Differential Equations, Geometric Theory, 2ed

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Author: Solomon Lefschetz

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DIFFERENTIAL EQUATIONS: GEOMETRIC THEORY SECOND EDITION

PURE AND APPLIED MATHEMATICS A Series of Texts and Monographs Edited

by: R. COURANT • L. BERS

VOLUME VI

•

J. J. STOKER

DIFFERENTIAL EQUATIONS: GEOMETRIC THEORY SECOND EDITION

SOLOMON LEFSCHETZ PRINCETON UNIVERSITY

THE NATIONAL UNIVERSITY OF MEXICO AND

RIAS, BALTIMORE

U BLISHERS

a division of John Wiley & Sons, New York • London

ALL RIGHTS RESERVED Library of Congress Catalog Card Number 62-21884

PRINTED IN U.S.A.

Preface The surprisingly warm reception which greeted the author's monograph Lectures on Differential Equations (Princeton University Press 1946) provided the incentive to enlarge the monograph to the present volume. However, the general plan has remained

the same, and readers acquainted with the monograph will readily find their way here. We continue to lean more heavily

than ever on vectors and matrices and to utilize topological notions. In relation to the monograph the present scheme is the following. The first four chapters are about as before: more or less pre-

paratory and standard. The next three deal with point stability and contain much additional information on the work of Liapunov and his Soviet successors, notably on the so-called direct method of Liapunov. With Chapter VIII — periodic solutions — the n dimensional part is concluded. This chapter is admittedly

fragmentary, but then this is unavoidable, given the present state of this general question. The remaining chapters deal with twodimensional systems. Two chapters are devoted to The results of and Bendixson: critical points, the index, behavior at infinity, some special systems. An introduction is also given to the important notion of structural stability and the contributions of Andronov-Pontrjagin and DeBaggis. The last two chapters deal with equations of the second order: notably the recent work. of Cartwright-Littlewood, of Levinson-Smith and of Levinson, the application of the perturbation method, and other related topics. After two appendices on vectors and matrices, and on topology, the volume concludes with a small list of problems which are not exercises but rather topics appropriate for further discussion and which had to be omitted owing to lack of space. Once in possession of the genera! theory two roads lay open V

VI

PREFACE

before us. We could follow Poincaré, Levi-Civitá, Birkhoff and

study "two degrees of freedom" and the extensive doctrine

centering around the three body problem. The second and more modest road, which we have selected, led to nonlinear differential equations of the second order: the group of problems stirred up a generation ago by Van der Pol. No applications whatsoever are taken up in the volume, but it is hoped that mathematically inclined physicists and engineers may find much to interest them within the covers of this book. In a work of this scope it has not been possible to range the topics according to difficulty. For the convenience of the reader we have marked with an asterisk the more arduous parts which may be reserved for a second reading. The "first reading" parts include the existence theorem and a few consequences in Chapter II; all of Chapter HI and IV; the first halves of Chapter V and VI; the two dimensional part: from Chapter IX to the end.

The author wishes to express his appreciation to Messers. Courtney Coleman, Ralph Gomory, and Felix Haas who have read

significant portions of the manuscript; also, to Messers. Robert Bass who read part, and Henry Antosiewicz who read all the proof and made many valuable suggestions. I also wish to express my thanks to the Office of Naval Research as preparation of the manuscript was done in part under Office of Naval Research Contract N6-ORI-105, Task Order V. S. LEFSCRETZ

Preface to the Second Edition The main differences between the two editions are these: correction of many misprints and defective proof pointed out by Jose Massera; improvement in exposition suggested by Jane Scanlon; considerable extension of the material on Liapunov's

direct method and its converse as required by its growing

importance, in the revision of which Antosiewicz was most help—

ful; revision of the treatment of Chapter VI, Stability in Product Spaces, with the constant assistance of my colleague J. K. Hale

of RIAS. To all these friendly critics the author wishes to express his deep appreciation. S. LEFSCHETZ

Contents V

Preface

I. Preliminary Questions 1. Elements of topology 2. Vectors and matrices 3. Analytic functions of several variables 4. Differentiable manifolds

IL Existence Theorems. General Properties of the Solutions 1.

2. The fundamental existence theorem *3• Continuity properties Differentiability properties *5 Analyticity properties 6. Equations of higher order 7. Autonomous systems

1

1

9 21

27

29

29 30 36 40 43 45 46

III. Linear Systems 1. Various types of linear systems 2. Homogeneous systems 3. Non-homogeneous systems 4. Linear systems with constant coefficients 5. Linear systems with periodic coefficients: Theory of Floquet

55

IV. Stability 1. Historical considerations 2. Stability of critical points 3. Stability in linear homogeneous systems 4. Uniformly regular transformations 5. Stability of trajectories 6. Stability of mappings 7. Further deñnitions of stability

76

VI'

55 57 68 69 73

76 78 79 80 83 84

84

CONTENTS

VIII

V. The Differential Equation

86

= Px + q(x;t)

(P a constant matrix; q(0;t) =

0)

87 88 95 96

1. General remarks 2. The general non-analytic system Analytic systems: Generalities The expansion theorem of Liapunov

107

VI. The Differential Equation dx

= Px + q(x;y)

(P a constant matrix; q(0; t) =

0)

(continued) 107 1. The method of Poincaré 112 2. The direct stability theorems of LiapunoV 122 Stability in product spaces 126 An existence theorem 130 *5• Stability in product spaces: Analytical case *6. System with a single characteristic root zero and the 133 rest with negative real parts 137 7. The converse of Liapunov's theorems 142

VII. The Differential Equation = P(1)z + q(x; I) (P (t) a variable matrix; q (0; 0

0)

Perron's reduction theorem *2. Various stability criteria The Liapnnov numbers. Application to stability....

142 145 152

VIII. Periodic Systems and Their Stability

*1. Linear homogeneous systems with ,periodic

co-

efficients

*2. Analytic systems with periodic coefficients *3• Stability of periodic solutions Stability of the closed paths of autonomous systems. The method of sections of Poincaré Systems of periodic solutions

155 159 160

160 164

CONTENTS

Quasi-linear systems and their periodic solutions. .. 7. A class of periodic solutions studied byLiapunov.... *8. Complete families of periodic solutions

IX 170

172 174

IX. Two Dimensional Systems. Simple Critical Points. The Index. Behavior at Infinity 1. Generalities 2. Critical points of linear homogeneous systems 3. Elementary critical points in the general case The index. Application to differential equations .5. Behavior of the paths at infinity

X. Two Dimensional Systems (continued) 1, General critical points 2. Local phase-portrait at a critical point 3. The limiting sets of the paths as! ± 4. The theorem of Bendixson 5. Some complements on limit-cycles 6. On path-polygons 7. Some properties of div (X, Y) 8. Critical points with a single non-zero characteristic root 9. Structural stability 10. Non-analytical systems XI. Differential Equations of the Second Order 1. Non-dissipative systems equation 2. 3. The equation of van der Pol: Phase-portrait 4. The equation of Cartwright-Littlewood 5. Applications and complements

6. The differential equation x" + f(x, x')x' + g(x) = e(t)

7. A special differential equation x" + g(x) = sin wt.. 8. A special differential equation x" + f(x)x' + g(x) = e(!)

9. Certain periodic systems investigated by Gomory

181 182 183

188 195 201

209 209 214 225

230 235 237 238 241

250 257 264 266 267 272 279 286 292

301 306

307

XII. Oscillations in Systems of the Second Order. Methods of Approximation 1. Self-excited systems 2. Forced oscillations

3. Approximations for quasi-harmonic systems 4. Equations of Mathieu and of Hill 5. The limiting position of limit-cycles

312 313 321 331 337 342

X

CONTENTS

Appendix I. Complement on Matrices 1. Reduction to normal form 2. Normal form for real matrices 3. Normal form of the inverse of a matrix 4. Determination of log A 5. A certain matrix equation 6. Another matrix problem

347 352 354 355 356 358

Appendix II. Some Topological Complements 1. The index in the plane 2. The index of a surface 3. A property of planar Jordan curves

360 360 366 370

Problems

373

Bibliography

376

List of Principal Symbols

386

Index

387

347

CHAPTER 1

Preflminary Questions in the present chapter we summarize briefly a number of questions which will play a basic role throughout the present book: elementary notions of topology, vectors and matrices, functions of several variables. References: For topology, Lefschetz [1]; for vectors and matrices, Bellman [4]; Gantmacher [1]; Wedderburn [1]; for functions of several variables, Boclrner-Martin [1]. throughout of the standard set Notations. We shall make theoretic symbols: a E A: a is a member of the set A; A u B: union of the sets A, B or set of elements contained in one or the other; union of the sets A0; UA A n B: intersection of the sets A, B or set of all elements contained in both; fl A0: intersection of the sets A0;

A cB,Bz'A: A is contained in B;

A — B: the complement of B in A or set of all elements of A which are not in B.

The symbol

stands for "implies."

§ 1. Elements of Topology 1. By a space one generally understands a set with some structural properties. Since all our spaces will be endowed with a distance, in fact, will be subsets of Euclidean spaces, it will be most appropriate to take distance as the basic concept and define the other classical concepts such as open sets, closed sets, etc., in terms of distance. is metrized whenever (1 .1) Metric spaces. A set or space 1

2

DIFFERENTIAL EQUATIONS: GEOMETRIC THEORY

there is defined among its points x, y, ..., a real valued function d (x, y), the distance with the following properties: (a) d (x, y) 0 and d (x, y) 0 when and only when x and y coincide; (b) d (x, z) d (y, x) + d (y, z) (triangle axiom). From (a) and (b) follows d (x, y) d (y, x) and since the converse manifestly holds we have: (c) d (x, y) = d (y, x) (symmetry law). A metrizable space 91, i.e., on which a distance may be defined, is said to be metric. We also say: a metric is defined on 91. The distance on 91 assigns of course a distance on every subset of 91. Thus when 91 is metric so is every subset of 91. In terms of distances one may define various simple notions: radius the set of all points Spheroid (xo, of center x0 x at a distance less than e from x0; distance d (x, A) from a point x to a set A: inf d (x, y) for all

yinA;

diameter of a set A, written diarn A: sup d (x, y) for all x,

yinA;

limit of a sequence convergence of the sequence, Cauchysequence are defined in the standard manner of real analysis. An example: The space is the Euclidean plane with its usual distance and the spheroids are the circular regions. When a set is metrizable its metrization is not unique. Thus if d (x, y) is a distance so is 2 d (x, y). Two metrizations say by (x, a) are and d (x, y) and d' (x, y) with spheroids (x, equivalent whenever every (x, contains and is contained in (x, a). This guarantees among other things that both an distances specify the same sequences as convergent and assign the same limits to convergent sequences. (1.2) Product spaces. Let 91k, 912 be two metric spaces with distances d1 (x, y), d2 (u, v). Let the pairs of points (x, u), (y, v), one from each space be considered as points of a new space 91.

It is easy to verify that d ((x, u), (y, v)) = d1 (x, y) + d2 (u, v) is a suitable distance in 91. The space 91 with this or any equivalent written distance is known as the product of by x This extends at once to a product ... x x

I.

PRELIMINARY QUESTIONS

3

The following are simple examples: (a) The product of two segments is a square and of three segments a cube. (b) The product C1 x C2 of two circumferences is a torus. This is seen and C2 by at once: — If C1 is parametrized by 0: 0 0 < 0 < 2ir, then C1 x C2 is parametrized by 0, varying between the same limits. Thus 0 corresponds to the meridian angle and to the parallel angle of the torus. (c) The product w x C of a closed circular region w: x2 + y2 1, by a circumference C is a solid torus. (1.3) Euclidean metrized by d (x, y)

The line 1: — 00 < X < + 00

is

which is a suitable distance. It is Ix — Euclidean n-space thus turned into Eucidean one-space is merely the product of n lines assigned the "product" metric Ii 1, 2, of (1.2) or any equivalent metric. Let the n lines n, be parametrized by coordinates xh. The points of the product 4 x ... x are the sets of n real coGrdinates x1, and the product metric assigns to (i" the distance .

d(x,y)

E IXh—yhl.

An equivalent metric is the more commonly used distance d' (x, y)

—

=

Another is the very convenient metric based on the distance d" (x, y) = sup

{

J

— Yh J }.

A further metric may be obtained as follQws. Apply to ti" the

affine transformation (1.4)

Xlh

where the determinant Jahkj the two functions Ixlh — yihI,

Xk + b4

0. Then take as distance one of (Xlh — yui)2J"

both of which are equivalent to one another and to those defined by d (x, y), d' (x, y), ci" (x, y).

(1.5) Other

Consider first the plane S of a complex

____ 4


variable z. One may metrize S by assigning to it the distance now a product of p planes S1, ..., S, of complex ..., and q lines •.., tq of real variables x1, Xq. This space = S1 x ... x x x ... x lq is assigned through the product metric the distance Consider

variables z1,

=

E

+E

Ixk—xk'I. The space occurs naturally in the study of functions of / complex and q real variables which we will have to consider later.

2. Topological considerations. The more strictly topological notions are best defined in terms of open sets. An open set U is simply a union of spheroids. Observe now the following two properties.

(a) If x

(x0,

then

by the triangle axiom there is an

(x, a) c (x0, Hence this characterization of an open set U: if x U then some spheroid (x, c U. (b) If d' (x, y) and (x, ci) are a distance equivalent to d (x, y) (x, a) c and its spberoids then some (x, Hence the open

set U is likewise a union of spheroids Conversely if it is a union of spheroids then it is likewise a union of spheroids Since all the topological properties to be discussed are expressed in terms of open sets they are the same for equivalent metrics. The open sets U containing a point or more generally a set are called neighborhoods of the point or set. We now define a closed set F as the complement — U of an open set. It is convenient to include the whole space and the null-set among the open and hence also among the closed sets. One verifies then that the union of any number and the intersection of a finite number of open sets is open. Hence the intersection of any number and the union of a unite number of closed sets is closed. Two more notions are now to be defined: the closure is the closed set which is the intersection of all the closed sets containing A; it is thus the "smallest" closed set containing A; and it is manifestly the boundary A n closed.

I.

5

PRELIMINARY QUESTIONS

The closure A is the set of all points at zero distance from A while the boundary A is the set of all points at zero distance from both A and its complement. A condensation point x of A is defined by the condition that every neighborhood U of x contains points of A — x. The closure A is the union of A and all its condensation points.

If A c

the open or closed sets of A are the intersections

with A of the corresponding sets of 91. Let 91, 91' be metric spaces. A function Jon 91 to 91', or trans/or91', is said to be continuous of 91 into 91', written f: 91

whenever if U' is an open set of 91' then

U' is an open set

of 91. This definition is readily shown to be equivalent to the well known e, ö definition. A continuous transformation is also known

as a If f is one-one and bicontinuous (both f and

continuous) then f is said to be topological or a homeomor/thism. One says also that 91 and 91' are topologically equivalent or homeotnorphic.

Example. Consider the system (1 .4) as assigning to the point x with the coordinates Xj a new point x1, whose coordinates in the same system as the are the given by (1.4). There is thus obtained a one-one transformation T of onto itself and T is topological. Cells and spheres. These sets are of particular importance. The n-cell is the topological image of the set of points of

represented

by (2.1)

The Euclidean space The (n — 1)-sphere

itself is an n-cell. is the topological image of the bounda-

ry of the set (2.1), i.e. of the subset of (2.2)

=

given by

1.

The n-cell is then the topological image of a spherical region in I

It is also the topological image for example of the sets: 0)

is orienkible, otherwise it is non-orientable. If the

are analytic throughout is said to be analytic. If M' is compact one may suppose that {U0} is finite. A differentiable M1 is always orientable. It is a Jordan curve if it is compact and an arc otherwise. (15.3) An example. The point-set of an Euclidean n-space defined by relations

28


h=1,2,...,n where ç' (u) is of class Cr, r>, 0 for lull 0.

r (a))


32

0 there exists a ô > 0 such that a trajectory crossing E (M, 6) intersects the hyperplane 1 = within (xo; e) (i.e., it meets (xO; 8) x to). (8. Ia) Referring to (6) and in its notations let y denote the part and let G = — y. IfG 00 of the trajectory rsay fort 1° let it contain the point Q (x"; t") not in the boundary of Since passes arbitrarily near Q by (8. 1) x (1) must cross 1", which is Thus as £ tends to one of ruled out since 1" is an end-point of

the endpoints of h the trajectory r can only tend to infinity or to the boundary of the domain An extension of the continuity theorem in a new direction is the following. Suppose that X is in fact a function X (x; 1; y) which is continuous and bounded in a region Q of x that with respect to .Q the Lipschitz condition Suppose is still fulfilled in the same form as earlier. The domain of continuity zi is defined as before as a component of the union of all the sets I?. We have now the following stronger result which

extends (7.3) and may understandably be stated in the brief form: (8.2) The solution is continuous in (t; x°; to; y) when (X0; to; y) ranges over zI.

The proof may be related to (7.3) by a well known device. Enlarge (1.2) by adding the differential equation (8.3)

dyfdt==0

so that (1.2) and (8.3) form a system such as (1.2) for the vector (x; y). Owing to the special form of (8.3) the Lipschitz condition

still suffices to prove the existence theorem and hence all its corollaries, including among them (7.3), which in the present instance becomes (8.2). and an arc MN of J'O, where Consider again the trajectory

EXISTENCE THEOREMS. PROPERTIES OF SOLUTIONS

U.

(x°;

to)

39

are the coordinates of M and (x'; t') are those of N.

Introduce the two sets S0 = (x0;

x

S1 =

(xi;

x 4,

Let are so chosen that both sets are in M' S0 and let F': x (1; M') be the trajectory through M'. It is clear from the argument proving (7.3) that for small enough x (t; M') may be extended throughout the whole closed interval 1: £0 £1. Consider a mapping p of the compact into product 7 x sending I x M into MN and I x M' into MW'. This mapping is one-one, since (x'; t') and (x"; t") certainly have different images if t' t" or x' x" (otherwise is compact q' distinct trajectories would meet). Since I x where

£

is topological. Take now a small n-cell

Applying q' merely to I x

t=

containing M and contained in

and denoting the hyperplanes

we have:

by

(8.4) STRUCTURAL THEOREM. Given a sufficiently small n-cell

in

containing M, a trajectory through a point M' of E0" intersects in a single point N' such that M' N' defines a containing N topological onto a similar E1" c 0 of and of course N = OM. Moreover let A be the closed arc M'N' of the trajectory through M', and let A (t) be the point of A corresponding to any I 1,1: to £ t'. Then (M', t) —. A (t) defines a

logical mapping

of the cylinder I x

x M') =A.

such that

(I x M) =

(8.5) General solution. This concept may now be introduced with reasonable clarity. Let be n-dimensional and I (I; c) a function such that: (a) for c in a certain region A of f is a solution of (1.2) in the domain (b) if M0 =f(t°; 00), M f (1°; c), where c, c° are in A, there are n-cells in x to and in A, such that for c A the correspondence c -÷ M is a topologicai mapping of e" onto such that q' c° M0. In other words, c may be chosen in e" so as to yield any solution with its initial value at in The functionf (I; c) is known as a general solution. By (8.4) the concept of general solution is manifestly independent of the particular point M0 chosen on the trajectory f (I; Co), i.e.,

it does not depend upon t° but solely upon the

trajectory itself.

40


§ 4. DIfferentiability Properties 9. These properties are embodied in the following very comprehensive proposition: (9. 1)r. Let X (x; t; y) and related designations be as before, and let X be of class Cr in x, y and t in a certain regiOn 1? of the space x x %. Then the solution x (t; to; x°; y) of (1.2) such that x (to; to; x°; y) = x°, (to; x°; y) .Q, is of class C' in tO; x°; y, and of class Cr + 1 in t. In the proof we follow more or less Gronwall [1] and Sansone [13, Vol. I, p. 27. We shall require the following: (9.2) GRONWALL'S LEMMA. Letf (t) be a scalar function such that

(9.2a)

0 f(t)

+ v) dt',

A+

where A, 1u, v are positive constants, t to, and f (t) is continuous Ift'_to = Tthen in the range t° t (9. 2b)

/ (1)

jl•

(ii T + A)

(Gronwall 111).

Set f(1) = so that g(t) is continuous in the same interval as f(t). If y sup g(t) in t° t t' then g(t) attains the value y for some

1°

t'. Now (9. 2a) yields:

+v)dt' From follows y A + v T and this implies (9.2b). An immediate consequence is: (9.3) Let f, A, v be n-vectors and an n x n matrix, where A, p, v are constant and f has the same continuity property as before. If (9. 3a)

then also

1(t) =

2

+

(4u / (t') + v) dl'

II.

EXISTENCE THEOREMS. PROPERTiES OF SOLUTIONS

(9.3b)

Uf(t)II

41

+

(HvII T

For (9. 3a) yields 0

+ f (n

IIf(t)II

JIf(t')H +

jIvjDdt'

of the lemma.

from which (9. 3b) follows by a direct

is merely (8.2) as regards

Proof of (9.1). Property (9.

We t°, x°, y, and obvious as regards t, so let us consider (9. proceed first with regard to y. It is to be proved that the vector (9xlayk exists and is continuous in y. It will be assumed that all

functional values are taken in Q and for the present I will T. For any f(y) let Jf = remain in a finite range — ..., 0, Jyk, 0, . . ., 0). We (0, where 4 y = I (y + 4 y) — f (y), start from the integral equation

x=

(2.2)

x°

+

X (x; 1'; y) dt'.

Owing to the continuity of x in y, 4 x -+ 0 with 4 y, and for 4 y small (x + 4 x; I; y + A y) remains in £1. Thus (2.2) yields for A

(94)

small

iix

=jt

Ax

/JYk

4Yk

\øYk

and where it is sufficient to know that uniformly in t for It — 101 T. Moreover

/ —÷

0

with 4

aX ox

the

kax8I'

Jacobian matrix of the vector X as to the vector x.

Consider now the system (9.5)

dz/dI=OX/8x•z+OX/Oyk,

with the initial condition z (to) = 0, it being understood that x stands for the solution x (I) of (1 . 2). Since the system is linear with continuous coefficients (by the class C' hypothesis for X)

____ 42


it has a unique solution and this solution is continuous in t. has an upper bound e in Hence T. The solution of (9.5) under consideration satisfies

z=

(9.6)

+

ft

Ax

Upon subtracting (9.6) from (9.4) and setting u (t)

= '-' Yk — z

we find for u (t) the relation (9 . 7)

u

(1)

rtIIøX

I—+

=J

) /

U

+

z + s2) dl'

Owing to the continuity of aX/ay and boundedness of

the

to( norm of their sum has an upper bound in T. z ± e211 < where —*0 with zI Yk. Hence (9.7) Clearly also

yields in view of (9. 3): which

0

with A

This shows that A x/A

z

as A yk

0.

T, and is Hence 8x/ayk exists, is continuous in I in — the solution of the differential equation (9.5) which is zero for t = to. The extension step by step from the limited to a larger I interval is an immediate consequence of the fact that ax/ayk satisfies (9.5). Thus ax/ayk exists throughout Q and satisfies (9.5).

Let us regard now x° as a parameter. The same reasoning will apply with the following minor modifications. Instead of (9.4) we will have Ax (EJX Ax (9.4') =e;—f-to\bX /ixjo / where e5 is the unit vector with components ôhJ. The system (9.5) is now replaced by (9.5')

dz/dt

z,

the desired solution being this time such that z (1°) = ej. Instead

of (9.6) we have now (9.6')

II.

