPACE

asymptotically stable. 4.

REMARKS

A.

The two conditions given in the theorem for asymptotic stability are usually easy to apply. The first Is always easy, and the second condition can be verified without knowing either a precise expression for a periodic solution of (4b) or its exact period — as was seen in the above example.

B.

The two conditions are in a certain sense nec­ essary for asymptotic stability. That Is, if either of the Inequalities Is reversed, the critical point of (2) at the origin is not even stable. It is evident that if one of the two inequalities is replaced by an equality, the trivial solution of (1 ) may or may not be asymptotically stable, depending on the form of g(x, t).

C.

If m = 1, the linear case, the two conditions are equivalent to the condition that the characteristic roots of the matrix of coefficients of the linear terms have negative real parts. The proof of this fact is not difficult, but is a little long because of the necessity of considering the various possible canonical forms for the matrix. The proof will be omitted.

D.

Zubov [5] has given conditions for asymptotic stability for m = p/q(p, q integers, q odd) where n is arbitrary. His conditions do not seem easy to use Inasmuch as one must have a priori bounds on the solutions x(t) of (1 ) for which ||x(o)|| = 1, namely ||x(t)|| < At-Qr, A, a positive constants.

RIAS, Baltimore BIBLIOGRAPHY [1]

Malkin, I. G., MA theorem on stability in the first approximation,” (in Russian), Doklady Akademii Nauk SSSR, 7 6 , (1951), pp. 783-784.

[2]

Massera, J.L., "Contributions to stability theory,” Annals of Math. 64, (1956), pp. 1 8 2 -2 0 5 .

[3]

Coleman, C.S., "A certain class of integral curves Annals of Math., 6 9 , May, 1959-

in 3 -space,"

268

COLEMAN

[k]

Forster, H., "Uber das Verhalten der Integralkurven einer Gewohnlichen Differentialgleichung erster Ordnung in der Umgebung eines singularen Punktes,!f Mathematische Zeitschrift, k-3, (1937 pp. 2 7 1 -3 2 0 .

[5]

Zubov, V. I., "An investigation of the stability problem for a system of equations with homogeneous first terms," Doklady Akademii Nauk (In Russian), 114, (1957), pp« 9^2-944.

XIII.

EXISTENCE AND UNIQUENESS OP THE PERIODIC SOLUTION OP AN EQUATION FOR AUTONOMOUS OSCILLATIONS Rui Pacheco de FIgueiredo I.

PRELIMINARY REMARKS

Consider the differential equation (1 )

x + R(x) + x = 0

where R(x) Is a real-valued function having a piecewise continuous de** rivative and from now on the dot on a variable denotes differentiation with respect to t. Both (1) and its alternate form (2 )

y + r(y)y + y = 0

where y = x and r(y) = R 1(y), have been widely used to represent the be­ havior of physical systems undergoing self-sustained oscillatory motion. In this paper we propose a general set of conditions for the ex­ istence and uniqueness of the periodic solution of (1 ). *

The results of this paper are from a part of the author!s doctorate thesis at Harvard University and were presented at the annual meeting of the American Mathematical Society In Philadelphia on January 2 0 , 195 9* It is a pleasure to thank Professor P. Le Corbeiller of Harvard for his very valuable advice throughout this reasearch, and Dr. P. Huckemann, now at the Mathematical Institute of the University of Giessen, Giessen-Lahn, Germany, for his careful reading of, and very useful remarks on the original manuscript of this thesis. This work was supported in part by a grant from the Junta de Investiga o, the point (o, y) , labeled F in Figures 1 (a), (b) and (c), is a stable focus or node de­ pending on whether p2 < or > 2 . Let us assume first that p2 < 2 and refer to Figure 1 (a). As we proceed clockwise on the orbit mentioned

Limit Cycle,

(a)

^2 ~ 2* y FIGURE 1 (b)

FIGURE 1 (a)

above byond B, we shall meet the positive x-axis at some point C to the right of F. Then again from the well-known results alluded above we deduce the relationship between the distances OC and OB

*

(9)

FIGURE 1 (c)

OC OB + 7

2

AN EQUATION FOR AUTONOMOUS OSCILLATIONS

273

where (1 0 )

We next assume thatp2 > 2 (see Figures 1(b) and (c)).Then, whatever the position of the point B on the negative x-axis, the orbit through AB always trends to F when followed clockwise from B, since now F acts as a stable nodefor the arc EF. Thus as A variesits position on the positive x-axis from 0 to there Is only one position of A corresponding to a closed orbit of the system, namely when A co­ incides with F. All other orbits of the system tend to F (and hence approach the closed orbit FBF) as t -- > °°. The above conclusions may be summarized as follows: LEMMA 2 . Under the conditions stated, the system (5 ) has in the (x, y) plane one and only one closed

DE FIGUEIREDO orbit which is approached by all the other orbits of the system as t -- > °°. 3.