EXISTENCE THEOREMS. PROPERTIES OF SOLUTIONS

43

The same argument as for x° holds for £° also. The initial step yields however a slight variant. We have

+62+f tfdX

Ax

\zlx

At°. The equation in z is (9.5') but the solution to be envisaged is such that z (i°) = X (X0; to; y). The rest of the argument goes through with insignificant modffiwhere

o with

cations. One proceeds from class C' to class C2 by replacing the given

equation by (9.5) or its parallels such as (9.5'), etc., down to class Cr. The class as to t follows by direct application of the rules for derivatives. Similarly for mixed derivatives. This completes the proof of (9.1). (9.8) Remark. It is a trivial observation that even under the

mere conditions of the fundamental existence theorem dx/dt exists and is continuous along a trajectory. It implies however the non-trivial geometrical facts that in a real system a trajectory possesses a continuously turning tangent and that its arc length ds is defined.

§ 5. Analyticity Properties 10. The argument in deriving the continuity properties of x (1) rests essentially upon the following three propositions: (a) A continuous function of a continuous function is continuous.

(b) If f (x; t) is continuous in (x; t) over a suitable range so is

f0f(x; t) dt. (c) A uniformly convergent series of continuous functions is continuous.

These thiee propositions made it possible to "transfer" continuity from the approximations xm (t) to the solution x (t) itself. Since these three properties hold also with "continuous"

replaced by "analytic" or "holomorphic" the same transfer will operate for analyticity or holomorphism. It will be necessary

however to distinguish carefully between analyticity as to x°, as to t, t°, or as to all three. The difference arises from the fact that we may wish to consider not only X (x; t) analytic in (x; t)

44


but also merely in x alone, or in certain additional parameters

that may be present in X. Suppose X (x; 1) in (1.2) analytic in x and continuous in (x; t) in a certain region .Q. Since the partial derivatives exist and are continuous in Q the Lipschitz condition holds in Q (5.1) and so the existence theorem is applicable. We now proceed as before replacing continuity by analyticity wherever need be and obtain first an analogue 4 of which we call domain of analyticity. Then we have the following properties, which we merely state since the proofs are unmodified. The space and the vakies of the variables and functions may be real or complex unless otherwise restricted. (10.1) If X (x; t) is analytic in both variables and zl is the domain of analyticity then the solution x (t; x°; 10) such that (x (1); t) A, x (to; x°; to) = x° is analytic in all three arguments (see (7.3)). (10.2) If X (x; 1) is merely continuous in I (I real) then x (I; x°; tO)

is merely analytic in x° (see (7. 3)).

(10.3) If X (x; I; y) is analytic in y also and 4 is defined accordingly as in (7), then x (I; x°; to; y) is analytic in all arguments

in the case (10. 1) and in (xo; yj y) alone in the case (10.2). 11. (11.1) P0INcARE's EXPANSION THEOREM. Let the differential

equation with t real and the domain of analyticity /1: (11. la)

X (x; t; y),

dx/dt

possess for y = 0 a solution (I) on the closed interval I: 1° t such that for fixed t the Xj (x; t; y) may be expanded in tower series of the — (I)) and of the uniformly convergent in some range

2r, are real. 10. Consider now a non-homogeneous system (8.1) where A

is a constant matrix. Since X, Y in (5.4) are any two nonsingular solutions of (4.1) and (5.2) we may assume C = E, and here X = eAt, hence Y = eAt, Y—1 = eAt. Thus the solution (8.3) assumes now the form (10.1)

b (u) du + eAt x (t0)

=

As an application consider the equation (10.2)

d2x/d12 + x =

b (t)

or equivalently

dx/dt—y,dy/dt=—x+b(i).

(10.3)

We have here

y

—

(sin t,

cos t

\cos€,

—sin t

and therefore {x, y}

or explicitly

(10.4) x

=

f

b (u) {sin (t — u), cos

b (u) sin (t — u) du;

=J

— u)} du

b (u) cos (t — u) du.

The general solution, is (10.5)

x = C cos (t — a)

+f

b

(u) sin

— u)

du,

with C, a as arbitrary constants. This is the well known form of the general solution of (10.2).

III.

73

LINEAR SYSTEMS

§ ö. Linear Systems with Periodic Coefficients: Theory of Floquet 11. Consider again our basic system (1.3), or

dx/dt=Ax.

(11.1)

We suppose t real and A real, continuous for all t, and with the real non-zero period z,. Thus A (t + w) = A (1). We will allow however for convenience complex solutions of (11.1), and so complex scalars. The domain of (11.1) is x

Let X =

0, be a non-singular solution of

IXI

DX = dX/dt = AX.

(11.2)

then (xi} is a base for the solutions of (11.1). EvidentlyX(t + w) satislIes(11.2) forthevaluet + w of t and as we know IX (t + co)I 0 also. Now Thus if x1

.

. .,

DX(t+w)=A(t+co)X(t+w)=A(t)X(t+w). Hence X (t + co) is a non-singular solution of (11.2) with non zero determinant, for the value t itself. From this follows

X (t + o4 = X (t)

(11.3) where

C is constant. Since IX

(1) X(t + cv) and CI

C

0, X' (1) exists, C =

0. In fact by (4.3): (trace A) cit

ICI=e

(11.4)

If we replace {xh} by another base the effect is to replace X by XP, IPI 0, and hence C by P'X' (t) X (t + cv) P = P is an arbitrary non-singular matrix and

the choice of base is essentially immaterial, we may assume X such that C is a matrix in normal form: C = diag (C1, . ., Cr), .

C (;t). In particular if the characteristic

where

roots The

(11.1).

. . ., hun.

of

C are all distinct C

d1ag

.

. .,

are known as the characteristic exponents of the system

74


Hence (1, 10. 10) there exists of thc same order as Cf such that a matrix and if B = diag (B,, . . ., Br), then ewB = C. Since

0,

likewise

0.

that if the characteristic roots are all distinct, or if they are not and still C = diag (1u,, . ., then choosing for any determination of i/co log we may take B = diag (2k, . ., are all distinct, the same will be If the true as regards the Notice

.

12.

Having chosen B consider the matrix

(12.1)

if we recall that B Z (t + co) =

and its power series commute we find

e(t + w)B. (X—' (t + co)

etB. ewB. C' X' = etB

)

ewB. etB (X (t) C)—'

Z (t)

(t)

In other words Z (t) has the period co. Moreover by (I, 10. 11): JZ

= IetBl

(ifl

0

for all values of t. Setting X* = ZX = etB then dX*/dt = BX*. (12.3) The relations between the coordinates assume the form: Xffi*

=

(1)

Xjh.

are the coordinates of the vector X*h referred to the coordinate system (xt*, . . ., deduced from the initial system Here

by the linear transformation (t) xj,

(12.4)

which is non-singular owing to (12. 2). The relation (12.3) yields dxth*/dt =

Thus x*4 = (xlh*, . . ., is a solution of the linear homogeneous system with constant coefficients,

III. (12

75

LINEAR SYSTEMS

=

.5)

xJ*

0, the set {X*h} is a base = Since = IzI for (12.5). The elements of this base may be written

(I)

XUi*

exit

where the are the characteristic roots of B, and the q,'s are are all distinct then the p's are polynomials. Whenever the constant. It follows that the initial base {xh (t)} for (11 .1) is of the form (1)

exit

where the *p's are polynomials in I with coefficients periodic and merely of period w, or else if the A5 are all distinct mod periodic functions of t.

We note then the following properties: (12 .6) THEOREM. By a transformation of variables (12.4) the periodic system (11. 1) may be reduced to a linear homogeneous system with constant coefficients. (12. 7)

There is a base {x") for the solutions of the periodic system (11 . 1) whose n elements are of the form

=

{v'lb (I)

.

.

are as above. (12. 8) When the characteristic

.

,

(t) eant}

where the

mod

exponents

the solutions are of the form periodic and of period

(12.9) Real

solutions. All

are all distinct

(t) eAit, where

is

that is required is to obtain a real

base, and a process for the purpose has been developed in (9). (12. 10) Remark. Generally the determination of the charac-

teristic exponents is difficult. For it necessitates the determination of a base for the solutions of (11.1) and following the elements of that base, for instance by analytical continuation, throughout a whole period co. And no one has ever viewed analytical continuation as a practical procedure.

CHAPTER IV

Stability

This short chapter is devoted to the fundamental concept of stability. Various types of "point" and "trajectory" stabilities are discussed and carefully defined and certain transformations which preserve stability properties are also discussed. References: Bellman [3]; Dykhman [1]; Krassovskü ti];

Lefschetz [2]; Levinson [3]; Liapunov [1]; Malkin [1, 2, 3, 7];

Perron [3]; Persidski [1, 2]; Poincaré [4].

§ 1. Historical Considerations 1. Historically, stability seems to have been first discussed by Lagrange in connection with the equilibrium of conservative systems. We merely recall that the state of a conservative system

depends upon a certain real vector x, its derivative x' (t) and two real continuous functions, the kineiic energy K (x; x') which is positive save that K =

0

when and only when x' = 0, the potential energy V (x), known only up to an arbitrary constant,

which satisfy the law of conservation of energy

K+V==const. If K and V are of class C' the positions of equilibrium may be defined as those where

0, 1= 1, 2, .. 76

IV.

STABILITY

77

where ii is the dimension of x. However, without referring to differentiability one may define equilibrium as an extremum of V, a definition which will be ample for our purpose. The following proposition first proved (later) by Dirichiet, was formulated by Lagrange: ThEOREM. Whenever in a certain position of the system the potential energy V is a minimum, the position is an equilibrium and that equilibrium is stable. (1 . 1)

Behind this theorem there lies the following definition of stability and it is this stability that we shall prove: (1.2) Corresponding to any e > 0 there is an > 0 such that

if e

Ix ever after.

f

+ x' fJ

at the beginning of the motion then:

J

We may as well assume that the position of equilibrium is x = 0 and that V (0) = 0. Since V is continuous there is a

c >0 such that in fjxjf 0 save at x = 0 where V = 0. Take now any e 0 there is an 2>0 such that at any E (s) ever after. We may suppose s so time t T implies X chosen that (2€) C U. Now corresponding to e there is an for x U — G (e) and all t ; (s) > 0 such that T xli so chosen that (2 c V. Hence for all and we suppose t r and y (si), T' y is not in U — (e). Since the origin like goes into the origin and T' (es) is connected, and since also

meets

(s), it cannot have points outside U. (e). The uniform stability assump-.

y Hence for all I 'r: implies that there is an (er) > 0 such that if y tion for (er) ever after. Owing to the (m) at time I x then y

restriction on the mapping T there is an ye V —

at any time t r implies J J T1

> 0 such that

yI

J

ever after.

By the same reasoning as before if x at any time t r at the same time, which implies Tx then Tx (s) ever after. Hence the origin in is uniformly hence x e stable.

Let now the origin be asymptotically stable in To show we must show that if > 0 there that the same holds in is an s (er) > 0 with such that T' (er) c (s) for all t If this is false there is an a > 0 such that T' (si) has points outside (a) for all E1 > 0. By hypothesis there is an >0 such that if I r and x e U — (a) then Tx II We may c (a). suppose (2a) c U and it follows as above that T—'

82


This contradiction proves the assertion as to asymptotic stability, and completes the proof of (5. 1). to 6. (6. 1) The following of the T in from the same in deduce uniform stability of the and conwhen for all t = then 0 versely.

Then given any e > 0 there Assume uniform stability in exist > 0 such that e1 e2 for lixil = and all implies that there is an (e') > 0, r. The assumption as to t then by the restriction on T that there is an > 0 such that if

for all t t. it is then 0 < a1 a2 clear that tc'gether with e establishes uniform stability of the

=

and

A similar proof holds with

origin in

interchanged.

7. The verification of uniform regularity is generally by no means simple. The following property characterizes a type of transformation which will cover many future requirements. (7.1) Let T be given by a relation y = A (t) x + a (x; t) where

for t

exists and A and A -' are continuous and bounded, (x; t) are tower series in the xj beand where furthermore the ginning with terms of degree at least two, with coefficients continuous

and uniformly bounded in I for t

and convergent in a fixed

spheroid xli 0 such that whenever jIxoII< x (t) —+ 0 as t + 00 uniformly in x0.

then

(11.2) We shall say that the origin is exponentially asympto-

tically stable (Malkin [1]) if there is a A > 0 and, given any.

r then

e > 0, a 6 (e) > 0 such that whenever x (t)

s exp [—

A

(t — t0)]

for all t to.

Exponential asymptotic stability is obviously the strongest of

all the types of stability that we have introduced. It is clear,

moreover, that equi-asymptotic stability is implied by uniform asymptotic stability and implies, in turn, ordinary asymptotic stability. However, it does not in general imply uniform (nonasymptotic) stability, not even in the simplest case of a single first order linear equation. Linear homogeneous systems have the following interesting property: (11.3) THEOREM. If the origin of a system (11.4)

r, is asymptotically stable then it where P (t) is continuous for t stable; if it is uniformly asymptotically stable is then it is exponentially asymptotically stable.

The first part of the assertion follows trivially from the fact that any solution of (11.4) has the form (11.6)

x (t) = X (t) X' (t0) x0

where X (1) is a (non-singular) solution of the matrix equation associated with the system (11.4). As to the second part, recall the remark made at the end of the proof of (3.3) which implies that II X (t) X' (ta) II < M for

t t0

where M is a constant independent of t0. By the uniform asymptotic stability there is a T > such that < 1/2 for t t0 + T. Hence we find that IIX(t) X'(t0) 0, it is convergent. Hence x (t) represents then a function which satisfies (1. 1). We will set up a system of successive approximations (t) based on (7.2) which will be in £1 (A, r) throughout and thus will

converge to an appropriate solution of (1.1). The first k columns of Z1 (t) represent solutions xO', of (2.1) which -÷0 as t -÷ + oo and we take

V.

x° (1) = d1 X01 + ••• +

(7.3)

93

NON-LINEAR SYSTEMS. I XOk.

Thus x° (1) 0 as t + Notice also that if in the notations of (6)

lix (t)II < 2

(7.4)

I

then also

0,

liz2 (1)11 < y2e°t, I < 0,

e

>

Our successive approximation reads now

(7.8)

Xm+l (I)

= x° (I) +

fz1 (1 — I') q (Xm (1'); 1') dl' Z2 (I — I') q (Xm (i'); I') dl'.

—f

The treatment is now the same as in (6) and details may be omitted. One shows first that if the d,, are small (7.4) m holds: i.e. if it is satisfied by xm then (7.4)m+i holds. Hence if one chooses the dh small, thus making hold, (7.4) will be true — for all m. Then the series E is shown to converge uniformly. The fact that the second integral in (7.8) has an infinite upper limit calls for a special argument in proving that x (t) = urn xm (t) solves (1.1). This is done as follows. Take some positive > 1. Then (7.9)

Xm+l (I) — x° (t)

=

f + f —

—

tl

Z1 (1 — t') q (Xm

(I'); t') dl'

(t — t') q (Xm

(1'); t')dl'

Z2 (t — I') q (Xm (I'); t')dt'.

94


Now r+0o

IiJ

Z2(t_t')q(xm(t'); t')dt'II

gl

<
1,

z,n=Mcm(z,, where M is a fixed positive constant. Then

(9.5) LEMMA. There is a positive constant y (M, A) depending solely upon M and A such that if 0. To prove the lemma it is sufficient to show that

(9.

(9.

holds

h <m. Then

z,mMcm(z*i, .. .,

.

= M cm (z*1 Z,, .. .,

Z*m

z,m

1),

(by the homogenity pro-

perty)

=

Mrpm (z,,

(9.

.

. .,

_,) (by induction)

(by definition).

V.

NON-LINEAR SYSTEMS. 1

Thus (9. 7)m holds. Hence (z1) = z = z1 + converges absolutely when Izil

z2

+

1, (10.

dx'/di

are polynomials in those of the where the components of m0: 2*/rn < o/2. Then for rn > in0 y > (1/2. Since according to (V, 8.5) the numerators in (2.1) are never zero y has a positive lower bound 81 for 1 1. Returning to the functionf, it has the formf = — a x1 + v (x), v = [x]2, where a is arbitrary. Let us choose a> 0. Upon substituting in (1.5) there is obtained for v the equation (2.2)

E

Xh +

=

øv/øxb — a

v.

Let M be an upper bound for all the Xh' in D (A) and set

ip(u)=M{(1—u/A)'—l—u/A}. One may determine V (x) from the relation (2.3)

(e Xh — q'

(u)) ØV/ôxk — a q' (u)

8

V,

like one may determine v (x) formally from (2.2), and it yields for V (x) a formal power series whose coefficients are all positive, which begins with terms of degree at least two and is a majorante of the series v (x).

Let us endeavor to satisfy (2.3) by a series V (u) in powers of u. If this is possible V (u) will satisfy the relation (2.4)

(8

u—n

(u)) dV/du — e V

=

aq

(u).

0 and is holo(u), where v' (0) Notice that q' (u) = morphic at the origin. Setting now V = uW, we find for W an equation

dW/du + j9(u) W = y(u)

(2.5)

where fi,

are

holomorphic at the origin. If

110


ö (u)

=

(u) du

then by a well known process (2.5) has a solution

W=

e 6(U) . v (u)

du.

Thus W (u) is holomorphic at the origin and W (0) = 0. Hence V = uW is a solution of (2.4) holomorphic at the origin and whose series in powers of u begins with terms of degree at least two. Since the formal series solution V (x1 + ... + x,1) has the same properties and is unique V (u) is that series. This proves the conVergence of the majorante, and hence of the formal power series solution, absolutely and uniformly in a suitable 9 (A).

3. From this point on we follow an argument due to Du.lac [1].

To begin with it has been shown that there exists a function which satisfies (1.5) and is of the form x1 + [x]2. Similarly for every j there is a function, now written z5 such that (3.1)

and that (3.2)

=

Now since the Jacobian of the right hand sides in (3.1) is equal

to 1 for x = 0, one may solve (3.1) for x as a power series in z. The identification of the first degree terms yields immediately for the solution (3.3)

x=z+[z]2.

Observe now generally that, with z5 replaced by ç' (x), (3.2) is (3.4)

dx/dt=X,

(3.5)

Now under the change of variables (3.3), the system (3.4) goes into

dz/dt = Z (z),

VI.

NON-LINEAR SYSTEMS. II

but (3.5) remains the same. Replacing back

111

by z5, (3.5)

becomes

(3.6)

dz;/di = Ajzj.

and so the system (3.4) goes Therefore Z (z) = (A1 z1, . . ., into (3.6) under our change of variables.

Now (3.6) is merely the first approximation to (3.4). Its general solution is z = (a1 exit, .. ., eant). Hence the general solution of (3.4) is (3. 7)

x = (a1 eAlt, + [a1 elzt,

.

.

. .,

. .,

eAnt)

elnt]2.

Of course this solution is only valid within a range for which the transformation (3. 3) is applicable, i.e. for I I (a1exit, .. ., II small enough. If the AA all have negative real parts, the solution

(3.7) of (3.4) will be valid for hail sufficiently small and t 0. However if some of the Ah have positive real parts the solution (3.7) will only be valid within a certain finite time interval:

tl t /2.

(3.8) Remark. The above result is to be compared with (V, 14.2) where the same result was obtained, following Liapunov, by means of series solutions of the differential equation. The Liapunov restriction—the real parts of the characteristic roots are all negative—does not differ very much from Poincaré's convexity condition. For admitting the latter there is a line D in the complex plane such that all the are on one side of D. Now the change of variables x —+ xeiwt replaces by = — ico. The new system has thus the form (3.9)

dxj/d/ =

4u;

x5 + [x

Since the ,a,s are merely the Ak rotated by the fixed angle — co one may choose the latter so that D becomes vertical to the left of the origin and with the to its left. "bus the will have

negative real parts and so (3.9) is amenable to the Liapunov method. The resulting series solutions differ from (3.7) but are valid for all time /.


112

2.

§

The Direct Stability Theorems of Liapunov

4. in his great Mémoire [1] Liapunov gave several theorems attacking directly the problem of stability. His method, Inspired

by Dirichiet's proof of Lagrange's theorem on the stability of equilibrium, is referred to by Russian authors as Liapunov's second method; his first method is to use the series solution of the differential system as exposed in (V, § 4).

Generally speaking Liapunov aims to set up something

resembling a potential function which has an extremun at the given critical point and to which trajectories do or do not tend. Let us examine the question in more detail. Take a real vector system dx/dt

(4.1)

X(x;t)

where X is continuous and satisfies a Lipschitz condition in a set .Q (A, r) and X (0; t) = 0 for t r. Thus the origin is a critical point and its stability is to be discussed. Let V(x; i) be a scalar function such that V(0; 1) = 0 whatever

r and. that V is of Class C' in Q(A, T). Let I': x(t) be a trajectory. Thus on P the function V becomes a function V(t) = V(x(t); 1) and on 1' its derivative is I

(4.2)

dVfdt = V'

(t)

ØV/ø1 +

ØV/ôXh

where the Xh are to be replaced by the components Xh (I) of x (1).

The whole argument of Liapunov rests upon the comparative signs of suitable functions V and their time derivatives V'. The idea is more or less that if one may choose V so that V = const. represent tubes surrounding the line x 0 such that all the F's cross through the tubes toward the line, the system is stable,

while if they proceed in the other direction, the system is

unstable.

5. The proper description of the functions V rests upon a certain number of definitions. In these definitions the scalar functions V(x; 1) and W(x) under consideration will be defined and, continuous, respectively, in Q(A, r) and 11(A) with

VI.


113

V(O; 1) = 0 for t x and W(0) 0. We say that W(x) .. [negative] sigh . IQ(A) of fixed positive V(x; t) Q(A, T) whenever it is 0 0] there; W(x) is positive [negative] definite in .Q(A) whenever it is > 0 .

in (1(A)

.

.

0;

for x

V(x; t) is positive [negative] definite in Q(A, r) whenever it dominates [is dominated by] there [by] a positive [negative]

definite function W(x); IQ(A,r)1

V(x;t)L . function ;is a Lia/unov W (x) .

(A)

j

whenever,

in that set,

it is positive definite, and with a derivative along the trajectories

IdV/dtl1which is continuous and of fixed negative sign. .

.

•

dW/dl

Observe that the restriction to class C1 imposed upon V is not essential but merely convenient. Of course when V is of class one may calculate V' directly from (4.2) and without any knowledge of the solutions. There are cases, however, when the weaker restriction on the Liapunov functions is decidedly convenient (see notably 24). As an example, of the two functions over Q(A, 2), x12+x22—2x1x2 cos t, t(x12 + — 2x1x2 cos t, the first is merely of fixed positive sign, but since the second dominates x12 + x22, it is

positive definite. 6. We shall now

consider Liapunov's theorems.

STABILITY THEOREM. If there exists a Liapunov function V(x; 1) over .Q(AI r) then the origin is stable. By hypothesis in Q(A, r), V (x; t) dominates a certain positive (6.1)

definite function W(x). Let 0 < s 0. Take any t0 r. Since V(x; t) set lixil = e let W(x) is continuous in x and V(0; = 0 there is a. 0 (e, t0) such that V(x0; t0) < a for J1x011 < Let F be the trajectory of the solution x(t) such that x(t0) Along

x0. Set also V(x0;

=

V0.

I'

(6.2)

V

=

V0

+5

V'dt,

to

Since V'

0 we see that along I', V

V0 < a. Hence I' can

114


never reach I xI I

e since on that set V

W

a. This proves

the theorem.

(6. 3) ASYMPTOTIC STABILITY THEOREM. If there exists in Q (A.

a Liapunov function V (x; t) dominated by a positive definite

W1(x) and such that V' is negative definite then the origin is asymptotically stable.

By hypothesis there are three positive definite functions W(x), W1(x), W2(x) defined in Q (A) such that for (x; t) ED(A, r) —V'(x; t). V(x; I) W1(x), W2(x) W(x)

At all events the origin is stable. With all quantities as before, one must show that the solution x(t) given any 0 < < is such that for t large enough < Let A be the lowest bound (positive) of W (x) on the compact set E xli such that W, < A in lxii < One may choose 0 < and hence V(x; t) < A on the same set and for all t i. Let

v be lowest bounds (positive) for W(x), W2(x) on

e

and suppose that the trajectory r remains in that set for I large. Then on F from some point (x0; t0) on we have from (6. 2) t above a certain value t1. At V0 —v(t — V that moment F is in i lxii < Since V < 2 and decreases for t

F cannot enter the set

Uxu e and the theorem

follows. (6. 4) INSTABILITY THEOREM. Let there exist a function U(x; t)

defined, bounded, and of class C' in Q(A, r). Thus dU/dt is defined along the trajectories in Q(A, r). Let U >. 0 in a certain subregion Q1 of whose portion B of the boundary contains the ray T: x =0, I r, and let U = 0 on B. Suppose that: (a) whatever there exist points (x0, t0) Q1 arbitrarily close to T;

(b) for every smallh> Othereisah(h) > Osuch that U in k(h) in the same set. implies U'

h

Then the origin is unstable.

Given any 0 < e < A and whatever 0 < tj < e and I9

< such that U (x0; = h > 0. Let I' be a trajectory issuing from (x0; ta). Since U' (x9;10) > 0, U increases along fandso U' (x; I) k (h) along the trajectory. Let A be a (positive) upper bound of U in the closure of the set {(lx(l < e} x 7'. Since U is bounded in Q(A, 'c), A is finite. Hence U, Now along F by (6, 2) for U: U h + k there is a point (x0; t0), Ilxoll

VI.


115

along F, will sometime exceed Hence F will leave the set U h, lxii < e. Since this can only be through the sphere I

lxii = e, the origin is unstable.

(6.5) COROLLARY. Under the same conditions for U but merely with

U'=1uU+U*

where ,u is a positive constant and U*

0, the origin is again

unstable.

For along F at once U

from which instability follows. (6. 6) Cetaev's generalization of Liapunov's instability theorem.

observed that actually all that the theorem requires is not the whole of Q(A, r) but merely a subregion containing SI1 u B.. Otherwise the statement of the theorem is unchanged. (6.7) Autonomous case. One may then state the corresponding theorems, but with functions W(x) replacing V(x; t) in the two stability theorems, and a function U(x) in place of U(x; t) in the instability theorems. Moreover, since one may replace everywhere t by t + k, one may take throughout r = t0 = 0. There are several interesting consequences: (a) Stability theorem. This time will be a function (e), independent of t0: the stability is uniform (see IV, 2).

(b) Asymptotic stability. We have here V(x; t) = W(x) = W1(x) and — Vl = W2(x), but otherwise there is no change. (c) Instability. The conditions U(x) bounded, and those related to h, k(h) follow naturally from the compactness of lixil

8), Geometric interpretation. It is most convenient to describe it for n 2 and functions W(x1, x2). Suppose that W is definite (6.

positive. Introduce a third coordinate y and consider the surface F: y = W(x1, x2) over Q(A), that is within the cylinder x12 + x22 = A2.

We think of it as an inverted cup over the

region. In that region the surface is above the horizontal plane y = 0, except for touching it at the origin. The curves W(x1, x2)

=

e

are the projections of the horizontal sections

H(e) of F by the planes y =

e.

The paths P in 51(A) are imaged

116


I

I. STABLE

U. ASYMPTOTICALLY UNSTABLE

Fig. 1

into paths A on F. in any case the orthogonal projection between F and 12(A) is actually topological. Now let us examine how the sign of W' affects stabilfty. To

say that W'

0 along a path is merely to affirm that on the

surface F the path (I of figure 1) goes down sluggishly and need not reach the origin. This corresponds to mere stability. If W' is actually ncgative and below a certain —a < 0 then the


VI.

117

path has a downward slope bounded away from zero and tends briskly to 0. This is the case for path ir and we have asymptotic

stability. Finally, if along some path such as III: W' > fi> 0 the reverse takes place: the path actually ascends at a rate bounded away from zero, and reaches H(s) no matter how low its starting point. Here we have instability. One may observe that if W is of class Ct and the surface F is tangent to the horizontal plane at the origin, then the curves H(s) for e small enough comprise a set of ovals surrounding the origin.

The above representation is easily carried out for any dimen-

sion. For V(x; t) something analogous may be done but the resulting inverted cup will touch y = 0 along the ray t r. 6.9 In the treatment of the theorems of Liapunov the coordinates have been (tacitly) assumed real. Suppose, however, that there are conjugate complex pairs of coordinates and ii — 2 real coordinates as

Xi,Xi,...,Xp,Xp,X2p+1,...,Xn. Then, •

., x

in the notations of (I, 13.10), Visa function V (1; x1, x1*,

...) which is real at the real points x1,

+

Thus .

...)

.

==

...)

..

and we have V'

=

+

+

+

t9V/t3xh ±

X2

+

0 V/Ox2

+ j,

Xh* has the meaning fully described in (I, 13.10). (6. 10) A/plications. As a first application of the theorems of Liapunov let us prove a partly more general stability property than (V, 7. 10). Of course the proof is startingly simpler and more direct. where

(6. 11) Given the quasi-linear

system

dx/dt= Bx+q(x;i)


118

where, in I? (A, T), q is continuous and satisfies a Lipschitz condition, q(x; t) = a(lIxIl), where a 0 with lxii, and B is a constant matrix whose characteristic roots 2h are all distinct. Then if the 2h all have negative real tarts the system is asymptotically stable; if they all have positive real tarts the system is unstable.

Choose coordinates such that B is in normal form. If the are all real take V= Then

V' = 2Et,lxh2 + and so for x small V' has the sign of the (7.2) is a consequence of (6.3), (6.4).

Since

V is positive

Suppose now that 'ab, /i = 1, .. ., are complex and = 1, 2, .., — 2 p are real. Then choose + .

V= E

if

=

4äh'

+i

+ E X22p.I.j.

we now find

V' = 2 E Xh'

+2E

+ o(IIxII 2)

and the conclusion is the same.

(6. 12) As a second application let us give a second proof of Lagrange's theorem (IV, 1.1) on the stability of equilibrium. Let us suppose that we have a system with n degrees of freedom

and in the notation of (IV, 1) let H K + V. Suppose also that the system depends on the usual variables q1, .. ., the the kinetic variables. The positional variables, and p1, .. ., potential energy V depends solely on q and we suppose that V (0) = 0 and that it is a minimum of V. Otherwise let V be holomorphic at the origin. Regarding K it is a positive definite quadratic form in the Ph with coefficients holomorphic in q. Thus H is a positive function in the sense of Liapunov. The equations of motion are dq8/di =

ØH/0p8,

= — 0H/0q3.

If we take H as the V-function of (6) we find dH/dt =

0.

Therefore

the origin is stable. That is to say static equilibrium at q

0

VI.


119

is stable. This is (IV, 1.1). The proof has been given under certain

restrictive conditions which are however readily removed. Liapunov has proved the counterpart of Lagrange's theorem for an isolated maximum of the potential energy. He restricted himself, however, to the analytical case. The complete statement is:

Let the Hamiltonian H(p; q) be analytic at q = 0. If q = 0 is an isolated maximum of is an hunstable energy, then this

(6.13)

the origin 0: /' = of the

0,

equilibrium.

With the notations remaining the same, this time

= K2r(P) +

.

.

.,

V(q)

=

—V2(q)

+...

V3 are positive definite forms of degrees 2r, 2 in their

where

variables and . .. represent, as usual, terms small relative to

those written in a certain neighborhood 9 of 0. Choose now

U=

Then

dU

=

+

+... = 2(rK2 + V2) +...

(by Euler's relation for forms). Thus in .Q: dU/dt> 0, except

that it is zero at 0. On the other hand in

qh> 0 (all h) we also have U> 0. Hence the equilibrium where V is a maximum, is unstable.

The simplest example is that of a vertical bar fixed at one end. If z is the distance from the centroid to the point of suspension, the potential energy is V = —mgz. It is a minimum where the bar hangs downward — stable equilibrium; a maximum • when the bar stands upward — unstable equilibrium.

7. We have dealt (6.11) with the stability of a quasi-linear system in the case of distinct characteristic roots. We shall now attack the more general case of a non-critical matrix B, that is, one which has no zero or pure complex roots. The construction of suitable functions V is then definitely more arduous.

120


Consider again a system (7.1) dx/dt = Bx + q(x; t) where B is non-critical and as be fore in Q (A, r): q(x; t) = a lxii, where a -+ 0 with lxii. The stability behavior of the first approximation I

(7.2)

dx/dt = Bx

is known, (see IV, 4): the origin is asymptotically stable if B is stable, and unstable otherwise. We propose to prove: (7.3) THEOREM. If the matrix B is non-critical the stability j.roperties of (7.1) and of its first

(7.2) are the same.

The method will consist of constructing simple functions V for (7.2) and showing that they also serve for (7.1). Let us agree for the present that all vectors are columnvectors (one column matrices). If x is such a vector, x' stands for the row-vector (one-row matrix) with the same components.

As a consequence, a quadratic form

P(x) = with the matrix F symmetrical: F = F' may be written simply x'Fx. In particular, the Euclidean distance squared is x'x (inner vector product). If our quadratic form is positive [negative] definite we shall

denote the fact by F> 0 [F < 0].

If f(x) is a scalar function øf/øx = grad f will stand, for the present, for the row-vector with components af/øxh.

Let P(x) be the same quadratic form as above. Then its derivative along the paths of (7.2) is (7.4) dP(x)/dt = (e9Pfax)Bx = Q(x) where Q (x) is likewise a quadratic form. We propose to prove the

following property due more or less to Liapunov: (7.5) If the characteristic roots of B all have positive [negative] real parts and Q is positive [negative] definite, then (7.4) has a

unique solution for P (x) and it is a positive definite quadratic form.

VI.


121

It is evident that this property for the "negative" case is an inverse of Liapunov's asymptotic stability theorem for the linear system (7.2), and that the "positive" part is a partial inverse

of his instability .theorem for (7.2) also. Suppose that P(x) = x'Fx, F' = F, Q(x) = x'Gx, G' =

G.

Then

dPfdt = x'(B'F + FB)x. Hence (7.4) yields

B'F+FB=G.

But then it is clear that (7.5) is an immediate consequence of (App. I, 8).

Suppose now that the characteristic roots A do not have negative real parts, and that 0 have positive real parts. One may then choose coordinates Yi' . ., Z1, ., 2g such that B = diag (B1, B2) where the characteristic roots of B1 all have .

.

.

positive real parts and those of B2 all have negative real parts (App. I, 11). Note that if q = 0, then B2 0, but the modifications required in this case are obvious enough. At all events, (7.1) assumes now the form dy/dt

-

=

B1y,

dz/dt =

B2z.

Take

Q,(z) = ZZk2, Q(y; z) = Q1 + Q2. According to (7.5) we can find positive definite quadratic forms P1 (y), P2 (z) such that Q1(y)

=

B3y = Q1(y), (aP2/az)B2(z) = —Q2(z). Let then P(y; z) = P1(y) — P2(z). Thus P> 0 in the region P1> P2 of the (y; z) space. This region exists since it contains the set z = 0. In the whole space however (origin excepted) z)

= Q1(y) + Q2(z) > 0.

Thus we have an inversion of the instability theorem (6.5).

122


We shall now apply the preceding results to the quasilinear system (7.1).

If B is stable take Q(x) = and P(x) as the unique positive definite quadratic form solution of (7.4). We have then in Q(A, r) dP(x)/dt = Q(x) +. along the solutions of (7.1) (the dots are as before). Hence in Q(A, r): P'(x) < 0, Q(x) > 0, except at the origin where both vanish. Therefore, by (6.3) the origin is asymptotically stable for (7. 1). Suppose now that among the real parts of the characteristic roots some are positive. Under the same choice of coordinates

as before (7.1) may be replaced by a system dy dz

where

=

+

= B2z+

Tizil)

q2(y; z; t)

and a:

0

with

+

Hence

with P(y; z), as before, we have again dP/dt = Ql(y) + + so that in a certain region 0: P> 0 and in Q(A, r) itself (the ray T excepted) dP/dt> 0. Hence we have instability. This completes the proof of Theorem (7.3). Concluding remarks about

direct method. The few

applications that have been discussed suffice to indicate the scope of the method. We emphasize the important fact that if one can obtain suitable functions V one may obtain stability information about the solutions directly from the equation itself, and without any knowledge of the solutions. §

3.

Stability In Product Spaces

8. The title is somewhat misleading. What we have in mind is the discussion of the stability of certain systems of the form 1(a) dy/dt (y; z), Y (y) + 8 k (b) dz/dt = Z (z) + (y; z).

VI.


123

Here y and z are a p-vector and a q-vector; the functions Y, Y*, Z, Z vanish at the origin which is thus a critical point for the system; and Z* are small in some sense relative to Y and Z.

The question arises as to what extent is the stability of the complete system governed by that of the partial systems (8.2b) (8.2a) dyfdt=—Y(y), dz/dt—Z(z). This problem has been dealt with at length by Liapunov and his successors in the Soviet Union. However, except for a general

result due to Persidskii they usually confine their attention to Z = Q z, where Q is a constant or variable matrix such that (8. 2b) is stable. Generally also they assume that Y, . . ., contain

the time t. We feel, however, that even the autonomous case,

upon which we mostly concentrate, will give a respectable notion

of the scope of their work. The norms for 9. Let = x those induced by the norm lix Ii = sup

=

sup {iYiI} and lIzil

{IxhI}

sup {IzkI}. The

and (V. 4) are £1,, (A), Q,, (A, t) for Thus (A, r) is the set: liz Ii A,

are naturally

and

for

namely

basic Q regions of

(A),

(A, r) for

We first treat a noteworthy result due to Persidskii [1] . He actually established it for vectors with countable components but for our purpose it will be sufficient to deal with finite dimensional vector spaces. Consider a system in a p-vector y and a q-vector z 1(a) dy/dI = Y (y; z; t) (9 1) (b) dz,'dt = Z (y; z; 1) in a closed region Q (A, r). It is assumed that the system is continuous and satisfies a Lipscbitz condition and that Y (0; 0; t), Z (0; 0; t) = 0 for t r. Thus the origin is a critical point for the system. Let now (t) be a continuous q-vector. Take the associated system

dy/dI = Y(y; flt); t). Let 0,,, 0 denote the origins y = 0 and z 0 for (9. la) and (0. lb), and 0 the origin y 0, z = 0 for the complete system (9.2)


124

is quasi-stable for (9. la) whenever, given any 0 <e < A and 4, r, there exists 0 e such (e, 4,) then any solution y(i) of (9.2) with that if II
0. This relation is well adapted to the use

of the generalized Gronwall inequality (10.4b). We set then (.t

IIz(t)II =

k

B IIzolI + C I

Ca,

Jo

Hence (l0.4b): (BIIZOII

+

Ca J 0 fBzo

+

± cf0

evt'Uti(u)IIdu) du)

C $0

After some simplifications and an integration by parts one obtains

+C or,

ds,

finally returning to z(t):

(10.7)

B

1141)11

ds.

we dispose of a let it be chosen 1. This is assumed henceforth for (11.1) itself. 12. We must now prove that the formal power series represents a function holomorphic at the origin. To that end we fall back upon the standard device of the majorante.

Let us set

z

=

F(Y,Z)=

11

l—A(Y+Z) G(Y, Z)

=1

(Y

(1+ + Z)

+

.

. .

Let 2 = inf {—Re 2,). We recall that according to (App. I,

3.3), one may take as normal form for Q one in which the subsidiary diagonals 1,.. ., 1 may be replaced by —e,..., —e where e> 0. As a consequence our system -is seen to have the majorante, in which M is suitably large: (12. 1)*

Zk_l —

MF(Y, Z)} = MG(Y, Z),

where z1 = 0. This system manifestly possesses a formal power (z) whose coefficients are all positive. Let series solution (12. 1)* be modified to (12. 1)** by making z_1 = The analogue Q** of (12. l)** has for equation of characteristic roots for Q (12.2)

(A—x)fl—efl=0.

Thus the characteristic roots of Q** are 2 — k = 0, 1, = Their real parts are 2 — e cos 2— e. fl — 1,

Hence if 0 < e < 2, the real parts all have the same sign. l)** will have a unique formal power series Consequently (z). We shall prove: solution

VI.

NON-LINEAR SYSTEMS. ii

129

(12. 3) The series * (z) is holomorphic at the origin and its coefficients are all positive and at least as great as the corresponding (z). Hence y** (z) is a majorante for y (z). As a coefficients of consequence y (z) represents a solution of (ii . 1) of the desired type. Assume that it has already been proved that all the coefficients of

(z)

be the corresponding

are positive. Let

coefficients of y*, y** and let the analogue of (11 .4) for y* be (12.4)

I'h(m) (.

m A q,h(m)

..,

)

Here Fh(m) is a polynomial with all coefficients positive. For

there will result relations (12.5)

= f'h(m) (.

.) + £1 (ifl) (. . where LI is like 1' save that (m)' may also be posterior to (m) in the ordering of {(m)}. Suppose that it has been proved that q,h(8) for all (s) preceding (in). This follows at once for the first (in) from the comparison of (12.5) and (12. 4). The in A

.

.,

. .

.

,

same comparison yields then the same result for all (m) in succes-

sion. All that remains then is to show that (12. 1)** has a holo-

morphic solution whose series expansion has only positive coefficients.

Let us endeavor to find a solution of (12.1) * * in which = a function of Z. For such a solution Yi ... = yp we will have

Z—qMF (Y, Z)} = MG(Y, Z).

(12.6)

The change of variables Y = ZY1 reduces (12.6) to a form readily shown to be — + aZ + [Y1, Zj2 . ) ( 127

dZ

—

Z(1

+

Z]1)

where a = 0 if r> 2. With an auxiliary variable 1, (12.7) is equivalent to a system. (12.7a)

[Y1, Z]2;

+ [Y1, Z]1).

Anticipating on a later theme (IX, 4) independent of the present

argument we have here a saddle point. It has two analytic

130


solutions as curves through the origin, the one Z =

0,

and the

other with a different tangent, of the form Y1 +flZ + [Y1, Z]2=0.

This one gives a solution Y1 = [Z]1 or [Z]2 to which corresponds a solution Y = [Z]2 or [Z]3 of (12.6). Since — e> 0 upon substituting a power series

Y=Z,)mZm

(12.8)

in (12.6) and identifying coefficients, we find that the are uniquely determined and they are in fact zero for m 0 there is a unique form of degree ni satisfying (12. 9a); (b) if f (z) is a polynomial and f(z) =fr(z) +fr +1(z) ± .. +18(z) where r >0 and is a form of degree h, there is a unique solution W (z) which is a poly.. nomial such that W = IV,. ± .. ± W6 where Wh is a form of degree h. (Liapunov [1], p. 276). Regarding (a) it is at once seen that the formal solution W (z) has only terms of degree m and so it is an actual solution. Thus no majorante argument is needed. As for (b) Wh is merely the solution of (12. 9a) corresponding to f = .

.

§ 5. Stability In Product Spaces: Analytical Case 13. Take an analytical p-vector system (13.1) dy/dt F3(y) + + where the components of Fh (and similarly later) are forms of degree Ii in the Yi'• ., y,, and the right-hand side is convergent in Q(A). (13.2) Definition. (Malkin[9], No. 91). The origin is said to .

be stable for (13. 1) regardless of the terms of degree > N when-

VI.

131


ever it is stable for (1&2a)

in the following sense: whatever the continuous function

K> 0, in Q(A), and cor-

G(y; t) such that uGh

responding to any 0 <s

A, there is a 0< 77(e,K), such that

for any solution y(t) of (13.2a) such that iIy(°)h1 N, whenever under the same circumstances as above and no matter how small fly (0)11 for some y(t) sometimes IIy(t)II =e. Consider

now an analytic system

F(y) + Y(y;z) (13.3)

dz

Qz+ Z(y;z)

with the following properties all valid in some I.

II. The stability properties of dy/cit = F(y)

(13.4)

are independent of the choice of terms of degree > N in the sen of definition (13.2); III. Q is a constant stable matrix; [V. Y(y; z) = [y; z]2, Y(y; 0) = [Y]N÷1; V. Z(0; z) = [z]2; Z(y; 0) = [YIN.

We have then with Dychman [1]:

(13.5) THEOREM. Under the preceding conditions the stability properties of the complete system (13. 3) and of the reduced system

(1:3.4) [in the sense of (13.2)] are the sawe. The proof will essentially consist in applying a regular transfor-

mation of coordinates suppressing in Y(y; z) the terms in y whose degree

and then applying Dychman's criterium (10.3).

There are (trivially) no terms of degree —1 in y in Y (y; z). Let us suppose that there are no terms of degree <m N in y in Y (y; z) and consider a regular transformation

...

(13.5) (m)

Emj=m,


132

any combination of the summands m, and the urn (z) are non-units, holomorphic in (A). Such a transformation has an inverse where (m)

is

(13.6)

where v 0) = 0, v z) = and v (0; z) = 0. By substituting the expansion of v in (13.5) it is found that v = We find now from (13.5) that dy7dt = F (y) + Y (y; z)

(13.7)

mj u(m) y1mi ... y/fl, —'

— (rn),j

...

(Y5 (y) + l'j (y; z))

... (m)

14.

We may write

(14.1)

where the components of Yh are forms of degree Ii in those

of y with coefficients non-units in z for h powers y1mi. m = ni1 +

N. Let the

be ranged in lexicographic order and let

.

... +

(m) = {m1, .. ., m,,}. Suppose

that by

successive application of regular transformations (13.6) we have succeeded in reducing (13. 1) to the same form with the same F and Q, but with Y(y; z) lacking all terms corresponding to an

(s) preceding (m). Upon applying a regular transformation (14.2)

z

z,

y

y + y1m'. .

.

we find that in addition the term corresponding to (m) will disappear provided that v satisfies a relation (14.3)

+ Z(0;z)] = B(v;z)

B behaves like the analogous term of (11.1). Hence, (14.3) has a unique, non-unit solution v(z) and we have the

where

desired transformation (14.2). By repetition of the process just described we will finally

133


VI.

reduce the system (13.1) to the same form, with F and Q unchanged but with Y(y; z) = Under the circumstances, in view of the convergency .of the N* (requiring perhaps repetition of the operation). In other words our transformation preserves the form of (15.1) but nmy change N and the new system will have a T N. where Z* is a non-unit and Z* (0; 2*) = [z*]2,

Let us assume then that (15. 1) already satisfies this condition.