ON OP A PERIODIC SOLUTION ON THE THE EXISTENCE OF A EXISTENCE PERIODIC SOLUTION

We nowstate state the the following following result result for for the the original original equation equation ((11)) :: THEOREM 1. The equation (1 ) has at least one periodic solution of (a) (b)

R(0) = 0 R*(0) R*(0) exists and is negative

and providedthere is a yQ > 0

such that

(c)

RRf(x) f(x) > 0, min (x

(d)

22 >> -- min R !(x) < R !(- x), (x < -- yQ yQ))

except for the values of simple discontinuities.

> yQ )

x

at which

R'(x)

undergoes

REMARK. Below it will be necessary to compare the slopes of the elements of orbits, through a point P in the (x, y) plane, of two systems z1 and z2 of the form (13)

21 21 :: x = y, y = - P1(y) - x = y,

y = - P1(y) - x

(1^)

z2 z2 :: x = y, y = - P2(y) - x = y,

y = - P2(y) - x

where F P 1 and F2 are real, continuous (piecewise differentiable) func­ tions. For this purpose we let z denote the distance from the origin of the (x, y) plane to a point on an orbit of and z2, i.e.,

z-

,

which after differentiation with respect to or (14) gives (1 5 )

z = - 1 P1(y)y, z 1

t

and combination with (1 3 )

i = 1, 2

This shows that at P, if y > 0 and F^ < F2, z is larger along the element of orbit of Z1 than along that of z2, In other words, when taken in the direction of increasing t, the first of these elements intersects the second one outward or away from the origin. The same is

AN EQUATION FOR AUTONOMOUS OSCILLATIONS

275

the case if y < o and F1 > F2, the opposite being true if y > 0 and F1 > F2 or y < 0 and F1 < Fg. PROOF.

Consider the two-dimensional system (3) equivalent to

( 1 ). Since R(y) is continuous on by a positive constant M there.

- yQ Pl < p2 p, > - min R'(y),

(y < - J Q)

.

Finally, let

(2 0 )

7 = p2 yQ + 2 M

.

p^, p2 and y thus defined satisfy the conditions assumed for these constants in (6) and hence by Lemma 2, the system (21)

(22a) (22b)

u = y,

y = - R 1(y) - u

r -

Ply,

R, (y) = 1

•- p2y - 7 ,

has a unique orbit In the (u, y) plane. We label this orbit r2 *

(y < o) (y > 0 )

plane and therefore In the

(x, y)

Now consider the annular region K (see Figure 2 ) in the (x, y) plane bounded outside by r2 and inside by the circle r1 defined by 2 . 2

where 6 Is a sufficiently small positive constant such that in the interior of and on r ^.

R*(y) < 0

EE FIGUEIREDO

s.

\ >

X,

-R(y), -R,(y)

FIGURE 2 Let (21) and (2 2 ), after combination with (1 6 ) and (2 0 ), be rewritten as (24a, b)

X = y,

(2 5 a) (25 b)

y = - F(y) - x

- P,y + M,

(y £ °)

p2(y - yQ) - M>

(y > °)

F(y) =

Then since and (1 8 ) we have

|R(y)| p2(y - yQ) - M = F(y)

where the rightmost terms in (26 a) and (26 b) arise from the definitions (2 5 a) and (2 5 b). It follows, from (26 a) and (26 b) (when considered together with (3 ) and { 2 k )) and by virtue of the remark just preceding this proof, that the elements of orbits of (3 ) on r2, taken in the direction of in­ creasing t, remain in K. Also, according to the conditions (a) and (b) of the theorem,

AN EQUATION FOR AUTONOMOUS OSCILLATIONS

277

R(y) < o for y > 0 and R(y) > o for y < o. Moreover (2 3 ) is a solution of the system x = y, y = - x. Hence, by the remark just re­ ferred to, the elements of orbits of (3 ) on r^, when taken in the direction of increasing t, intersect r1 outward from the origin and hence into K. Thus an arc of orbit of leave this region for increasing plane, which is the only critical by Lemma 1, K contains at least cludes the proof. k.

(3 )starting at a point on K cannot t. Moreover, the origin of the (x, y) point of (3 ), is external to K. Hence oneperiodic orbit of (3 ). This con­

UNIQUENESS OF THE PERIODIC SOLUTION

We now make the following proposition: THEOREM 2. Assume that the conditions of Theorem 1 are satisfied for the equation (1 ) and let there •£ exist a y1 > 0 such that (a)

R(y1) = R(o) = 0

(b)

x R(x) 0,

(d)

R* (x) > i R(x), x

(0< |x | < y1

)

(x > y1) (x y1 )

except at the values of x where R*(x) undergoes simple discontinuities. Then (3 ) has a periodic solution which is unique except for the translations in t. REMARKS. As before, Instead of (1 ), the equivalent system (3 ) will be considered. Figure 3 illustrates the direction, for increasing t, of the elements of orbits of (3 ) under the conditions of this theorem. In this and the following figures, - R(y) is also shown plotted along the axis of the abscissae. It follows from (3 ) that, for increasing t, the ele­ ments of orbits of (3 ) above the x-axis are directed rightward and those below It leftward; the ones at the left of - R(y) are directed upward and those at its right, downward. The curve - R(y) itself Is intersected By evident changes In the proof it can be shown that the result of this theorem holds if the conditions (a), (c) and (d) are replaced respectively by (a) R(- y1 )= R(o) = 0, (x < — y,, x > 0).

(c) R(x) > 0,

(x < - y1),(d)

R !(x)>-1R(x), x

2 78

DE FIGUEIREDO

by every orbit horizontally and the x-axis is crossed vertically. Let r in Figure b represent an arc of orbit of (3) that describes a complete clockwise turn around the origin, leaving the positive x-axis at A and meeting this axis again for the first time at some point I. Then nowhere on r is an alement of r along a radial direction. In fact, this follows directly from the discussion in the preceding paragraph, for that part of r that does not lie in the region between the curve - R(y) and the y-axis (indicated by the hatched area in the Figure). To show that the same holds for the arcs FG and BC of r that lie in this region, consider first FG. In the y-interval on which FG is de­ fined there is a N > o such that R(y)/y £ N. Since an orbit through F of the linear system x = y, y = - Ny - x, when followed clockwise from F, does not approach a radial direction in this y-interval, the arc FG cannot certainly approach a radial direction (since R(y) ^

0 a=l P=1

t

+J

T\

m ez ^t'"r^ua (t)dT]dz-- — 2 C L

Y

ua^t ) f

a - l

285

dz

0

C is a contour enclosing the zeros of D(z) Then A

(T )dT]

Jr,

[ % a (z )/D(z)] dz G

.

,

APPENDIX

286

By (2.4) the first term on the right vanishes because its integrand has no poles inside C. Therefore V.

Replace equation (2 .1 3 ) by

(2.13)

Sia = - “

/ [DlaU)A>(z)] dz

2*1 c

.

VI. Replace the material below equation (2.14) and above equa­ tion (2 .1 6 ) by, "To evaluate Sia note that as z ----- >" VII.

Change formula number (2 .1 6 )

to (2.15)*

VIII. Line 1 1 , p. 36 should read, according as ... ."

"degree m - 1 or < m -

2

IX. Replace "inserting (2 .1 6 ) into (2 .1 5 )," by "inserting (2 .1 5 ) into (2 .1 3 ),"• X.

Omit lines 1 5 -1 8 , p. 3 6 .

XI. Replace line 19, P« 36 by "Solving (2 .5 )for inserting in (2 .1 1 )," XII.

y (t) 1

E>iQf(z)

and

Replace lines 4 and 5, p. 37 by

-- L.

f Dl(2)—

2*1

[eztB(z) +

[

ez(t—0 TT/

y, . ,

{

D(z)B(z)

•

The integrand has poles at the characteristic roots only, so by the theory of residues and (2 .9 ),

yi(t)=Z

"r1 r

. . M

{Gij (2j)[e J BCz j ]

Sj

+J

eZj(t-T)U(z.,

T)dT]

}

.