An exceptional case is when identically 0* &) = 0(y) +

Y(y; C(y)) = 0, (y; 0) = 0. When this happens the system (15.4) has the solution z* = 0, y c, where c is an arbitrary

constant. In other words the y axis is a line of critical points. We shall assume that we are not in presence of this case. Thus effectively N 2. 16. Let 0(y) gyN + . . . then (16. 1) The stability

of

dy/di = O(y) are independent of the terms of degree > N.

That is, if

(16.2)

+ h(y; t)y]

where h is continuous and Ih (y; t)$ 0 (6.4) and asyrnptotically stable when g < 0 (6. 3). Needless to say these results agree with (16. 4).

VI.

NON-LINEAR SYSTEMS. Ii

137

7. The converse of Liapunov's theorems. 18. rs the stability situation of the origin completely characterized by the existence of suitable functions V, W or U? The

problem has been dealt with by many authors and largely solved, in the affirmative, by Krassovskii [l}. There are also notable contributions by Persidskii [1] (first to deal with the question), Malkin [9], Massera [2, 3], Kurzweil [1], and others. Especially ample references are given by Krassovskii [1] and Antosiewicz [2].

The complete discussion of this converse problem is quite complicated and difficult. We shall do no more here than give a taste of the situation and deal with the converse for stability and for asymptotic stability. 19. Take our usual system (19.1)

dx/dt

X(x; t),

0 for t r in a region Q (A, where we assume, however, that X is of class C'. Prelimina?y observation: It is evident that the proofs of the two Liapunov theorems on stability and asymptotic stability with X(O; t)

are still valid under much weaker assumptions on the functions W(x). In particular the following type is quite sufficient. Let

stand for the Euclidean modulus and choose as W(x), a step function F(r) defined thus: Divide the interval

r(x)

.=

lxii

A] into a sequence of subintervals by means of a decreasing sequence A0 = A, A1, A2, . . ., where is the midpoint of —,- 0 with 1/i. As a matter of fact, any [0, As]. Thus decreasing sequence -÷ 0 with the preceding property would define F(r) = do. On 0, where < and —. 0 with lfi. It is clear that the adequacy of the new functions W(x) is much more easily verified than that of the original functions.

Notice furthermore, that as a consequence one has the following characterization (due to Antosiewicz [2]) of the function V(x; 1)

of Liapunov's theorem: V is of class C' in (1 (A, r); it is zero for every 0 < E 0 for x = 0 and t 0 in Q(A, r). It is such that V in Mxli [e,A]; V' such a function that is henceforth called a Liapunov function. 20. We shall now take up the proof given by Persidskii [1]

138


of the inverse of Liapunov's stability theorem (6.1). We foJiow more or less the detailed formulation of Antosiewicz [2) p. 146. It will be convenient to write F(t; x0; t0) for the solution of (19.1) with trajectory F passing through x0 at time We also assume that the domain of existence is of the form I x 0, HxII 0 for e> 0 and G (0) =

For I> 11 we have is a decreasing function, ç,*(t) 1. Then if the Perron conditions hold there exist /ositive constants a, fi such that for all I

Let us write everywhere G, ..., for /.s, G1 Proof of (6.2). It follows from the expressions of the Yss in (4)

that as a consequence of the second Persidski condition G(t0 + T, <M for all r. Hence

G(10+mT,to+(m—1)T)<M. As a consequence WI

G

+ mT, t0) = H G,

+ (r + 1) T,

0

<Mm G (ta, t),

148


t + T. Hence if y=sup G(to,T) for t0 in the

for r

segment just stated:

G(t,r) 0.

Since

Consider now the expression f°°Gm (I', t) di'

Hm(t

t + oo it takes the form 0/0. By the rule of 1' Hospital its value as I —k + 00 is the same as for the quotient of the derivatives

or of the expression

__.Gm(t, r) —,nG' (I, r)

—

Hm±l(t,r)

—

in

+1

Hence

as I -÷

+

oo

K(m) (I, i)

H(tr) -k-. in 1

Since H (I, i) is bounded so is K(m) (1, v) and this completes the

proof of (6.3).

150


7. The results just obtained will now be applied to the proof of the following proposition also due to Malkin [1]: (7. 1) If the Perron conditions hold then the system (1. 1) *ossesses function which is a quadratic farm in the a strong

Observe that the existence of such a Liapunov function is not affected by a linear homogeneous transformation of coordinates (with constant coefficients). Consider a transformation — (7*

—

'Yi

= (7* — 2Yz ..

., Xg = y,,.

The system for y will be dy1/dt

= Pu Yi

dy2fdt =

a P21 Yi +

= a"

Y2

yi + ... +

y,,.

By taking a small enough we may dispose of the situation so that the coefficients below the main diagonal are in absolute value arbitrarily small for t r. Let us suppose then that in the initial system (1. 1) the j k, are suitably small. The Perron conditions still hold and so we can define

V(x;t) =EK82)(t,r)x.2, and we note that since there are positive constants a, fi such that

fl,s=1,2,...,n we will have W (x)

= fi

V (x; t)

a Ex82

=

W1

(x).

Hence V is positive definite, and -÷ 0 with x uniformly in t. Moreover

E (2p,, (t)

V'

+

= — E X82 +

(1, r) —

1)

x.'

VII.

NON-LINEAR SYSTEMS. III

151

where the qsr may be assumed arbitrarily small. Let them be chosen such that for Ex,2 = 1 the maximum of E x, x, is < 1/2. Under these conditions V' 0" and < 1 or > 1" are equivalent properties. Hence we deduce from (V, 14 .8) (2.5) STABILITY THEOREM FOR LINEAR PERIODIC SYSTEMS. If the

characteristic exponents are all < 1 [> 1] in absolute value the system

VIII.

PERIODIC SYSTEMS AND THEIR STABILITY

159

(1. 1) is asymptotically stable [is unstable] at the origin. If k of the exponents are z 1 [> 1] in absolute value there is a linear k-dimensional family of asymptotically stable [unstable] solutions at the origin.

§ 2. Analytic Systems with Periodic Coefficients 3. We turn now our attention to real systems in an n-vector x, of type (notations of (I, 14 . 5))

dx/dt=P(t)x+[x;t]2

(3.1)

where [. . .] is holomorphic in x for all I in a fixed Q (A). Further-

more the terms of P (1) and the coefficients of the series [...J are continuous and of period cv in t. Thus the question of the

stability of the origin arises again and there are now two possibilities.

I. The matrix P is constant. The system falls then under the theory already treated (see V, 14. II. The matrix P is periodic but not constant. Then the change of variables (1 .3) will reduce the system (3. 1) to one of the same form but with P replaced by B. system falls under the type studied under (V, § 4) and (VI, § 1).

Observe now that property (V, 8.5) for the 4 is equivalent to the following for the characteristic exponents (3. 2) There exists no combination H /Lhmk, mh

0, Em,j > 1.

It may be observed that since the Aj are only determined condition (V, 8.5) for them reads: (3.3) The A5 satisfy no relation

mod

Emh 4, mod

0,

> 1.

We have now from (V, 9.1, (3.4) EXISTENCE THEOREM. If the characteristic exponents are all and satisfy (3 . 2) then theorem (V. 9. 1) holds for the system (3. 1) provided that the (1; a) are now

< I in absolute

in I with coefficients periodic and of period cv in t. The 4 occurring in the series are the characteristic roots of the matrix B.

160


(3.5) STABILITY THEOREM. If the characteristic exponents are all

< 1 in absolute value the system (3. 1) is asymptotically stable at the origin. If of them are < 1 in absolute value there is a family

of solutions depending analytically on k parameters which is asymptotically stable at the origin. §

3.

Stability of Periodic Solutions

4. The preceding results have an immediate application to the stability of a periodic solution (t) of a system (4.1)

where X is holomorphic in x in a certain and is continuous and periodic with period w in

of

and

for all t (t)

Suppose in addition that if x = + y, then for y small enough and t arbitrary X + y; t) may be expanded in power series of y. Upon substituting in (4.1) there results an equation in y of the form dy/dt = P (1) y + {y; t]2. (4.2) The matrix (4.3) is

P(t)

=

e9x

merely the Jacobian matrix of the X5 as to the Xk taken on the

known closed trajectory. If P (t)

0 and is non-constant we

are under Case III. There will be a set of characteristic exponents {4u,} and we may state: (4.4) If the Jacobian matrix (4.3) is not constant and are its characteristic then theorem (3.5) holds with orbital stability of the trajectory (t) instead of stability. §

Stability of the Closed Paths of Autonomous Systems. The Method of Sections of Poincaré 5. It so happens that autonomous systems escape the result

4.

just obtained. Let indeed (5.1)

dxJdt=X(x)

VIII.


161

be such a system, and let it behave otherwise like (4. 1) and in particular let it possess a solution (t) of period cv. We form again (4.2) but unfortunately: (5. 2) At least one of the characteristic ext'onents (Poincaré)

is now unity.

We notice first that

dy/dt=P(t)y

(5.3)

is the variation equation of the periodic solution (i). Since (t + T)/t9T is likewise a (t + r) is also a solution of (5. 1), solution of (5.3). Since this solution is periodic of period cv, (5.2) is a consequence of (2.1). Thus in the all important autonomous case one cannot apply the full stability part of the general stability theorem and one must look elsewhere to complement the argument. This is where Poincaré's method of sections enters the problem. ft may be said that this method was extensively utilized by G. D. Birkhoff ([1,1 II, 268). Speaking first in very general terms suppose that the paths of an autonomous system, say (5. 1), intersect repeatedly a differentiable manifold —1, and do so "without contact," i.e. without — 1• ever being tangent to This means really that the vector X (x) is never a differential vector of Let a path y intersect —1 consecutively in points Q, Q'. Then Q -+ Q' defines a trans-1 into elf. The fixed points of S correspond formation S of to closed paths. Moreover if F is such a point and the corresponding closed path it follows readily from the definitions in (IV, § 6) that the stability behavior of the transformation S in the neighborhood of F characterizes the orbjtal stability properties —

of y. We will observe the application of this principle in a 1 moment. The manifold is called manifold of section of the system, and the method just exposed is Poincaré's method of sections. —

6. Returning now to our problem let y be the closed path of (1) and let Q be any pOint of y. Take Q as origin and coordinates such that the axis is the tangent to y at Q. Thus in the new coordinates X(Q) = (0, ..., 0, (Q)). We may further choose

units such

(Q)

= 1. Thus X(Q) is mcrely the unit vector


162

axis. To simplify matters we will also suppose that the period is and that Q is (0) so that (0) = 0. Let now the present disposition be identified with the dispoalong the

1 there sition of (11, 16) and so that the transverse cell considered is in the hyperplane H; = 0. One may now take —

as the parameters uh the coordinates x1, . . ., H. However it will be actually more convenient to preserve for those coordinates the designations uh. —

According to (II, 16) the path y' passing through the point u (u

small)

will meet H next at a point u'

where

(6.1) is a constant matrix of order n — 1. Now the general solution, near E(1), starting at the point u

where D

0 is analytic in u and so it has the form (notations

in H at time t

of (I, 14.5)) (6.2) x (t; u) =

(t)u + n (t) +

(t;

u),

ço =

[u]2

an n x n — 1 matrix. If y = ., is an n-vector or A an n x m matrix, denote the (n 1)-vector (y1, . ., let and A* the (n — 1) x in matrix of the first n — 1 rows of A. Let also where

is

.

.

denote the rn-vector of the last row of A. With these notations we obtain from (6.2): x*(0; u) = u —

=

Hence

±

+

u).

Observing now the relations

= 0;

a ± 0

=

+ s is the next time that y' crosses H, we have

and if

+

+ e, u) =

±

ae±

As a consequence, e (u) is holomorphic at u

Observing also the relations (0)

(2'r)

=

0;

=

u),

=

[e; u]2.

= 0 and e (0) = =

X* (0)

= 0,

0.

VIII.


163

we find again from (6.2):

+ e; u) = U' = r (2ic + e) +

+ €) u

= D.

and therefore

Recall now that according to (III, 3.8) and the remark leading

to (5.2), the variation equation of (5.1) has the fundamental matrix solution

I

+ r)

*

-

C (t) = a

(t + r)

the characteristic roots of C the characteristic exponents. Now Since C (0)

C are

0 —

1

Hence the n — 1 characteristic exponents, other than unity corresponding to the solution (t), are the characteristic roots of the matrix D. That is: (6.3) The characteristic exponents of the autonomous system (5. 1)

other than one equal to unity are merely the characteristic roots of the constant matrix D of the transformation (6. 1).

7. By applying if need be a linear transformation to the

variables Uh (a linear transformation in the hyperplane H) we may reduce D to normal form. Let this be done and let us suppose

that the characteristic exponents are all less than unity in absolute value. Let S denote (6.1) and let its linear approximation

S1be (7.1)

u'=Du.

Take one of the constituent blocks of D say

/4tL0...


164

and if k is its order let Rk = — 1u Ek. We recall that RkI is merely Rk with its diagonal of units lowered by j steps. Hence for s sufficiently high

0,...,

0

1

D18

=

+ Rk)' (k!1)

(k!2)

•

•,

so that < 1, is the largest then the terms of D8 are in absolute value < = a Sn/.t8 where a is independent of s. Also log = log a + n log S From this we conclude that if

s log (I/'u) -*

follows that if

00 with —s. Hence

<e

then IJSi8uII

Now referring to (6. 1): take so small that when u < may then choose s so large that stances if U is the spheroid (0;

0

fis

+

00.

It

n

with u. Hence one may 10. One then I [u]2 I < 1/2. Under the circum0

I

I

H then c U. Therefore the path y is orbitally stable. Moreover diam SSU -÷ 0 as s —÷ + oo and hence the orbital stability is of the

asymptotic.

< 1. One may assume Suppose that only h 1 for j k then the same family is negatively orbitally asymptotically stable and therefore positively unstable. —

To sum up then we have: (7 .2) The stability theorem (3. 5) is a closed /.ath y of an autonomous system provided that the characteristic exponents n — 1 other than unity.

in the statement are the

§ 5. Systems of Periodic Solutions now turn our attention to another problem. Consider an analytical system S.

We

VIII.


165

dx/dl = X (x; v; t)

(8. 1)

of the same type as discussed in (II, 11. 1) except that in addition X has the period cv in 1. Let us suppose that for v = 0 there is a known solution (1) of period cv. One asks whether this periodic

solution is member of an analytic family of solutions with the same period cv. The method of attack goes back to Poincaré ([4],

vol. I, Ch. IV). We first apply his theorem (II, 11.1). The solution which starts at time t = 0 at (0) + is analytic in cv, andvforO 0.

Hence first A0 (t) =

e1t.

The periodicity condition yields then x (16.4)

A1

From (16.3) we obtain

+ A2

= x (0), or

+ ... = 0.

180


A1 (t)

e4 (t

=

g

eit', eit', eit', 0)

cIt1

and hence A1

g

=

eft, eft,

where F is a polynomial in

F

(16.5)

0)

dt = F

Thus (16.4) becomes

+ /4F1

+ •..

=0

where the Fh are readily shown to be polynomials. If they are

all identically zero then the general solution for small is periodic. Let this not be the case and let F5 be the first 0. Writing now for simplicity F, F1, . . ., for F5, F5 + we will still have (16.5). With this convention if Cu) is a solution and (0) = then must be a root of the equation

Fh

(16.6)

Let be a root of order p. We have then from the Weierstrass theorem

F

+ F1

+

+

•.

where the fh (1u) are non-units. Hence the solution of (16.5) for reduces to that of (eu) such that (0) = (16. 7)

The required solutions may be obtained in a systematic manner by the Puiseux process (see for instance Picard [1] vol. II). In the present case there will be s so-called circular systems each consisting of q conjugate sets (16.8) The values

—

,2r/q E (1u

(eu) defined by (16.8) correspond to a single periodic ('u), t) such that x (eu), 0) = and q = p. Thus

family x we have obtained a complete solution of our problem.

CHAPTER IX

Two Dimensional Systems. Simple Critical Points. The Index. Behavior at Infinity

Hereafter we confine our attention to two dimensional systems

and equations of the second order. Abandoning the vector notation we shall consider more particularly in the present chapter autonomous systems

dx/dt = X (x, y), dy/dt = Y (x, y). The principal goal is to obtain the jhase-fortrai€, or the global qualitative description of the totality of the paths of the systems in the x, y plane or /thase-plane. All that one can do in general is to provide a good deal of information about the paths and

it is only in some cases that one may really obtain the full phase-portrait. The index, the closed paths, the behavior at infinity all contribute important, if not final, material to the general objective. Our more detailed program is as follows: First there is given

the "local" phase-portrait near a critical point when the first degree terms in the expansions of X, Y are present in its vicinity.

Some details regarding the index of a closed curve and of a point are then given, the topological fine points being relegated to Appendix II.

To determine the behavior at infinity when X and Y are polynomials one must extend the concept of index to the sphere and the projective plane, a task carried out at length in Appendix IL Regarding infinity the given system is imbedded as it were into a broader system valid for the projective plane. As a conse-

quence of this the infinite region may be treated like the rest 181

182

DIFFERENTIAL EQUATiONS: GEOMETRIC THEORY

of the plane. This is explicitly brought out by an example at the end of the chapter. References: Bendixson [1]; Coddington, Levinson [1]; Hurewicz [1]; Poincaré [2]. §

1.

Generalities

1. Consider a real analytical system (1. 1)

dx/di

= X (x, y),

dy/dt

y)

Y (x,

valid in a region .Q and possessing only a finite number of critical points there. If 0 is such a point one may choose it as origin and in its neighborhood

(1.2) X = ax + by + Xe (x, y), dy/di = cx + dy + Y2 (x, y) where X2, Y2 = [x, y]2. The characteristic equation of (1.2) is (1.3)

(a—A)(d--—,%)—-bc=O.

We shall consider more particularly the case when the characteristic roots 0, i.e. when ad — be 0. The corresponding critical points are known as elementary. As the topological character of the critical point is the same, except in one case, as for the first approximation, which is linear homogeneous, we shall first discuss such systems. (1.4) Path-rectangle. Referring to (II, 16.4) if y is a true If I

1fl1

I

I

I

cr Fig.

1

IX.

TWO DIMENSIONAL SYSTEMS. i

183

path (not a critical point) and A is a point of y, a certain neighborhood of A may be mapped topologically on a rectangle so as to produce the configuration of fig. 1: (for convenience the paths and their arcs are identified with their topological images) where BC, B'C' are arcs of paths and EH, FG are subarcs of

preassigned arcs transverse to y at B, C. Moreover every path passing sufficiently near A contains an arc such as B'C'. The dotted rectangle EFGH is a path rectangle and the arc BC of y is its axis. (1.5) Closed baths. According to (II, 15. 1) the path v is closed whenever it is a Jordan curve containing no critical point or equivalently whenever its solution (x (t), y (t)) is periodic. §

2.

Critical Points of Linear Homogeneous Systems

2. Consider the real system (2.1)

dxfdt=ax±by, dy/dt=cx+dy, ad—bc

The characteristic roots

2.2 are still the solutions of (1.3) and the reduced normal form of the coefficient matnx Iab\ will

\cd/ completely determine the behavior of the paths, and more particularly their behavior in the neighborhood of the critical point at the origin. If the characteristic roots are real then a real transformation of coordinates will reduce the system to one of the same form but with coefficient matrix of one of the two types A:

O\

\o

)

(

O\ ;

B:

(

\1

while if the roots are complex a complex transformation will

reduce the matrix to the form

C:(

\o

O\

A/

with a certain subcase. All told there are five cases which we shall now discuss separately.

184


(2.2) First case. Real roots of same sign, matrix of type A. The reduced form is (2.3)

du/dt=21u, dv/dt=22v.

Suppose first 21, 22 both negative: 2i = — general solution is

u=a

v=

4uj > 0. Then the

j9

where a, are arbitrary constants. Evidently the path y tends to the origin as t —- + oo. It reduces to the u axis when jI = 0, to the v axis when a = 0. Supposing a fi 0 and the ratio v/u —÷ 0 as t oo. Hence y is tangent to the u axis at + the origin. If a 0 = 0] y is the v axis [the u axis]. The form of the characteristics is that shown in fig. 2. The arrows indicate

the direction of motion on the paths. The critical point thus arising is called a stable node. V

Fig.2

are real and positive, the preceding behavior corresponds to i -÷ — oo. Hence assuming again

the role of the two axes is reversed but the general aspect of the paths is the same. When = A2, hence all the paths are straight lines through the origin. When

(2.4) Second case, matrix of type B. There is only one characteristic root A and it is of course real. The reduced form is

du/dt=Au, dv/dt=u+Av.

(2.5)

Suppose first —2 =

4u

> 0. The general solution is

= ae"t, v =(at+ fl)e'it. For a = 0 it represents the positive v axis if fi > 0, the negative v axis if < 0. Whatever a, fi both u, v 0 as t + 00.

tends to the origin as t -÷ + 00. Assume now Since V/U-+ + 00 whent-÷ + 00, y is tangent to the v axis the origin. It also crosses the u axis at t = —a/fl. The co-

Hence

at

the path

ordinate v has an extremum when dv/dt = 0 or t = (a—fl4u)/cçu. Hence the paths behave as indicated in fig. 4. When A > 0 the situation is that of fig. 5, the arrows being merely reversed. I,

U

Fig. 4

Fig. 5

The critical point is still called a stable or unstable node. Thus the earmark of the node is a set of paths tending to the

+

the stable node or 1 — oo for the unstable node. (2.6) Third case. Roots real and of opposite sign. The matrix is then necessarily of type A. The reduced form is still (2. 3). Assuming A and > 0, the paths are origin when I —k

(2.7)

00 for

y: u =

v = fi

186


The semi-axes u, v are still the paths corresponding to

=

0,

a = 0. If ajfl 0 then u —*0, v + 00, as I —÷ + 00. Hence the paths have the general form of fig. 6. The origin is then called a saddle point.

Fig. 6

When the signs of "2 are reversed the roie of the axes is reversed, or equivalently all the arrows in fig. 6 are reversed. However the essential aspect of the critical point remains the same.

(2.8) Fourth case: Complex roots with non-zero real tart. The reduction is to the form

du/dt =

(2.9)

2u,

du/dt =

The transformation of coordinates

u+u

u—_u

2

2i

is real and the real points of the initial system correspond to conjugate of u. This is then assumed throughout. Let us suppose first A — 1u + iw, where w and are positive. The general solution of (2.9) is

u=y Setting u =

=

a

a and fi real.

we have then

r__aept, O=fl+cot which represents a logarithmic spiral. Thus the aspect of the paths is that of fig. 7. The critical point is then called a stable

IX.

187

TWO DIMENSIONAL SYSTEMS. I

focus. When A = 1u + ico, ,s > 0, the situation is the same with arrows reversed (fig. 8) and we have the unstable focus. U

u plane

plane

II

Fig.

Fig. 8

7

(2. 10) Fifth case: Pure complex characteristic roots. The situation is the same save that 0. Hence the paths are given by

r = a, 0 = + cot. In other words in the (u, v) plane they are circles with the critical point as center and all described with the same angular velocity co. The critica' point is then known as a center (fig. 9).

It should be kept in mind that the terms logarithmic spirals, circles, applied to the paths in the (u, v) plane are only partly appropriate. For even if the original x, y coordinates had been chosen rectangular, the transformations utilized were not orthogonal but merely linear homogeneous. Thus the circles in the (u, v) plane of fig. 9 would merely correspond in the original (x, y) plane to a family of concentric similar ellipses.

cane Fig. 9


188 §

3.

Elementary Critical Points in the General Case

& Before taking up our main theme let us say a few words regarding a transformation S regular at the origin:

x=f(u,v), y=g(u,v), f(O, 0) = g(0, 0) =

0,

a (u,

We note the following properties:

(3. 1) There exist neighborhoods M, N of the origins in the jlanes 17, 17* of x, y and u, v, which are 2-cells mapped to logically into one another by S.

(3.2) S is regular on N and S tranforms (1.1) into a new system and in such a manner

that paths and (isolated) critical points go into the same for the new system. Moreover S does not change the matrix in normal form to which

(a

may be reduced.

In substance then we may freely use transformations such as S without affecting the behavior of the paths about the origin. We shall assume now that we have the general system (1.1) with ad — bc 0. A linear transformation of variables will reduce it to a system whose first approximation is of one of the forms considered in (2). The same terminology: node, focus, etc., is used as before. The basic result is: (3.4) THEOREM. The behavior of the paths in the neighborhood of a critical point is the same as for the first except that when the characteristic roots are pure complex there may arise a center or a focus. The nature of the coefficient matrix of (2. 1) and of the roots gives rise to the same classification into five cases as before and we must again examine each case separately.

4. (4. 1) First case. Let the reduced system be written (4.2)

dx/dt—A1x+p2(x,y), dy/dt=A2y+q2(x,y)

where A1, A2 are real and of the same sign and generally PA(X, y), are series [x, y]4. The first approximation is (2. According

IX.


189

to the complements (V, 14.1, 14.2) to Liapunov's expansion theorem we notice this: the first approximation has the special solutions

u=

(4.3a)

Hence, if

6Ait, v

0; (4.3b) u =

v=

0,

eAat.

22 < 0, there correspond for t sufficiently large,

and respectively to (4.3a, 4.3b), the solutions

x= x=

(4.4a) (4.4b)

u + {u]2, y [v]2,

=

u=

[u]2,

y = v + [v]2, v =

Now (4. 4a) represents an analytical arc through the origin which is an integral of (4.2) and is tangent to the x axis at the origin since dy/dx = 0 for u = 0. Hence the arc has an equation

(x, y) = 0, = [x, Similarly (4. 4b) yields an arc solution

y+

x+

(x, y)

= 0,

=

[x,

Yb•

Y]B•

Now the regular transformation reduces (4

(4.2) to the form f

= X + FI(X, Y), F1 =

[X, Y]2, 4Y/dt = 22Y + G1(X, Y), G1 = [X, Y]2. dX/dt

Since (4.5) has for integrals X = 0 and Y = 0, necessarily F1(0, Y)=0,G1(X,0) = 0. Hence F1 = A1XF, G1 =A2YG, where F, G are [X, Y]1. Thus (4.5) has the actual form (4.6)

X E1(X, 0,

Y E2(X, Y),

X decreases in the first and fourth

and increases in the second and third. Since 22 < 0, Y decreases

in the first two quadrants and increases in the last two. The semiaxes are paths which tend to the origin. Hence all the paths tend to the origin and we have a stable node. 22> 0 the result is the same, save that in (4.4) one If


190

assumes —1 large. One arrives again at (4.6), the paths all move away from the origin and one has an unstable node. (4.7) Second case. The basic system is now (4.8) dx/dt = Ax + p2 (x, y), dy/di = x + 2y + q2 (x, y) and the first approximation is (2. 5). Suppose first —A = u = aept, v = (at + b) be the general solution of (2.5). Then according to Theorem (V, 9. 1) the general solution of (4.4) is given in the neighborhood of the origin by expressions X

=

+00

Pm (a, b; t) e

y

=

+00

E Qm (a, b; I)

rn—i

where Pm, Qm are polynomials in t whose coefficients are forms of degree m in a, b. The first terms of the x, y series are respectively equal to u, v. Hence we may write x

=

E Pm e-mpt, y = + m>i

This shows that as t —÷ lim y/x

+

(at + b) e—pt + £ Qm e-mpt. rn>i oo (x, y) tends to the origin and

urn t + b/a

=+

00.

From this follows readily that the behavior of the paths is the

same as for the first approximation and is represented near the origin by fig. 4 when A < 0. If A> 0 we replace again by t' —t, and as in the preceding case, show that the behavior of the paths the same as in fig. 4 with arrows reversed, i.e., the same as in fig. 5. Thus we have the same stable or unstable node as for the first approximation. Third case. The basic system is still (4.2) with (4. 0. The treatment is the same as for the first 2 and = —A, 22 = in (4.6). Thus we have case save that (X, Y), (4.10) dX/dt = —2 XE1 (X, Y), dY/dt = E, (0, 0) = 1. Thus for (4.10) the axes are paths, and X decreases andY increases

in the first quadrant. Now near any segment of the axes near the origin (origin excepted) the system of paths forms a path rectangle. Hence in the first quadrant, near the origin, the general behavior can only be in accordance with fig. 6. Similarly for the

IX.

191


other three quadrants. Hence the complete behavior in the X, Y plane, and hence also in the x, y plane is that of fig. 6, i.e., we have a saddle point as in the first approximation.

5. Fourth and fifth cases. These last two cases may be conveniently taken up together. The fundamental system is now dx/dt =

(5.1)

Ax

dx/dt =

+

+

and the first approximation is (2.9). The most convenient method for dealing with the paths is to pass to polar coordinates r, 0. Write explicitly A = + iw. Then (5.1) yields 6° dr/dt + iret8 dO/dt =

Cu

+ iw) ret° +

(re'°,

ret0)

Upon dividing by 6° and equating real and complex parts we obtain relations (5.2)

dr/ill

r

+ a1 (0) r + a2 (0)

+

(5.3)

where

(0),

(0) are polynomials in cos 0, sin 0.

The series converge for r sufficiently small and any 0. Since w 0, we obtain by division a relation dr/dO = rcu/co

(5.4)

±

(0)

r + ...)

where the coefficients and convergence are as before. Since the

system is analytic the solution, taking the value may be represented in the form (see II, 10. 1) r (0,

=

c1

(0,

.4- c2 (0,

e2

for 0 =

00,

+

the series being valid for an arbitrary 0 range and small. The term independent of is missing since e = 0 must yield the 0, solution r = 0. Since the series may be assumed valid for the solutions near the origin may be assumed to have their = 0. Hence we may write our solutions initial value for and choose for them a representation r (0,


192

(5.5)

= c1 (0) e + c2

r (9,

(0)

+

are determined by substituting in (5.4) and identifying \Ve thus obtain a system

The

equal powers of

dc1/dO = (5. 6)

Since

dc2/dO = r

(0,

c1 /2/co

= e identically

(5.7)

c1

(0)

=

1,

c2 +

(0) C12.

we must have (0)

=

for n

0

>1

The differentkal equations (5.6) together with the initial conditions (5.7) one to determine the (0) one at a time. In particular (5.8)

c1 (0) =

ePOIW.

or which is the In order that r (0, be periodic of period same in order that its path be an oval surrounding the origin we must have r = or (5.9)

e

=

=

There are now two possibilities:

(5. 10) All the paThs for s4iciently small are ovals surrounding the origin, i.e. the origin is a center.Then (5.9) is satisfied identically. Since (e) — e is holomorphic at e = 0, all its coefficients

vanish. Hence

c1

=

1,

The first relation yields yield the condition that every

=

0

for n > 1.

1, hence 1u = 0 and the rest (0) has the period Thus the

center can only arise when the characteristic roots are pure complex and the

(0) are all periodic. Conversely when these two conditions hold q' — = 0, the paths near the origin are ovals surrounding it and the origin is a center. Notice that when the origin is a center the time period T for the description of the path r (0, calculated by means of (5.3) is

IX.


193

r=1' J0 co+fi1(O)r(O,e)+ Thus it is not necessarily and will generally depend upon constant, contrary to what happens in the linear case. Expressed

also in another way the point r (t), 0 (t) of a given path y describes

it with an instantaneous or even an average angular velocity which are not generally independent of y. (0) do not all have the period The closed paths, if any exist in the range considered, will correspond to the solutions in of (5.9). Under our hypothesis the coefficients (5. 11) The coefficients

0 is an isolated root of (5.9). This means that a a > 0 may be selected such that (5.9) has no roots in the interval 0 <e > ... Thus 0. Since

Then necessarily 0

.

has a limits

the

is continuous at = 0 or (5.9) has a root in the interval the situation is analogous save sequence {(q — we have

—+ 0.

Since

p

—

that we merely conclude that the spirals diverge from the origin. The latter is then an unstable focus. Suppose in particular 0. If < 0 then for r small dr/dO will steadily decrease and we have the stable focus, while for > 0 it will be the reverse with an unstable focus. Thus in the fourth case—real parts of the characteristic roots non-zero—the situation is the same as for the first approximation. This completes the proof of (3.1). (5. 12) There still remains the following important question to be settled: If 4a = 0, to recognize from the given differential system whether the critical point is a stable or unstable focus or a center. The system (5.6) assumes now the form

DIFFERENTIAL EQUATIONS: GEOMETRiC THEORY

194

0,

dc1/dO

=

(c1,

.

. .,

—

with the same initial conditions as before. Thus c1 (0) = 1. the expresSuppose that up to and including _i, but not e'°) have no constant which are polynomials sions terms. If n = co, i.e. if no *ph have constant terms then we have the critical a center. If n is finite and is the constant term of

point is a focus. In the series (5.5) the first non-periodic term will have the will be (0). Thus for small r (2n, e) — r (0, Hence the focus is stable if ô, 0.

6. Final remark. The preceding results have confirmed the property already known that our critical point is stable if the real parts of the characteristic roots are negative, unstable if they are positive, conditionally stable if they are of opposite signs. Now in the applications it is often important to have at hand rapid criteria to detect which one of the three cases arises. This is very easily done in terms of the initial system (1. 1). Let us suppose then that the point A (x0, Yo) is an elementary critical point. The expansions of X, Y in its neighborhood are X

(x -— x0)

Y=

Yx0 (x — x0)

...,

are

where

+ +

(y — Yo) +

(y —

Yo)

+

the partial derivatives. The characteristic

equation is ¶XXO

— r, I

=—

+

Hence for the characteristic roots

r+

Yy0 —

r2:

J= 13(X-! Y)

=.r1r2.

Yx0 =0.

IX.

TWG DIMENSiONAL SYSTEMS. I

195

Hence the following comprehensive property: < 0 the critical point is (6.1) If J(x0, yo) > 0 and X,0 + stable. If J (x0, yo) > 0 and + Yy0 > 0 it is unstable. If J (x0, Yo) < 0 the point is conditionally stable (it is a saddle point).

§ 4. The Index. Application to Differential Equations 7. Let us recall briefly the definitions and simple properties of the index. For full details the reader is referred to Appendix II. Let be a field defined over a Jordan curve J in the Euclidean plane II. It is supposed that has no critical points (no vanishing times the angular variation points) on J. Then Index (J, is I of the vector V (M) applied at M J as M describes J once.

Let the field be defined over a region .Q and let A Q be at worst an isolated critical point. Let also y be a small Jordan curve surrounding A but no other critical point and oriented is independent concordantly with the plane 17. Then Index (y, of y and is by definition Index (A, Noteworthy properties of the index are: defined over a Jordan curve J have no (7. 1) If two fields critical points on J, and their vectors are never in on J then Index (J, Index (J, a'). (7 .2) The index of a non-critical point is zero. (7.3) If is defined in a region Q and the Jordan curve J c Q surrounds a finite set of critical points A1, . .., A8 then Index (J,

—

E Index (A1,

(7.4) The index according to Poincaré. In his mémoire [2], p. 29, Poincaré gave a definition of the index of a Jordan curve J which

may be formulated as follows. It is supposed that the field crosses a given fixed direction D in at most a finite number of points. By crossing one means definitely that the direction of the

vector passes through the direction D and one excludes the places where the vector, say rotating forward just reaches D and then retrocedes. Let p be the number of crossings as the vector V revolves positively and n the number as it revolves negatively. Then Index (J,

This definition is highly

convenient in the applications, where frequently p

and

n are

196


easily calculated. That it is equivalent to the one by angular variation is seen as follows: the crossings of D by the vector V count off the multiples of 2T in the angular variation from D. Hence the total angular variation is — n) n and so the index is (P — n) 8.

=

The application to differential equations is immediate. To

the system (1. 1) defined over a closed region Q there corresponds

of the vectors (X, Y) defined over Q. Let J be an oriented rectifiable Jordan curve, so that one may integrate along J. Then at once if J contains no critical point: the field

(8.1)

Index J =

1

d

arc tan

Y

1

XdY—YdX X2 +

This expression makes it evident that the index varies continuously with J when J is continuously deformed without crossing critical points. Since the integral is an integer its value under the circumstances is constant. (8.2) Let A be an isolated critical point and J a positively oriented small circle surrounding A so that Index J Index A. Apply an affine transformation of coordinates, and take (8.1) in the new coordinates. Using the definition of the index of Appendix 11(4.5), and referring to Appendix 11(4.1) it is readily seen that the value of the index in the new coordinates is the same as in the old. Thus, one may conveniently use (8.1) to calculate Index A in any afilne coordinate system. 9. Index of the elementary critical points. We may suppose as

usual that the critical point is at the origin and that the index is given by (8. 1) where J is a small circle of radius r. We first have: (9. 1) The index of the origin relative to the system (1 . 2) is the same as for its first approximation. The components of the vector V for (1.2) and of the vector for its first approximation are respectively

IX.


197

X =ax+by+X2, Y ==cx+dy+Y2, y*cx+dy, X*=ax+by, 0.

ad — bc

By (7. 1) we merely have to show that on J, V and V* are never in opposition or that we cannot have

X+kX*=0,

k>0,

or explicitly that one must rule out relations (9.2)

(1 + k) (ax + by) + X2 =

0,

(1 + k) (cx + dy) ± Y2

0.

As a consequence of (9.2) we find

(1 +k)2[(ax+ by)2+(cx+dy)2] ==Z4(x,y). Introducing polar coordinates r, 8 we obtain (9.3) (1 + k)2[(acosO + bsinO)2 + (ccosO + dsinO)2] r2

(0) +

(0)

+

a form of degree n in sin 8, cos 0 and the series at the right converges for r small and any 0, uniformly in 0.

where

As a consequence the right hand side of (9.3)

0

with r wJ atever

0. On the other hand since ad — bc 0, we can only have ax + by = 0, cx + dy = 0 if x = y = 0. Hence the square Thus bracket at the left 0 whatever 0, i.e., on 0 0 it is continuous and positive on a closed interval and hence it has a positive lower bound Since k > 0, the left hand side whatever 0. Hence for r sufficiently small the two sides of (9.3) are different. This proves (9. 1).

10. We shall now deal directly with the first approximation.

Let the reduced forms be those of (2) save that we still use coordinates x, y instead of u, v. Here again each case requires special examination. (10.1) Case I. (Node.) Here are real and of the same sign. Assuming we have on C:

DIFFERENTIAL EQUATIONS: GEOMETRIC THEORY x2

+ y2 =

X

21x,

Y

= lay, X2 +

= 212x2 + 222Y2•

It follows that if we vary

22 continuously, without vanishing, 1 when till they both become + 1 when they are positive, they are negative, the integral (10. 1) will vary continuously also. Since it is an integer it is then constant and so the modifications

do not change Index A. At the end the vector V will be along the radius AM and always pointing inward or always outward. In both cases the index is unity. of

(10.2) Case Ii. (Node.) Here = 22 1. Instead of the circle C we shall take the curve J indicated in fig. 10: J =

BCDEB'C'D'E'B.

D

Fig. 10

Here CDE, C'D'E' are portions of paths and the other parts of J are segments. The figure is symmetric with respect to the origin and it is drawn for 1 > 0 (unstable node). It is clear that

J is deformable continuously into a circumference without crossing A, and so it is admissible as curve J in (8. 1). The angular variation of V(M) along B'EDCB is manifestly

since

the vectors turn continuously forward from —,t/2 to + Similarly on the second part of J from B to B'. Hence the total angular variation of V(M) is and the index is again unity. If I < 0 all the vectors are reveised and the result is the same. 22 are real but (10.3) Case 111. (Saddle point.) This time of opposite signs. We may assume 0 and choose an integration curve J which is the arc BCDEF of fig. 11 repeated

by symmetry around the axes. Along that arc the angular hence its total angular variation is variation of V is Hence the index of the saddle point is —1.

__________ IX.


F

A

199

E

Fig. 11

(10.4) Cases IV and V (focus and center). Let the reduced form be this time

dx/dt=(iA+iw)x, The denominator in the integral (8. 1) is the squared length of the vector V(M). Its value here for M on the circle of radius one is 1u2 + a)2. Hence the same argument as before shows that we may vary 4u continuously to zero, and w continuously to unity without changing Index A. For w = I the vector V(M) is represented in the complex plane by ix, i.e., it is tangent to the circle. As M describes the latter V(M) rotates forward and so the angular variation is 2iv. Thus the index of a focus or a center is unity. To sum up we have proved: (10.5) THEOREM. The index of an

critical point other

than a saddle point is unity; the index of a saddle point is (—I). 11. We come now to an important proposition due to Poincaré ([2], p. 57): (11.1) THEOREM. The index of a positively oriented closed path

is unity. The following ingenious proof is due to Heinz Hopf. Let y be the closed path in the plane fl. Since our system is analytic y is rectifiable. Take a horizontal tangent to as low as possible and let Q be its point of contact. Let arc length s

200


Fig. 12

counted from Q in the positive direction and let the scale be such that is of length unity. on

be

Let now a new plane 17* be referred to coordinates s1, To a pair of points of y marked by arc lengths s1, s3: 0 s1

we assign the point (s1, s2) of 17* and determine at this point a vector V represented by the segment from to s2. This defines

in the closed triangle ABC a field

without critical points. Therefore the angular variation of V around ACB is zero. Now on AC the vector at (s', s') is simply the tangent vector to y at the associated point s' of y. Hence the angular variation along Index y. Along CB and BA it is merely —sr. Hence AC is Index y — = 0 and this proves the theorem. (11.2) COROLLARY. A closed j5ath must surround at least one criiical poini. (11.3) COROLLARY. If a closed path surrounds only elementary critical points they cannot all be saddle joinis.

The same argument yields: (11 .4) Let a positively oriented planar Jordan curve J be rectifiable and possess a continuously turning tangent. If the field

IX.

201


is tangent to J along J and has no critical point on the curve then

=+

1.

(1,1)

5,5)

SI

Fig. 13

§ 5. Behavior of the Paths at Infinity 12. Consider now a system (12.1)

dx/dt = P (x, y), dyfdt

= Q (x, y)

where this time P and Q are relatively prime polynomials. This

last condition is to avoid systems with an infinite number of critical points. We propose to discuss the behavior of the paths far out in the plane. Following Poincaré [2] we shall deal with the problem by completing the plane IT to a projective plane by addition of a certain line, the classical "line at infinity." This process has already been discussed in (I, 3.8). Let be referred

to homogeneous coordinates X, Y, Z. If the point M of is not in the line Z = 0 we assign to M the cartesian coordinates X/Z, y = Y/Z, and identify M with the point (x, y) of fl. x where L is the line Z= 0. Thus 17= intersects L in a unique Every line ).: bX — aY = 0 of

202


2 2* is one-one. point Since A determines uniquely the line bx — ay 0 of 17, there is a one-one correspondence between the directions of 17 and

the points of L. It follows in particular that in the model of the projective plane as a closed circular region Q with the diametral points of the boundary 1' matched, the set of "matched" points corresponds to the line L.

It is advisable to transfer the situation to an Euclidean S. Let it be divided into two open hemispheres H', H" by an equator E. One may identify Q say with H' = H' u E and transfer everything to the closed hemisphere H'. The matched

points become the diametral points of E. However if M 17 corresponds to M' H', one may equally well associate with M the diametral point M" of M'. Thus 17 corresponds to the pairs of diametral points of H', H" and to the pairs of diametrai points of the whole sphere. The sphere S is the so-called doubly covering sphere of the projective plane Let us return now to our special differential system which we write 13.

(13.1)

Let /,

Qdx — Pdy q

= 0.

be the degrees of P, Q and set p* (X, Y, Z) Q* (X,

=

Y, Z) =

Let also n be the largest of relation dX, X, Zn—PP*,

ZP

P (X/Z, Y/Z),

Zq Q

(X/Z, Y/Z).

and q and consider the differential

dY, dZ 0

At points not on Z = 0, i.e. in the Euclidean plane 17, we may take Z = 1, dZ = 0, X = x, Y = y, and then (13.2) reduces to (13. 1). Therefore (13.2) may he viewed as the extension of (13. 1)

to the projective plane 13.

IX.

TWO DIMENSIONAL SYSTEMS. i

203

Upon expanding the determinant, (13.2) becomes: (13.3) and Q*. Hence the Since P and Q are relatively prime so are coefficients of dX, dY have at most a power of Z in common. q no power of Z divides the coefficient of dZ and Now if (13.3k is to be left as it stands. On the other hand if = q = n (x, y), qn (x, y), the terms of highest degree in P, Q are and if then Z factors out of every term in (13.3) such that xqn

but no higher power does. Under the circumstances one must cancel out the term Z in (13.3). In one or the other case (13.3) assumes the form

(13.4) A (X, Y, Z) dX + B (X, Y, Z) dY + C (X, Y, Z) dZ =0 where A, B, C are forms (homogeneous polynomials) of the same degree.

14. The equivalent cartesian form of (13.4) remains of course (13. 1). This means that (13. 1) determines a line through each non-critical point M, namely the tangent to the path through M. This is in contrast with (12.1) which defines a definite ray through M. The opposite ray corresponds to a change of I into I.

Returning now to the situation in (12) the line through M determines arcs through M', M". Let us choose a definite time I,

and thus have definite rays. Let the directed arc through M'

be such that it projects into the ray through M. Then the associated arc through M" will project into the opposite ray through M. The arcs thus defined determine a vector field

on

S — E, and by continuous extension on the whole of S. The are associated in pairs with equal index. critical points of Such a pair K', K" gives rise to what we shall describe as a

critical point K of the system (13. 4), and its index is by definition

the common index of K' and K ". The sum of the indices I (K) is half the same as for the sphere, i.e. unity. The critical points are the places where the tangents defined by (13. 1) cease to exist, i.e. where A = B = C = 0.

204


To calculate an index I (K) one proceeds as follows: Select say projective coordinates X', Y', Z' such that K is the point (0, 0, 1) then pass to cartesian coordinates x' X'/Z', y' = Y'fZ'. As a result (13.4) will be replaced by a system Q' (x', y') dx' — P' (x', y') dy' = 0.

(14.1)

One passes now from (14. 1) to the associated system in the form (12.1)

dx'/dt=P', dy'/dt_—Q'.

(14.2)

This amounts to replacing the lines (14.1) through each point by a system of rays. This system has then the same index as one of the points K', K", i.e. its index is that of K. It is readily seen that for the critical points in the Euclidean plane 17 the procedure just described yields the usual index. For the points at infinity however it yields an index where none was at hand before. In other words the notion of index is thus brought to bear upon the system (13.4) throughout the projective plane 15.

As an application let us take the following system dis-

cussed by Poincaré ([2], p. 66):

1dx/dt=—x2+y2—1, 5(xy— 1).

(15.1)

The right hand sides equated to zero have no common solution

and so there are no critical points in the Eucidean plane. In homogeneous form (15.1) becomes dX,

dY, dZ

Y, Z=0,

X2 +

— Z2, 5 (XY — Z2),

0

or after expansion (15.2) —5 Z (XY — Z2) dX + Z (X2 +

— Z2)

dY +

IX.


+ (4 X2 Y —

205

—5 XZ2 + YZ2) dZ = 0.

The only critical points are given by Z = 0, Y(4X2_ Y2)

0.

They are thus the three points

A(1,0,0), B(1,2,0), C(1,—2,0). Since the system has no finite critical points in the Euclidean plane it possesses no closed paths in that plane. Let us now consider the nature of the critical points. Critical /.oint A. The appropriate cartesian coordinates are Y/X, z = Z/X. Making X = 1, dX = 0, Y = y, Z = z in y (15.2) we find z (1

y2

z2) dy + (4y — y3 — 5z2 + yz2) dz = 0.

The critical point is the same as for the pair of equations

dy/dt = — 4,' + 5z2 — yz2 + y3, dz/dt = z + z (y2 — z2)

As the characteristic roots are —

4,

1 the point A is a saddle

point, and its index is —- 1. Critical /oint B. The transformation

Z-*Z

X-+X,

sends B to the point (1, 0, 0). The same process applied to the transformed equation shows that B has the same behavior as for

dy/dt =

4y,

dz/dt = 5z.

As the characteristic roots are 4, 5 the point B is a node, its index being + 1, Critical point C. The transformation is now X —÷ X, Y -÷ Y — 2X, Z Z, but otherwise everything is as for B and C is also

a node, its index being again + 1.

206


Fig. 14

The sum of the indices is + 1 as it should be. As a complementary and useful observation let us point out that (15.2) is satisfied by Z = 0. Hence the line at infinity consists of arcs of paths. The preceding information will now be utilized to construct

the phase-portrait. To that end let the projective plane

'.—

4

Fig. 15

a

be

IX.

TWO DIMENSIONAL SYSTEMS. 1

207

identified with a closed circular region Q with diametral points of the boundary I' matched. Let us mark on P diametral points A, A'; B, B'; C, C' for each critical point. Thus A, A' are saddle points and the rest are nodes. Since AB, AC and A'B', A'C' are parts of paths, they are on paths tending to or away from the

two saddle points. Referring to (15.1) dx/dt> 0 outside the Hence the approach to A and removal from A' are as shown in fig. 14. Since the other critical points are

circle x2 + y2 =

1.

Fig. 16

nodes the paths tend to or away from them as shown in the figure.

Let P denote the circle x2 + y2 = 1 and J the hyperbola xy = 1. From (15. 1) we learn that along a path x decreases inside 1' and increases outside F, while y decreases outside J and increases in its two interior regions. This situation is described by horizontal and vertical pointers in fig. 15. Observe also that P is the locus of the points where the tangents to the paths are vertical and A the locus of the points where they are horizontal. The only crossings of F, A by paths consistent with these various

properties are as indicated by the arcs a, fl,

y, c5 in fig. 15.

208


If we recollect that within 1' the slopes dy/dx of the paths are always 0, we find that the paths crossing r have the general form of A in fig. 15. Returning now to the circular region of fig. 14 we find that A assumes the general form indicated in fig. 16. From this follows that the two paths issued from the two saddle points other than the equator are of the types A1, A2 of fig. 16. The other basic types are then represented by A3, A4.

CHAPTER X

Two Dimensional Systems (Continued)

In the present chapter one will find a description of the general critical points of analytical systems, and of the limiting sets of their paths as t -÷ + 00. The groundwork for this study was laid by Poincaré but his results were considerably extended, refined and given more precision by Bendixson.

A full treatment is also given of the behavior of the paths near a critical point with a single non-zero characteristic root, a topic already dealt with by Bendixson. Finally the chapter concludes with a consideration of structural stability, an important concept due to Andronov and Pontrjagin, but fully treated for the first time by De Baggis. References: De Baggis [1]; Bendixson [1]; Coddington, Levinson [1]; Lefschetz [4]; Niemitzki-Stepanow [1]; Poincaré [2].

§ 1. General Critical Points 1. Consider again an autonomous analytical system (1.1) dx/dt X (x, y), dy/dt = Y (x, y). and suppose that it has an isolated critical point 0 which one

may take as the origin. We will assume that X, Y are holomorphic in a closed circular region Q of center 0 and that 0 is the only critical point in Q. Thus X, Y vanish simultaneously in Q solely at the point 0. We will first prove several preliminary properties. (1.2) Let f (x, y) be a real function holomorjhic in Q with f (0, 0) = 0. If the radius Cl of Q is small then the curve / (x, y) = 0 has in Q at most a finite number of real branches and they are of one of the two types 209

210


(1.2a)

x=0,

(1 2b)

y

Xk/fl +

+

j

+

+ 1)/ti

where k, n are positive integers and the a; are all real. Moreover for a small enough the real branches intersect one another in £2 only at the origin.

By the Weierstrass preparation theorem

f

xm (yP + a1 (x) yP —1

+ ... +

(x) ) E (x, y),

where the parenthesis is a special polynomial in y and E is a unit (see I, 14. 1). Both are real when f is real (1, 14.2). The solutions of f = 0 in £2, for a small enough, are thus x = 0 if rn> 0 and those of (1.3)

yP + a1 (x) yP

1

+

+

(x) = 0.

Now the theory of the Puiseux series (see notably Picard [1], vol. II, p. 350) is fully applicable to (1.3) and so the solutions in question are of the type (1. 2b) with the a5 real or complex. The real solutions correspond evidently to systems (1. 2b) with the a; all real. Suppose now that there are two real branches. If one is x = 0 and the other is (1. 2b) they clearly intersect only at the origin in £2. Suppose that there are two branches, one of them (1. 2b) in place of k, n, ak. The and the other similar with k', n', intersections of the two correspond to the solutions in £2 of a relation + fl1

where the

+ ...)

=0

are not all zero. By a well known property of analytic

functions the solutions

0 are bounded away from zero and

hence correspond to points outside £2 for a small. This completes the proof of (1 . 2).

Let A and B be the end-points of an arc A. We shall conveniently use the designations (AB), [AB), (AB], [A.B] for the open arc A, the are A closed at A and open at B, the arc A open at A and closed at B, the closed arc A. We shall also say with Poincaré that a differentiable arc A of one of the above four types, and situated in the field of operation

X.

TWO DIMENSIONAL SYSTEMS. II

211

of the system (1.1), is without contact, whenever 2 contains no critical point, and the unique path y through any point M of 2

is never tangent to 2 at M. (1.4) Let 2 = [A A') and = [BB') be two arcs without contact intersecting at most in A (then A = B) and disposed as in fig. 1 so that A and B are on the same y. Given any point M €2 let the y (M) through M, followed forward first meet p. at a point M'. Then M -+ M' defines a topological of 2 ff) on a subarc [BA of [BB'). Moreover if B' A', then the assign-

Fig. 1

ment A" = The same

A' makes of a topological [A A'] -÷ [BA"]. hold if the paths are followed backward instead

of forward, and likewise even if A = B provided that A is not a critical point. If diam AB is sufficiently small this property is merely a special

case of (II, 17.3). If AB is large one decomposes it into a finite or countable set of consecutive small segments (countable set if A = B, non-critical) and applies the same property to each segment in turn, resulting in a mapping such as Consider now the curve (1.5)

rdr/dt = xX + yY = Z (x, y).

212


Let a be a branch of Z

0. Take a circle of radius and center 0 contained in Q. (1 . 6) If e is sufficiently small the branch a intersects C0 in a single A and the arc (OA] of the branch is an arc without contact.

Let the coordinates be so chosen that a is not tangent to one of the axes at the origin. If a is identified with the branch (1. 2b) its tangent at the origin is: x = 0 if k n; y = akx if k = n. Hence a has a representation

y= mx + axP + .. .,

rn

At any point (x, y) of (OA] the slope of the tangent to the branch is

=

m + /axP —

1

+ ...

while the path through the point has the same slope

as the

circle of center 0 through the point. Hence

=

x —y

—.1 m

+ paxP —1

+

and therefore

=

—1

+ 0(xP').

Thus for e small the path through any point of a within or on C0 and the branch a are nearly orthogonal and so the part of a in the closed interior of C0 is without contact. Suppose that a contains several disjoint arcs within C0. Then as decreases C0 would sometime become tangent to a. Hence the path through the corresponding point of a would become tangent to a, whereas, as we have seen, they are nearly orthogonal.

Thus a has a single arc in C0. Similarly, if this unique arc a intersects C0, for arbitrarily small, in more than one point then some a small, would be tangent to a and this is ruled out as before. To sum up when is sufficiently small a has a single

arc in the closed interior of C0 and this arc intersects C0 in a single point.

(1. 6a) Returning now to the curve (1.5), we conclude that

X.


213

if e is small enough each branch of the curve intersects CQ in

a single point and in CQ the only point common to all the branches is the origin. Hence the interior of CQ together with CQ is decom-

posed by the branches of (1.5) into a finite set of triangular sectors in each of which dr/dt has a fixed sign. Since the paths are nearly orthogonal to the branches in CQ, they all cross a given branch in a fixed direction: all inward or outward relative to a given sector. (1.7) If a path y enters heaves] a sector OAR and does not leave [enter] it through one of the sides OA, OB, then as t -÷ + 00 [as t -+ — oo] y must tend to the vertex 0 of the sector.

The second case: t change of variable t

— oo, is reducible to the first by the — t, so that we only need to consider

0

Fig. 2

the first case where y enters the sector. We must show that

along y as t —> + oo. If r0 = inf r (t) on in the sector then r0 is reached on in the sector and we must show that = 0. r (1) —* 0

Suppose r0> 0 and let [A 'B'] be the arc of the circle r = comprised in the sector. Since is compact it has a point P on

[A'B']. Since P is an ordinary point there is a path ö (not a point) through P. Upon constructing the path-rectangle whose axis is an arc A of ô containing P. it is seen that y —* P in the sector

is only compatible with y = ö. Thus y must leave the sector if P is A' or B'. Hence P is between A' and B'. Since however dr/cit < 0 or> 0 at P, y must cross A 'B' at P. Hence if r0 0 it is not inf r (i) on and therefore r0 = 0, proving (1.7). Let r, 0 denote polar coordinates. Then: (1 .8) Let the path y tend to the origin so that r (t) 0, either as oc — oo. Then or as I —+ along y, 0 (1) tends to a finite or I —' +

214


infinite limit, i.e. y 0 in a fixed direction or else it spirals around 0. (Bendixson [1], p. 34.) We deduce at once from (1 .

dO/dt = xY — yX

(1.9)

1)

= p411 +1

+i (0) + ram + 2 (0)+...)

where ah(O) is a form of degree h in sin 0, cos 0. Suppose first 0. Let r (t) —÷ 0 as t that xY — yX 0. Thus am + (0) + 00.

The case t -÷ — oo may be taken care of by the change of variable t -÷ —I. To prove that 0 (t) has a finite or infinite limit

we merely need to show that when 0 (t) remains bounded it cannot have two distinct limiting values 00, Suppose that this is the case. One may select a 02 between and 01 and such that 0. Thus for r below a certain dO/dt for 0 82 a111 + 1 will have the fixed sign of am + 1 (02). Hence the ray L: 0 = 02 will always be crossed by y in the same direction. This is however

ruled out since L is crossed by arcs of y arbitrarily near 0 and to points arbitrarily near joining points arbitrarily near in both possible directions. Thus in the present case 0 (1) has a limit. If xY — yX = 0 then dO/dt = 0, 0 = C and (1.8) obviously holds. §

2.

Local Phase-Portrait at a Critical Point

2. We shall now examine the various possible dispositions of the paths in the individual sectors and then combine the sectors to obtain the complete disposition around the critical point 0. Take then a sector OAB and suppose dr/dt < 0 in the sector. Thus the side AB is crossed inward by the paths. There are then various possibilities depending upon the modes of crossing of the other two sides. Each case must be examined separately. The radius e of the circle C0 is supposed throughout to be small enough to have the subdivision into sectors of (1. 6a). Case I. The sides OA and OB are both crossed outward. A

through a point M of (OA] when followed backward necessarily

goes farther from 0 throughout the triangle and must cut the arc [AB] at a point N. By (1.4) N -÷ M defines a topological mapping of a subarc [AC) of [AB] onto [A0). Similarly the paths

through (OB] when followed backward define a topological

X.


215

mapping of [BO) onto a subarc [BD) of [BC). Since the paths through the subarc [CD] of (AB) must remain in the sector, by (1.7) they au tend to 0 as t -+ + 00. We thus obtain the disposition of fig. 3 with the typical paths y, 6, s. For evident reasons

the paths crossing [CD] form a system referred to as a Jan. 8

£

0 Fig. 3

As a limiting case we might have C

D, so that the fan

consists of a single arc. This is shown in fig. 4.

Fig. 4

Case II. The sides OA and OB are both crossed inward. Thus the whole periphery of the triangle, 0 excepted, is crossed inward.

By (1.7) all the paths considered must tend to 0, giving the disposition of fig. 5.

216


Fig. 5

Case III. The two sides OA, OB are crossed in opposite manner.

Assume that OA is crossed inward and OB outward. The path 6

through A may then either remain in the triangle, and then it can only tend to 0, or else leave it and hence cross (OB). This gives rise to two subcases. A tends to the vertex 0 in the triangle. (a) The path 6 As under Case I, the paths through the points of (OB] give rise to a topological mapping of (OB] onto a subarc (CE] of [AB] (fig. 6). The paths through [AC] form a fan. The paths exhibit the three types of paths in relation to the triangle.

e

S

I Fig. 6

(b) The path 6 through A crosses (OB) and this no matter how small the radius of the circle CQ. We have then the situation

of fig. 7, with only two typical paths: y and e.

X.


217

I Fig. 7

If the crossings at OA and OB are reversed there will result cases 1111a, ..., derived from lila, ..., in the obvious way.

It has been assumed so far that (AB) is crossed inward by

the paths. If the crossing is outward the change of time variable I -+ — I will preserve the paths but make them cross (AR)

inward. The resulting configurations are the same as before but with all the arrows reversed. We shall refer to them as I' Suppose now that the curve Z = xX + yY = 0 has no real

branch through the origin. Thus in the vicinity of the origin

dr/cit has a constant sign. Let us suppose first dr/cit < 0 so that in the vicinity of 0,r is decreasing along any path Consequently along y the coordinate r tends to a limit r0 and r = r0 is a closed path of diameter Since this contradicts: dr/dt < 0 for r the only possibility is r0 = 0, tends to 0. Suppose that there is a ray L through the origin which is not a path. Then as regards the open sector of angle bounded by L the situation is that of Case III. We have now the two possibilities corresponding to ifla and IIib which we consider in turn.

Fig. 8

218

DIFFERENTiAL EQUATIONS: GEOMETRIC THEORY

Case IV. One of the paths ô crossing L reaches the origin in the

open sector. Then by our earlier discussion it will be seen that all the paths near enough to 0 behave like ô. Thus the point 0 is a stable node (fig. 8). Case V. Every path ô crossing L, say at a point M1, crosses it again at a point M2. For another path ô' as in fig. 9, the crossings will be M1', M2' and by (1.4) M1' —÷ M2' defines a topological mapping [0M1J -÷ [OM,J, which shrinks the interval 0M1. It follows that a, and hence every path sufficiently near the origin 0

Fig. 9

is a spiral tending to 0. Thus 0 is a stable focus. Exceptionally M'1 throughout 0M1 and 0 is a center. We have in addition Case VI. dr/dt = 0, hence all paths are circumferences of center 0. The origin 0 is then again a center. If every ray issued from 0 is a path, we have actually Case IV and 0 is a node.

Finally if dr/dt> 0 near 0 we have reversal of arrows and Case IV': unstable node, and Case V': unstable focus. (2.1) Local phase-portrait. To obtain the full configuration of the paths around the critical point 0 (the local phase-portrait around the point) one must combine adjacent sectors of the various types described in the various admissible ways.

X.


219

Let us observe at the outset that if all the sectors around 0 are of types Ilib (fig. 7) or 1111b limit (rotation reversed) or the

same with III' instead of HI, then all the paths near 0 are spirals and we merely have a stable or unstable focus. As this case will arise otherwise anyhow we may leave it out of consideration at present. On the other hand, a succession of sectors such as Ilib or 1111b not surrounding fully the point 0, has no topological effect on the configuration of the paths. Thus the situation of fig. 7 and related types need not be considered here. Upon matching the other types in the various admissible ways, and adding to them the focus and center we obtain the following list: 1. Fan. This may be attractive or accordingly as the paths all tend to the critical point or all away fr6m it (fig. 10). As a limiting case we have the nodes.

Fig. 10

II. Hyperbolic sector (fig. 11).

Fig. 11

Hi. Elliptic sector (fig. 12). This is the only truly new type.

220


Fig. 12

IV. Focus. V. Center. (2.2) Existence of the various types. All the types except III are known to occur as critical points of. linear equations. Re-

garding III the following very simple type was suggested by Gomory. Take the system dx/dt =

x,

dy/dt = — y

with a saddle point at the origin and no other critical point. The transformation by reciprocal vectors x—).

x

y —,y-÷ x2+y2 x2+ys

will replace the system by an analytical system with a single critical point at the origin which has four sets of elliptic sectors one in each quadrant. (2.3) Remark. While our analysis of the critical point has been developed for an analytical system (1.1) it is manifestly appli-

cable to an isolated critical point of a system which merely satisfies the fundamental existence theorem provided that the curve xX + yY = 0 divides a suitably small circular neighborhood of the origin into a finite number of triangular sectors such as those considered above, in each of which dr/dt has a fixed sign, and similarly for the curve Yx — Xy = 0 and the sign of dO/dt. One must also assume that the branches of the curves clustering around the point possess continuously turning tangents in the given neighborhood of the origin. (2.4) Index. Let us suppose that the critical point 0 consists

221


X.

of a hyperbolic sectors, v elliptic sectors and the rest fans. Let

the positive angles of the tangents to the hyperbolic sectors, the same for the elliptic . . ., . . ., fi, and sectors and the fans. Thus a1,

.

..,

be

+

+

=

27r

Let us draw now a small circuit y which coincides in a hyperbolic

\

/

\

/

/

\/0 Fig. 13

sector with most of a "hyperbolic" arc such as ABC of fig. 13,

and in an elliptic sector sector with most of an oval such as HKL of fig. 14.

Let us carry out the construction of fig. 13. The tangents at A and C meet at D and EFG is a circular tangent to the lines DA and DC. Consider now a vector distribution tangent throughout to ö ABCEFGA and coincident with the vectors of our system (1.1) along ABC. Since 5 is negatively oriented its

index is —1 (IX, 11.4). The angular variation V1 along ABC

222

EQUATIONS: GEOMETRIC THEORY

is that of our field and along CEFGA it is W1

—(a

+

"almost equal"). Hence V1 — W = —2ii — i.e. V1 is almost a — =a— For the elliptic sector the situation is very similar. The arc MNHKLM is positively MN is circular and the circuit oriented. Its index, relative to a tangent vector field, is thus + 1. The angular variation V2 along HKL is that of our field, and along LMNH it is W2 = — Hence V2 2n — W2 = 2,z — i.e. V2 is almost fi + (7r — = fi +

means (—a —

0

Fig. 14

Our constructions made for a, /1

only need minor changes

if a or fi and these are left to the reader. Since for a fan of opening angle the angular variation we have:

Index 0

(v—a)/2 +

+ P5+

where E is arbitrarily small. Since

+

is zero or a multiple of 1/2 it can only be zero. Since the sum is merely we arrive at the following formula due to Bendixson ([1], p. 39): (2.5)

Index 0 =

1

+ (v—a)/2

X.


223

Since the index of a focus or center is unity, (2.5) holds even in these cases. Hence it holds without exception. An obvious corollary of (2.5) is: (2.6) The numbers v of elliptic sectors and a of hyperbolic sectors have the same parity.

(2.7) Stability. It is at once apparent that if there is a hyperbolic sector or an elliptic sector the critical point is unstable. Thus: (2.8) A n.a.s.c. for stability is that the point be a stable node,

a stable focus or a center. In the first two cases the stability is (2.9) Limit-cycles. Separatrices. These exceptional paths will

play an important role later (see 8) and in general when one wishes to discuss the full phase-portrait. A limit-cycle is a closed

path which is not a member of a continuous family of closed paths, i.e. an isolated closed path. Roughly speaking a separatrix

is a true path (not a critical point) behaving topologically abnormally in comparison with neighboring paths. Explicitly a separatrix is a path which is either a limit-cycle or else a path terminating or beginning on the projective plane with a side of a hyperbolic sector. Thus the four arcs of a saddle point A which tend to A belong to separatrices. 3. The local phase-portrait may be completed by information

regarding the mode of approach of the paths to the origin. Suppose in fact that

(3.1) X=Xm+Xm÷i+..., Y=Yrn+Ym+l+..., where Xh, Yh are forms of degree h in x, y and one of Xm, Ym is not identically zero. (3.2) If the form Urn +1 XYm — is not identically zero then the directions of are among those represented by = 0, so that their number is at most 2nt + 2 and they are Urn opposite in pairs. The value of dO/dt is given by (1.9) with +i (0)

=

Um+i(X,

o.

224


In addition (3.3) where

= r" + '(Pm + i (0) + r Pm +2 (0) + ...), is like ab in (1 . 9). Along the path y tending to the

origin replace the variable t by r

=

f r" - 'dt. As a consequence

may tend to a finite value r1. (We shall see however that t —* ± 00 as before.) The new system for v instead of t —+ ± 00, is now /

dr/dr

= am + i (0) + ram +2 (0) + •.. = r {Pm +1 (0) + j9m +2 (8) + •.

.

Let r, 0 be considered in (3.4) as cartesian coordinates. The critical points of the new system are of two types. First the

images of the former critical points other than the origin. These

critical points are at a positive distance Then there are new critical points on r =

0 0,

from the 0 axis. (the 0 axis) located at

i.e. at the 0 places the discrete places where am +i (0) = corresponding to the directions represented by Urn +1 0. Now if f (T) is any solution of

dO/dr=am+i(0),

(3.5)

= f (t), r = 0)

is a solution of (3.4). Since there is a (nonpoint) solution of (3.4) starting at any non-critical point 0)

then (0

every arc of the axis r 0 is a path of (3.4). It follows that as r —÷ 0 along y this path can only tend to infinity or else to a critical point such as (0,, 0) (am + 1 (0,) = 0), so that at the same time T ± oo. Thus, as stated earlier, as (am +1

± oo along paths for which r —÷ 0, r —+ ± 00 likewise. Going back to the initial system it means that the directions t

of approach to the origin are restricted to those represented by 0 and this proves (3.2). Urn + 1 4. One may actually proceed a step further. Let be a simple —x sin be a simple root of am. + 1 (0) = 0, i.e. let I = y factor of Urn + 1 (x, y). Let also f?m + 1 XXm + yYrn. Then A: r = of Vm +1

0,

0, i.e. 1 is not a factor 0 = E is an elementary

X.

DIMENSIONAL SYSTEMS. ii

225

critical point of (3.4). Since the path r = 0 tends to A in a fixed direction, A is not a focus nor a center. Hence there are at least four directions of approach th A, and at least two distinct from the directions on r = 0. These two are opposite, hence one is in

the part r> 0 of the plane and it corresponds to at least one path of (1.1) tending to the origin in the direction 1 = 0. Since the same situation will hold regarding + x, paths approach the origin in both possible directions. Thus: (4.1) asimple factor of Um+i and does not divide Vm + ithen directions.

ap/roach the origin along I in both possible

§ 3. The Limiting Sets of the Paths as t

±

00

5. To study these limiting sets it is convenient to close the Euclidean plane at infinity by a point, thus turning it into a sphere. Analytically this is done by applying the standard

transformation

=

1/x,

y'

= l/y

which reduces the infinite region to the origin.

The reason for giving up the projective plane for the present is that the argument will rest heavily upon the Jordan curve theorem. As a matter of fact instead of passing to the sphere in the above manner, one could replace the projective plane by its doubly covering sphere. A certain number of preliminary properties are needed. (5.1) Let the Jordan curve J on the Euclidean plane IT or on the sphere S possess an open analytical arc 2. Let a path cross A at a point P where y and A are not tangent. Then, at P, y crosses from one of the two regions bounded by J into the other. Let P be taken as origin for local coordinates x, y and let x, y be so chosen that at P neither y nor 2 are tangent to one of the axes. Since A is analytical, a neighborhood A' (an arc) of P in A has a holomorphic representation (5.2)

where

ep

(0)

=

(0)

x = (u), y = (u) = 0 and owing to the choice of axes both

226


0. One may eliminate therefore u from (5.2) and obtain say an analytic representation for a suitable A' of the form (0),

(0)

/ (x, y) = y — g (x) = 0. The condition of no tangency at P means (5.3)

(5.4)

X (P)

0, Y (P) /X (P)

g' (0).

Fig. 15

Let now U, V be the two components of 1T—J and S—f as the case may be, which have J as common boundary. Consider also for x, y small the two sets

W1: y==g(x)+s W2:

y=g(x)—e }

The two sets Wj are connected and together with A they make up a full neighborhood of P in 17 or S. Hence say W1 c U, W2 c V.

Let us follow now y through P. We have on

df/dtfxX+fyYg'(x)X+ Y X(Y/X—g'(x))..

f

is monotonic through 0. It follows that along P and so varies say from — to +, i.e. along one passes from W2 to W1, that is from V to U, and this proves (5.1).

Hence (df/di)p

'X.

TWO DIMENSIONAL SYSTEMS. it

227

6. We shall now discuss some simple properties of pathrectangles (IX, 1.4). We have at once from their definition: (6. 1) The paths which traverse a path-rectangle all describe it in the same direction. A noteworthy path-rectangle (generalization of the earlier configuration of the same name) is associated with a closed arc without contact A. Let 2a be its length, A its midpoint, s the

arc length counted positively along A in a certain direction. If P is any point of A and y the path through P, let v be made to correspond to the solution (x (t), y (t)) of (1. '1) such that (x (0), y (0)) is the point P. With f as in (5) and changing if necessaryf into — f, we can. find a r> 0 such that on any y the

Fig. 16

functionfO for

Consider

now s, t as rectangular coQrdinates of an Euclidean plane 17, and let R* be the rectangle in 17 determined by Is I a, Ill r. Define a mapping q, of R* whereby the point M* (s, 1) goes into the point

M (x (t), y (1)) of the path y issued from the point P of A at a distance s from A. The mapping is manifestly continuous and one-one. Therefore since R is compact, is topological. We shall refer to the image R = pR* as a path-rectangle of median line 2. This mild generalization of the earlier path-rectangles has all

their properties. The distinction occurs primarily in the applications. The earlier path-rectangle (R1 in fig. 16) always has a preassigned axis but may be made very thin around that axis. The present type (R2 in fig. 16) has a preassigned basic line but may be very thin around that line, i.e. in the transverse direction relative to the paths.

228

DIPFERZNTThL EQUATIONS: GEOMETRIC THEORY

(6.2) LEMMA. If a y followed as t increases, has with an open arc without contact A three consecutive crossings M, M1, M2,

then M1 is between M and M2 on A. If the lemma is incorrect, then M2 is between M and M1 or

M between M2 and M1. The first case corresponds to fig. 17, and the second to the same figure followed backward. Taking fIg. 17 choose at all events A so small that there is a path-rectangle related to it as in the figure. Let J be the Jordan curve MSM1M2M.

Then J does and does not separate Q from R, a violation of (5.1) and the lemma follows.

S

Fig. 17

7. We come now to the limiting sets. Let y be a path, y+ (M) the subset of y consisting of M and of all points of v traversed after M, (M) the analogue referring to the points traversed before M. We refer to y+ (M) and r— (M) as the positive and negative half -paths determined by M. The closures

(M),

(M)

give rise through their intersections to two new sets A+ (y) =

(1

it

(M), A- (y) = fl

it

(M)

called the positive and negative limiting sets of y. (They are the w and a sets of G. D. Birkhoff.) (M) are connected, so are their closures. The Since y+ (M), latter being also compact and non-empty, we have since we are on the sphere or the projective plane (I, 4.6):

X.

TWO DIMENSIONAL SYSTEMS. n

(7. 1) The limiting sets

229

(y), A— (y) are compact, connected,

and non-empty.

(7. 2) If y is a critical point, then it coincides with its limiting sets.

(y). .Thus there Suppose that P is an ordinary point of say is a path ö through P. Construct a path-rectangle £1 with P on its axis and let a be a small segment transverse to ö at P. Since

P

(y), there are points of y+ (M) arbitrarily near P and the arcs

(M) through them will contain subarcs in Q. In particular, these arcs will all cross a in the same direction and will meet it in points R1, . . ., R8, ..., encountered in that order as (M) is described forward from M. By virtue of Lemma (6.2) the R, are in the same order on a. Hence they can have only one limitof

Fig. 18

point on a and since P is manifestly such a point, R1 -÷ P. Moreover, on a the points R, are all on the same side of P. Or explicitly:

(7.3) If P is an ordinary point of

(y), ô the path through a a suitably small segment transverse to ô at P, then (M) . followed forward meets a in successive points R1, R2, .., which are ranged in the same order on a, and tend to P on one side of

P on a.

The notations being as before, let P' be a point of in £1 and ô' its path. Since ö' meets Q it crosses a in a point P" which is a condensation point of the R8. Hence P" = F, ô' = and P' is in the axis of £?. Since Q is a neighborhood of P. we have: (7.4) Corresponding to any open arc of a path ô c (y) and P of there is a neighborhood U of P such that U n (y) c L

230


(7.5) A path which meets a limiting set is contained in that set. It is sufficient to consider (y), and to show that if it meets the path 13 then it contains 5. When S is merely a critical point this is obvious so this case may be dismissed. Let E = S n (y). Since S is a Jordan curve or an arc, if E S there is a boundary

point P of E in S. Since E is a closed subset of 5, P is in E and hence in (y). Identify now P with the point thus designated before. The arc 5* of S in Q (the axis of Q) consists of condensation-points of the arcs of

(M) through the points Rq, Rq +1,. for q above a certain value. Hence 5* c (M) whatever M, and

therefore 5* c A+ (y). Thus P is an interior point and not a boundary point of E. Therefore E = 5, 13 c A+ (p). The possibility that S = y is not excluded. In that case P be one of the points Rq. This can only occur when the all coincide with P and y is closed. Conversely y closed implies

that

A (y) = y and that all the Rq coincide with P. Notice also that a n.a.s.c. for 13 = y is that the two intersect, i.e.,

that

(y) meet y. Since the properties of A (y) follow from

by changing I into —1, we may state: those of (7.6) A n.a.s.c. for y to be closed is that one of A+ (y), A— meet y, and then both coincide with y.

(7.7) Path-polygon. Let E2 be a two-cell whose boundary consists of a finite number of separatrices and critical points. This boundary is called a path-polygon. It will be required in a moment. §

4.

The Theorem of Bendixson

8. We are now in position to give a complete description of the limiting sets and of the related behavior of the paths. (8.1) THEOREM. The limiting sets of a path y fall under the following mutually exclusive categories: (a) A which is critical and with

increasing time y spirals toward A

else tends to A in a fixed

direction.

(y) is a closed path S and either y = 5 = (y) or else with increasing time y spirals towards S on one side of 5. (b)

is contained in a 2-cell E2 whose boundary is (y) which (y) and any is now a path-polygon. The path y spirals towards (c) y

X.


231

Fig. 19

path which starts from a point of E2 sufficiently near

(y) spirals

towards the latter in the same direction as y. This last property holds also for (a) and all the paths starting sufficiently near A, and likewise for (b) and those starting sufficiently near on the same side as y. (d) A(y) is of the same nature as A+(y) save that the spiralling Ch. I.) and tending to a limit occur as t -÷ (Benclixson Fig. 19 illustrates the case of a graph and the related behavior of y.

Part (8. id) is obvious so that it is sufficient to examine the other parts.

(y) finite. Since it is connected, it must consist of a single point A. The path through A is in (y), hence it is A itself. Thus A is a critical point. Combining this with (1.8) we find that under the circumstances the behavior 9.

first

of y and (?) conforms with the theoiem.. It may be observed that obviously when y tends to a critical point A then A alone. Similarly when tends

toA with t—>—oo,thenA(y)=A. (9.1) SuI'pose now that

(y) contains no critical point. Since

the set is not empty it contains ordinary points and the paths through them. Let ô be one of these paths. Suppose (ö) —6 non-empty and let P be a point of the set. By hypothesis the path through P is distinct from 6. Since P is in the closure of subsets of 5 it is in contains. ô. Since P e

and hence in the closed set (y) which (6). there are points of 6, i.e. of (y), arbitrarily near P and not in As this contradicts (7. 4), we

232


(ô) = ö and hence a is closed. Thus every path must have (y) is closed. in (y) contains two distinct closed paths 6, 6'. Suppose that Since 6, 8' are closed and disjoint A4 (y) is not connected, (y) = 6. contrary to (7.1). Hence To sum up: (v) contains no critical point, it consists of (9.2) Whenever a single closed path.

A noteworthy incidental consequence is the following sition due to Poincaré: (9.3) THEOREM. A closed region Q which is free from critical points and contains a half-path contains also a closed path. At the cost of changing t into — t we may suppose that the (y) c (M) c Q. Hence (M). Then half-path is contains no critical point and so it is a closed path in Q.

10. Continuing with the analysis of the limiting sets let us take any closed path 6 and examine its relation to the paths issued from the neighboring points. If y = 6 it is already known 6. Choose any point (y) = 6, so we suppose (7.6) that

P of 6 and a segment a transverse to 6 at P and situated in a certain line L. Referring to (II, 14.8), it is known that starting from any point M a and following y+ (M) we first encounter L at a point M1 and M —+ M1 defines a topological mapping q,

of the segment a into another segment a1 in L. The point P is a fixed point of q'. Further properties will now be brought out. Since y+ (M) cannot cross 6, the point M1 is on the same side

of 6 as M, i.e. on the same side of P as M in L, and it is an analytical function of M (II, 18.3). This implies that for a sufficiently small, either M = M1 for all M or else only for M = P. In the first case all the paths meeting a on the side considered are closed; in the second none except 6 is closed. Suppose we have the second situation and for one M the point M1 is between M and P as in fig. 20. Then the same situation will hold for all points between P and M, for otherwise by a well-known continuity argument q' would possess a fixed point

between P and M. Under the circumstances, all the paths meeting a on the same side as M will spiral towards 6 with increasing. t. Let M, M1, M2, ...

be

the successive intersections

X.


233

Fig. 20

of any one of them, say y with a. By the above M, M1, M2, are met in that order on a and they are all on the segment MP. Hence has a limit P' on MP. Since P' is a fixed point of ip we must have F' = P, and so —+ P. Since this occurs on

any segment transverse to ö, as a point follows y4 (M) with (y) = ô, and t -÷ + 00, the point tends to ö. This means that that y spirals forward around ô towards ö. Moreover all the paths issued from points near enough to ô on the same side as behave likewise.

If the point M1 in the above argument is such that M is between P and M1, then following y(M1) backward, we would find that y and all the paths on the same side issued from points near enough to ö spiral towards ô as 1 -÷ — 00. In this case

= 8.

When the paths spiral towards [away from) 8 on a given side with increasing time, then they do the opposite with decreasing

time, and so they are then orbitally stable [unstable] on that side. Notice that the behavior on both sides of 8 need not be the same. If 8 is orbitally stable [unstable] on both sides, then it is orbitally stable [unstable] in our usual sense. [f it does not behave alike on both sides, then it is orbitally conditionally stable.

To sum up, then, when 8 is closed the only paths y which have condensation points on 8, i.e. whose limiting sets meet 8, are 8 itself or those which spiral towards or away from 8. In the first case A+(y), in the second A(y) is 8 itself. The behavior relative to 8 is clearly in accordance with the theorem.

234

DIFFERENTIAL EQUATIONs: GEOMETRIC THEORY

11. The remaining possibility for is to consist of critical points and non-closed paths. Let ö be one of the latter. Now we

may prove by the same arguments as at the beginning of (9) that if contains an ordinary point then ô must be closed. Since the ô here considered is not closed its Á+(ö) consists only of critical points, and since is connected it consists of a

single critical point, say A. The rest of the argument will be based upon the behavior of y and ô in the vicinity of A.

We shall first show that 8 cannot spiral around A. For passes arbitrarily near every point of a, and hence arbitrarily near A. Hence if ô spirals around A, so does y (3, case V) and therefore A

contrary to assumption. Thus 8 tends to A in a

definite direction (1 . 5).

Take on 8 a point P arbitrarily near to A. Since y passes arbitrarily near P on one side of 8 and does not tend to A, one side of a hyperbolic sector centered at A and so it is a separatrix. Let be the separatrix which contains the other side of the sector. As we follow away from A we shall reach a critical point A1 (which may be A itself), then leave it with a separatrix .3k, etc. Since y must return to the neighborhood of 0, we shall ultiniately return to after having described in the process the full set A+(y), which is thus a path-polygon 17. Part of the argument of (10) is applicable here to show that y and any path passing sufficiently near H on the same side as y o contains

all spiral around 17 in the same direction and it need not be repeated. Let U be the component of the complement of 17 in the sphere containing y. Since the possible "loops" in H are all

outside of U, U is a two-cell. This completes the proof of Bendixson's theorem. (11.1) R.mark. The proof of the theorem applies with hardly any modification to a basic system (1.1) within a closed region £2 where: (a) the system satisfies the conditions of the fundamental existence theorem; (b) in £2 there are at most a finite number of

critical points around each of which the property described under (3.3) holds. Inparticular, Poincaré'stheorem (9.3) on the existence of a closed path is valid without any reference to (3.3) (see 34).

X. §


235

Some Complements on Limit-Cycles

5.

12. Suppose that ô are closed paths bounding an annular region free from critical points or other closed paths. We say then that y, ô are consecutive. (12. 1) Two consecutive closed paths v1 a cannot both be stable or unstable on the sides facing one another. (Poincaré.)

That is to say, if say y is interior to ô then it is not possible to have y stable [unstable] outside and 3 stable [unstable] inside. Suppose, in fact, the assertion false. Replacing if need be t by we may dispose of the situation so that both y, ô are unstable

on the sides facing one another. Choosing now paths v" a' in the annular region respectively very near y, ö and suitable transverse segments, we will have the configuration of fig. 21. Let U be the inner annular region bounded by arcs of y', 6' together with the segments MM', NN'. Any path starting from a point of the boundary of U remains in U. Since U is free from critical points it must contain a new closed path e, and e must surround else it would bound a region free from critical points. the assumption that y, 6 are consecutive Since this (12.1) is proved. (12 .2) Examj.1e. Consider the general equation in polar coordinates (12. 3)

dr/dO =

rf (r2)

wheref is a real polynomial. Setting 0 t, and gular coordinates, we obtain the system '12

to rectan-

1dxfdt = — y + xf(x2 + y2) •

/

which is of the basic type (1. 1). The behavior of the paths is more conveniently investigated, however, in the polar form. Since only the positive roots of f(z) matter, let us set f(z) zkg(z)h(z) where g(z) >0 for z> 0 and h(z) has only positive roots

It is also not a genuine restriction to assume h(0)> 0, since this may be achieved by the unimportant change of 0 into —0. Now if a is a root of h(z) the circle Ca r2 = a is a closed path. There are two possibilities:

236


Fig. 21

(a) The root a is of odd order. Then h(r2) changes its sign as r2 crosses a. Suppose that it goes from — to +. Since dr/dt changes sign from — to + as r2 crosses a any half-path near Ca inside the circle will spiral away from Ca, and likewise for any half-path

near Ca outside the circle. Hence Ca is unstable (on both sides). If the change of sign of h(z) is in the opposite direction, then Ca is orbitally stable. (b) The root a is of even order. Then h(r2) does not change sign as r2 crosses a. If h(r2) 0 near a then along a path 'near must increase and Ca is orbitally stable inside but unstable

outside. If h(r2) 0 near a, Ca is orbitally stable outside and unstable inside. Thus is semi-stable in both cases.. If a, fi are consecutive roots of h(z), then along any path y in the region between Ca and Cft the sign of dr/dt is fixed and so 0 is monotone increasing or decreasing. Hence v spirals away

from one of the two circles and towards the other. Thus the

only closed paths in the finite plane are the circles Ca. Beyond the last circumference Ca the paths spiral to infinity if the leading term of h(z) is positive, away from infinity in the contrary case. The origin is manifestly a focus. The paths passing near the

X.


237

origin spiral away from the origin since g(O)h(O)> 0. Thus the origin is an unstable focus. There is no difficulty in writing down the general solution of (12.3) and it is a simple matter to verify the preceding properties by means of the solution. §

6.

On Path-Polygons

13. Let us consider again a path-polygon ii with the 2-cell Q, which it bounds. We suppose also that the two sides that termi-

44 11

P

Fig. 22

nate

at a vertex A always determine, in an evident sense, a

hyperbolic sector centered at A and contained in Q. Take now any point P on a side ô of Ii and draw in Q a transverse arc I to ö at P. A very elementary argument will then show that the paths originating very near P in Q intersect I in points which tend monotonically towards P or away from P. If the second case holds we replace t by t' = — t and the first case is then obtained. As before we may show that whichever of the two

possibilities takes place, it will take place for all the paths sufficiently near H. (13.1) The paths passing sufficiently near H in either all spiral toward 17 or away from II, or else again they are all cfosed.

DIPPERENTIAL EQUATIONS: GEOMETRIC THEORY

238

A(y) of

In the first case fl is the

all such paths. Let Q, R be two consecutive crossings of A by y with Q arbitrarily near P. Then a slight deviation from the proof of (IX, 11.1) will show that Index (y u RQ) = 1, it being assumed that y is

oriented concordantly with Q. We may take y so close to H close to P) that there are no critical points between y u RQ and H. Applying now (IX, 7.3) we have: (13.2) The 2-cell bounded by a contains critical points, and the sum of their indices is + 1. (Q so

§

7.

Some Properties of dlv (X, Y)

14. Let V designate the vector (X, Y) so that

div V =

(14.1)

aX/t9x

+

This divergence has two interesting properties. If y is a closed path one of the characteristic roots A2 is zero. Let A be the other root. Then by (VIII, 1 .9) if r is the period: (14.2)

Hence this property: (14.3) If the time average of the divergence on the closed path y is negative [positive] y is orbitally asymptotically stable [unstable].

15. A second property of the divergence is the following: (15.1) Criterium of Bendixson. If div V has a fixed sign (zero excluded) in a closed two-cell Q then Q contains no limit-cycle nor even an oval going from and to a critical point. For suppose that there is a limit-cycle y in Q and let it bound a region S Q. Applying Green's theorem we have

55

div V dxdy

=

(X dy — Y dx) = 0.

Hence div V cannot have a fixed sign in S, nor afortiori in Q.

The case of the oval was pointed out to us by Coleman. If ö is an oval in Q going to and coming from a critical point A, then

X.


239

Fig. 23

one may "round-off" ô near A so as to produce a continuously turning tangent on the resulting closed curve 61, and this by

an arc of arbitrarily smail length e. Let a = sup V'X2 + y2 = inf (div V( in Q, and let S, S1 be the regions bounded by 6, Assuming for convenience div V> 0 in Q — the case div V< 0 would be treated in the same way—we have

0 < ff div J7 dxdy

=f

Vds < a e.

Therefore if S, S1 denote also for convenience the areas of the two regions 0 0 there is one more TO-curve for Q* and it is the image of a unique TO-curve in the (x, y) plane

tending to the origin along the bisector L: x = y in the first quadrant. The change of variables y = 321/4 + 0*, discloses the presence of a similar TO-curve 4, tending to the origin along L in the third quadrant.

21. We now take up the study of the system (17.5). The Weierstrass preparation theorem yields

y + f(x, y) = {y — A(x)}Ei(x, y), y + g(x, y) = {y — B(x)} E2 (x, y), where by identification we find

Ei(0,0) = 1;A,B=[x],. Hence E (x, y) = E,/E1 is a unit such that again E (0, 0) = 1. Upon changing the time unit according to di -÷ E1 di, (17.5) is replaced by tI'e simpler (geometrically) equivalent system (21.1)

dx/di—y----A(x), dy/dt—_{y---B(x)}E(x,y).

The curves (21.2)

ry: y

A (x);

y = B(x),

244


will play an all important role in our discussion. They are both tangent to the x axis at the origin. ry is the locus of the points where the tangents to the paths are vertical, and r1q the same

where the tangents are horizontal. In the first and fourth pass below L. The jointly divide the quadrants rv and plane into two regions R1 below and R2 above. We shall deal directly with the phase-portrait in R1. The same for R3 is obtained through the change of variables x —x, y -+ —y in (21. 1), and

applying the R1 treatment to the new system. For the present we call I',, the branches of (21.2) to the right of the y axis, i.e. in R1.

Observe that

r,1. For if they were the same (19.1)

would yield dy/dx

E (x, y),

and hence the origin would not be a critical point. Thus rH is either above or below rv. 22. Suppose first l'H above ry. From the signs of dx/dt, dy/ill and rv are both above the x one infers at once that unless axis, the disposition is that of fig. 30: R1 is a nodal sector. In the exceptional case there arises the possible alternative indicated in fig. 26 and which might give rise to an elliptic sector. How-

ever along Ox the slope of the tangents to the paths is dy/dx = B(x)fA(x). Since fH is above l'y which is above Ox for x> 0 and small, we have here B(x)> A (x)> 0. Hence the slope in question > 1 and so the tangents in question make an angle > n/4 with Ox. Now under the situation of fig. 26 the angle would be nearly zero. Hence this case is eliminated and fig. 30 for R1 prevails in all cases.

now that ry is above I',,. This time the phase-portrait in R1 upon heeding the signs of dx fill and dy/ill is in accordance

with fig. 27 with two hyperbolic sectors, save that we have indicated a separatrix S. However a priori a fan could arise in place of S, and also (in R1) right below cl1. They are both eliminated by the same argument as follows. First since ry is above rH, C = A — B> 0 for x positive and small. Suppose that we

245


X.

/

/

/L I;

// Fig. 26

had a fan and let y(x). y(x) + a

a> 0, correspond to two of

its TO-curves with 8(x) of at least the same order in x as y. From

dy(y—B)E

(22.1)

dx

y—A

and denoting "approximately equal" by

de.

2'

dx

/ •

=

we

have

8(y—B)E øy y—A

1—CE

y —BØE

y—Aøy

Now we may write

E = 1 + g(x) + (y — A) h(x,y), where g, h are non-units. Hence de dx

. a

(— C + (y — A) (y

(y—A)2

B) m (x, y))

246


where m is a non-unit. Thus for

0, de/dx has the sign of

+ (y—A)(y--B)m(x,y). Let A = axP E1 (x), B = E2 (x), (0) = 1. We now distinguish three possibilities: I. p b. Then C is of order p and one of y — A, y — B is of order at least Hence q has the sign

of — C. Thus de/dx < 0 for e> 0. Since this contradicts the structure of a possible fan to the right of Oy, no such fan exists in this case. II. = q, a = b. We have then A(x) = D(x)

a'

so that s> p

+ a'x +

B(x) = D(x) + b'xs + b', D = axP + fix' ÷' + ... + and of course p

2, since rH, rv are tangent to

Ox. Making the regular change of variables y = we have (z—a'xS-—-

dx

Upon

— 1

z

+ D(x), x

...

applying the Weierstrass preparation theorem to the

numerator we find (22.3)

dz_ (z— dx

...)

(x, z)

,

E* (0,0) =

1

Since we have dealt with a regular transformation (22.3) may replace (22.1) as regards the search for the fan. Here s is the analogue of p before and since b' a' we are under the preceding case and so the fans are again eliminated.

Ill.

q. This is only consistent with rv above and rg

below Ox, with the possible fan likewise below Ox. Here C is of order q and one of y — A, y —B is of order at least q. Hence q has again the sign of —C, which leads once more to the elimination of the fans.

X.


247

Thus fig. 28 with just the one separatrix in R1 does represent the situation. One readily verifies that the position of PH, Pp relative to the x axis does not change the situation. Since the situation in R2 may be reduced to that in R1 by reversing the positive direction on the axes we conclude that if in R2, i.e. above L, PH is below f'v, R2 is a nodal sector, while it consists of two hyperbolic sectors separated if PH is above by a single separatrix.

23. Let C(x) = cx' E(x). Taking account of all the preceding remarks we have the following possibilities:

Fig. 27

I. c> 0, r odd. Then Pp is above PH in R1. below it in R2 and so we have four hyperbolic sectors separated by four separatrices, i.e. a saddle-point (fig. 27).

II. c> 0, r even. Here Pp is above

in both sectors and

so we have two hyperbolic sectors in R1 and R2 is a nodal sector (fig. 28).

III. c if U1 c U,1, and this makes of {A} what is known as a linearly ordered set. Consider They are all subsets of the compact set Q and the closures any finite subcollection < < .. 0, ut,> UA. Hence (2.9) Pa cannot be closed.

Suppose now that a a, i.e.,

is as in fig. 1. We may now

introduce

= 5 AD so that

iiu

+ 5 EC du,

du,

Along AD or CE we may write

= q1(a) + du

= 5 DBE

F dy dx

dx

—Fg

y—F

dx.

Since F < 0 for x 0 for a small and 3, drJdt is negative and so v rises but comes nearer to the origin till it reaches the line x = —3. On the other hand y

,dy

x

y—,u(x2f3----x) —

)2

276


Hence for y large and positive and x2 < 3, approximately y" = — 1/y 0 and endeavor to obtain results valid for the whole 4a range. Guided by the electrical circuit analogy, various restrictions will be imposed on the systems considered here and throughout the remainder of the chapter. By and large it may be said that while they are anything but minimal, they are such as to preserve

the essential features of the systems under consideration. In particular we will not impose any symmetry properties upon fand g. As in (2) we set: F(x) and

=

f:fx

dx, G(x)

=

g

(x) dir, u

= y2/2 + G(x),

in addition

E(t) =Je(t)dt. We make the following explicit assumptions: I. For all x f and g are continuous and g satisfies a Lipschitz condition.

II. There exist positive numbers a, a, /1 such that f(x) a for for III. E(i) is continuous and IE(t) is bounded for all t.

xIa, and that

280


Note that in view of our conditions F and G exist and even have continuous derivatives for all x. We will denote by E the bound of )E(t)I IE(t)I E. We note

also that in view of ii, IF(x)I and 1G(x)l -÷+oo with IxI. We may therefore suppose a so large that: IV. (a) There exists a positive y such that:

(b) G(x)>O for Finally since g is continuous

in the strip fxl a.

will have an upper bound ô> 0

We replace the system (7.1) by the equivalent system (7 2)

(7.3) The system (7.2) satisfies a condition for all x, y. Hence the existence theorem and trajectory uniqueness hold in full for the system.

The proof is the same as for (2. 4). 8. (8.1) THEOREM. There exists a bounded closed 2-cell

such

that a solution x(t), y(t) starting at time t0 from any M0 in the phase-plane, will after a time T, which generally depends on the solution, enter and remain permanently in 9. Let 2k,, be an upper bound for CFI + E in the strip a and let be a positive constant to be specified later. Consider the rectangle R:

+

if

s

(8.2) A trajectory I' which remains outside the open rectangle R necessarily spirals around the rectangle and each turn is accomplished in finite time.

The slope of a trajectory is •

3'

dx

y—

—g(x) (F(x) — E(t)

)

XI.

DIFFERENTIAL EQUATIONS OF THE SECOND ORDER

Within the strip and outside R: IF(x) Also

mi

(8.4)

— E(t)

281

hence

I

Hence

= m0.

Hence r must merely cross the strip in one direction or the other.

Since above the rectangle and within the strip y A > 0, dx/dt> 0 and so x increases. Thus the arc such as a = is

I I

I

/

Fig. 9

described as shown in fig. 9. Similarly for the arc

=

of

fig. 9.

Notice that outside the rectangle a trajectory can only leave the strip above the rectangle to the right and below the rectangle

to the left. (8.5) The

crossing

time of the strip by a trajectory above or

below the rectangle has a

bound r1.

Assume say that the crossing is above like a. On a dx = {y — (F(x) — E(t)

)}

di>

Hence 10 =

—

2a/A1 = r1.

di.

282

DIPFERENTIAL EQUATiONS: GEOMETRIC THEORY

9. We shall now discuss the parts of trajectories outside the strip and here we must have recourse to the energy u. We have (9.1)

du_—ydy+g(x)dx,

and hence from (7.2) along a trajectory

du—u(F(x)—E(t))dy. Now along a trajectory for x a: dy/dt =

(9.2)

F(x) — (9.3)

----g(x)

—fi, and

y. Hence

du ufi ? dt. Since u remains positive outside the strip it will reach the strip in a time 22k). Since for xa is at least

the loss in is along the arc

f (F — E(t) )

y,

and equally along e'• Hence —

—

This will be positive if we choose, as we may 21> Under the circumstances

=

y2(C) —

v—

84u

q= This proves (10.1).

8

y

— = 4uq,

284

DIFFERENTIAL EQUATIONS: GEOMETRIC ThEORY

11. We proceed now with the construction of the closed

region £2. In fig. 10 points such as B, B' or C, C', ..., are symmetrical with respect to Ox. The shaded rectangle is R and the slanted lines have slopes ± m0. The arcs such as DQD', ..., are parts of curves u const. The closed region £2 is the interior and boundary of the Jordan curve FPF'G'SGF. The open region

£ is the interior of the Jordan curve EDQD'H'K'RKE. Thus

G

x

Fig. 10

the closure of I is in the interior of £2 and the boundary of I is at a positive distance from the boundary of £2. It is already known from (10. 1) that every trajectory meets the open rectangle R after a finite time. To prove (8.1) it is thus more than sufficient to prove the following strong result needed later: (11.1) Every trajectory r leaving the open rectangle R remains in the region I. If r leaves say across BB', as it cannot cross the arc of curve

u = const. joining B to B' it remains in £ outside R or else returns to R (the second possibility alone takes place). Similarly if it leaves across LL'. If I' leaves across LB then it cannot reach

XI.


285

LC and so it must leave the strip across BC. Since it does not cross the arc DQD' it remains in E outside the strip and either returns to R or else re-enters the strip through the interval (B'C').

Then either F re-enters R or else crosses the strip above K'C' and below L'B'. Then F re-enters R or else, remaining in I, re-enters the strip across (LK). It must then re-enter R or else leave the strip below its first departure at T (10.1) and hence across (GB). From this point on the argument just repeats itself and so (11.1) follows. This completes the proof of theorem (8. 1). 12. (12.1) THEOREM. Every trajectory reaches and remains in a region e for all I above any specific value since is bounded. Hence either x(t) -÷ 0 as t and so the theorem holds, or else x'(t) oscillates indefinitely with increasing t. If is anyone of its maxima we will have x'(11) = 0. Hence from (7. 1): ,

— g (x(t1) )

f(x(t1))

is bounded independently of Hence since 1 As

so is g(x), say Ie(x) <M.

where K is independent of 4u. Thus x'(t) has a fixed upper bound independent of at all these maxima and hence for all I large enough. A similar remark holds for all negative values and (13.1)

follows.

§ 5. Applications and Complements 14. Autonomous oscillations (E(t) = 0). When E(t) = tion (7.1) becomes (14.1)

d2x/d12

0,

equa-

+ 1(x) dx/dt + g(x) = 0

which is a generalized form of Liénard's equation. The equivalent system reads

XI.


dx/dt

(14.2)

=

287

y — ti F(x), dy/dt = —g(x).

We make the following supplementary assumptions to those of (7): V. f(O) =

a

< 0;

VI,. y(x) = g(x)/x is continuous and> 0. Also to simplify matters: VII. f and g are analytic for all x. Except for analyticity this is still decidedly more liberal than the assumptions made regarding Liénard's equation. The price to pay is the weaker property: (14.3) Under our various assumptions the system (14.2) has at least one limit-cycle and hence (14. 1) has at least one periodic solution. As

already pointed out at the end of § 2 there might well

arise several limit-cycles. The proof of (14.3) is quite simple. Under our assumptions the origin is the only critical point.

The first approximation to (14.2) is dx/dt = y — a x, dyfdt = —fix,

fi

= y (0).

The characteristic equation is r2

+

+fl = 0.

The two characteristic roots have therefore positive real parts and so the origin is an unstable node or focus. Thus

increasing

time any path 1' will stay out of a suitably small circular region w surrounding the origin. On the other hand by (10.1) and (11.1) with increasing time r will remain in Z, and so a fortiori in — w. Hence Poincare's theorem (X, 9.3) is applicable here and so (14. 1) follows. 15. Periodic function E(t). Let E(t) be periodic and of period r. If e(t) is developable in Fourier series this means that the series

has no constant term or again that the mean value of e(t) for a period is zero. We have then:

288


(15.1) THEOREM. If E(t) has the period r there exists a periodic solution with period r.

Let S(t1) denote the transformation whereby if x(t; x0, Yo), y (t; x0, Yo) is the solution passing through the point P0 (x0, Yo) at time then S(t1) P0 = P1 (x1, Yi) where x

+ t1; X0,

Yo)

= X1, y (to + t1, Xo, Yo)

Yi•

Thus S(t1) is a topological mapping of the Euclidean plane 17 into itself. Let in particular S = S(r). Consider now any point M of the closed region Q and the trajectory 1' starting from M at time £0. According to (10.1)

at some time t1 r has a point M1 within the rectangle R.

Since S(t1) M is in R and S(t1) is continuous, there is a neighborhood U1 of M1 in R. Hence S (t1) U1 = U is a neighborhood of M such that if V = U n SI then S(11) V c R. It follows that S(t)

Let n(V) be the least multiple of r such that ii r t1. Then S' V c E for all v Since £1 is compact and {V} is an open covering of .Q there is a finite subcovering {V1, ..., V,}. Let N = fin (Vi). Then

S'QcZforallvN.

Going back now to the proof of (11.1) we observe that every trajectory issued from a point of the closed rectangle R remains in the open region I. Therefore S(t) R c I whatever 1. In particular SkR c Q whatever k. Consider now the successive transforms 52O = Q, Q1 = S SI,. = Sk Q, ..., and let Zo, Z1,. .., be the closed complements

of the unbounded components of the exteriors of S?0 u £11,... Since SkR c lb whatever k, every 11k meets Q. Referring to Appendix 11(9.1) if Jo., J1'..., are the boundary Jordan curves

Di,..., and H0, H1,..., those of Z0, Z1,..., then Let H = HN -1, Z = ZN -1. Since Hk c Jo U J1 U ... U of

.Q0,

SNQ0cQ0,wehaveSNJoc.Q0cZ.HenceS(J0U

...

Z and therefore SH c Z. It follows that if W is the closed interior

of SH then W c Z. But the closed interior of Sif is SZ. Thus SZ c Z. Since Z is a closed 2-cell, by Brouwer's fixed point theorem S has a fixed point P (x1, Yi) in Z. Thus if 1' (x(t), y(t))

is the trajectory starting at P at time £0 then x

+ r) =

XI.

y

(t0


289

+ r) = Yi• Since the system (7.2) is now periodic with

period r, the continuation of I' for time t + r, is the arc• of r decribed for t0 t 10 + r. Hence r is closed and x(t), y(t) is periodic. This completes the proof of (15. 1).

As an interesting consequence of (10. 1) one may assert the following:

(15.2) A closed trajectory (periodic solution) cannot surround the closed rectangle R. This holds especially for the autonomous oscillation of (14) and the forced oscillation of (15. 1). 16. We have already established a boundedness result, (13. 1),

for a strictly dissipative system. We shall now prove, under additional restrictive hypotheses, the following much more powerful property. (16.1) CONVERGENCE THEOREM. Let again as in (13. 1)1(x) A

> 0/or all x. By (13. 1) the solution (x(t), x'(t)) of (7. 1) enters a certain square R for large 1. Let in addition g"(x)exist and Ie"(x) be j

bounded in R. Furthermore let g'(x) B> 0 for all x. Then for sufficiently large all solutions x(t), x'(t) converge to one another as t -++ 00. (Cartwright [1] p. 186.) Noteworthy special case: F(x) and g(x) are monotone increasing and g(x) 'is twice continuously differentiable. Then for sufficiently large all solutions converge to one another as I -÷+ 00. (Cartwright bc. cit.) For all solutions enter a closed square in which is bounded

and under the assumptions f(x) A > 0 and g'(x) We also note at once the following.

B> 0.

(16.2) COROLLARY. Let E(t) have the period r, so that there exists a periodic solution x(t) of period r. Then under the conditions of the

theorem all solutions converge to x(t), x'(t). Hence this periodic solution is stable.

Proof of theorem (16.1). Let x(t), x1(t) be any two solutions and set z(I) = x1(t) — x(t), iF = F (x1(t) ) — F (x(t) ) zig

= g (x1(t) ) — g (x(t)).

From (7.1) there follows (16.3)

.z"(t)+1udAF/dt+Ag=0.

290


Multiplying by z'(i) and integrating from T to 1, 1 T

0, we

find:

z"ilF•

1/2 [Z'2]tT + /L[Z'LIF]tT_/4

+ JT I z'zlgdt=O. Upon substituting z" from (16.3) and integrating by parts we find

(16.4)

+ft

z

z

2dtz;

Denoting by 0(1) a function which is bounded in absolute value as I + 00 we have from (16.4) and since z, z', x, x', x1, x'1

are all 0(1): (16.5)

ft

iT

I,

z

2dtzJ

z

And now d g (x + z)

d zig

dtzdt

g(x)

z

x'(t) =—{g (x+z)—g(x)) z ,

—

{g (x +

z) — g(x)

— z g' (x + z)}

=x'(i)g"(x+ Oz)

O'z))

=0(1),

OO, O' 1. Thus say jcl/dt zig/zI 0, for I large. On the other hand

zlF/z ==f(x +

O,i,

zlg/z = g' (x .+

1.

z),

XI.


291

Hence for I large

iJF/z.iig/zAB. It follows that if AB> a, or ti> a/AB, the bracket in (16.5) is positive and O(1) Thus (16.5) yields

fz2d1 = 0(1). It follows that given any e> 0 there exists a T> 0 such that

for any t>T

ftz2dt<e.

(16.6)

(16.7) As a consequence of (16.6) z ÷ 0.

0 and a sequence For in the contrary case there exists that T > We may suppose holds.

Returning now to (16.4) it yields (16.8)

I

t

[z'2 T

=—2

2

Jot

Since z' = 0(1) and x1 —* x, one may take T so large that at the right [. .] I T, and by (16.7) so that the second term < 1/2 e in absolute value. As a consequence .

{z'2]tT <e. Thus z'2(i) tends to a limit as I

oo. Let z'2 -÷ k2

0.

292


For if not there is a sequence t1 small the solution (2.1) will be within a small neighborhood of a circumference—a path of the harmonic system for = 0: where

dx/dt==y,dy/dt——x.

(2.2)

If time be Hence the path will cross the y axis say at (0, counted from that moment then x0 = 0, Yo = and so (2.1) becomes

= A0

(2

I)

+ A1

I) +

•

where the At, At' are analytic in

x=

0,

y=

whatever

1A0 (,j,

(2

0)

For I =

0

we must have

Hence

= 0, •..

,

(,j, 0)

0,

•

A0

time I

is merely the solution through (0, 0, of the harmonic system (2. 2), we have I), A0' (ii, I)

at

XII.

A0

METHODS OF APPROXIMATION

sin 1, A0'

t)

315

= cos I

t)

so tl'at (2.3) reads now

Since the system is autonomous there is no reason to assume that a periodic solution very near the solution sin I, cos t 3.

has exactly the period

Let its period be 2r + r (,u) where, as we shall see, r is analytic in ji. It causes the following indirect attack which is simpler than the general method outlined in (VIII, 9.9). If we substitute x, y from (2.5) in and apply (lU, 10.5), obtain for the solution of (1.2) the relations x

sin I +

(x, y)

sin (I — u) du

y)

cos (t — u) du

0

(3.1)

y = cos t + ,i

where under the integration signs x, y are the solutions expressed as functions of u. Expressing the fact that (3.1) has the period 2ir + r we obtain the basic relations: (3.2)

H

i) =

,7sin

+

j24T +

(3.3)

Tj

(x, y) sin (t —u) du

0

f(x,y)cos(ir—u)du==0.

This is a real analytical system for determining r as functions of u. Let us suppose that (0) = a 0. Then (3.2) yields (0) = 0. It is readily found that the Jacobian

e(H,K) (a,o,o)

0

and so we cannot apply the implicit function theorem to obtain

316


solutions since

('u), r (p)

in the neighborhood of (a, 0, 0). However

(a, o, o)

may solve (3.2) for r as a holomorphic function r in the neighborhood of (a, 0). We find also directly from (3.2) that 0. Hence = 0 for one

(3.4) where the BA (ii) are holomorphic in a certain circular region

It will be convenient to have the value of B1 ('?) later. We find at once U

Now by differentiating (3.2) as to 4u with r = r

ar

8H

I'Zfl+T

Jo

there comes:

f(x,y)sin(r—u)du+Is[ ]=0,

where the value of the square bracket is omitted since it is not needed. Since r 0) = 0, we have (3.5)

=

B1

1 —

I

f

sin u,

cos

u) sin u du.

4. Once r 4u) is known it is substituted in (3.3) which is then replaced by (4.1)

The left-hand side is a series in term independent of Thus K and the solution (4.2)

— a) and

containing no

('1' Is)) = 1sK1 (ii,

(,u) of (4.1) such that

(0)

= a,

will satisfy

XII.


Sit

It also yields r (1u), as analytic in ,u. In order that (4.2) possess a solution of this nature we require

Kj(a,O)=O.

(4.3) Now

K1 (a, 0) =

=

(0K (ti'

r

1u)

/(a,o)

01u

sin u, a cos u) cos u du,

and so we must have (4.4)

(a)

f (a sin u, a cos u) cos u du = 0.

=

Since f is a polynomial so is 0. Thus a must be a real root of a certain algebraic equation. Notice that if a satisfies (4.4) so does —a. Therefore 0 (a) = a polynomial. Thus if 0 has any real root a a1 W (a2), with it also has the positive root lal. It will therefore be sufficient to confine our attention in the sequel to the positive roots of 0 (a).

5. Choosing now a definite positive root a of 0, we may expand K1 (ii'

(5.1)

K1

as a power series in — a and 4u and we will have

=

A (a)

—

a)

+ B (a) 1u +

and to discuss (4.2) and stability we require information about A (a). We have (5.2)

=

A (a) =

(OK

r (ii, 077

Now —

=

+

OK(77,4a,r)dr 3r

318


axf9T

+ ayeT\

— i— + — — Hence

.

— u) —f (x, y) sin (T — u)

cos

dii.

from (5.2) and since by (3.4) — = 0 for 1u = 0: A (a)

+

—

ayaa

U du,

/

where under the integration sign f = (x, y), and x

y = a cos ii. Therefore finally (5.3)

A

a sin U,

(a) =

now any simple positive root a of we suppose that (4.4) holds and that Consider

that is to say

(5.4)

and we have the corresponding development (5.1). Under the circumstances in view of (5.4), the implicit function theorem asserts that (4.2) has a unique analytic solution (p) in the neighborhood of = 0, such that (0) = a. Substituting then in (3.4) we obtain a similar series r ('u) such that r (0) = 0. The two functions ('u), r (1u) verily (3.2), (3.3) and hence the corresponding solution (x (1u), t) ), y Cu), 4u, t) ) represents a closed path of (1 .2) which tends to the circle Aa: x2 + y2 = a2 when —÷ 0. We may also say that x (u), 1) represents the oscillatory solution of (1. 1) which tends to a sin I when —. 0. In other words under the circumstances there is a unique periodic

solution of the desired type. The circle

is known as the

generating circle of the closed path. We have tacitly assumed the solutions to be real. It is only necessary to observe that the determination of the coefficients of the series involved never makes an appeal to any irrational operations. Hence the coefficients are all real and so are the series.

There remains to discuss the question of stability. Let yr.,.

XII.


319

be the path of (1; 2) passing through (0, and let us assume a and small. Under the circumstances the solution y,,, at a time will cross the positive axis at, a point (0, + 1u) is the solution (3.4) of H r 4u), where T a, t) =0. Let

L

=

ô

We have at once F, Q

=

A(a)

are

=

,j.

holomorphic in the neighborhood of (a, 0) and W(a)

0.

relative to the closed behavior of We are interested in i.e., when is arbitrarily path when the latter is very near small. At all events let us choose which is assumed positive — a)2. Then for — a) sufficiently small so that 0 0.

cos t, The fu,nction n> 0, is a polynomial in A0, . .., A, sin i, cos t; 1) one in the five sin is a polynomial in

variables, and A0

cos t + sin t.


322

7. The solution (6.3) just obtained is a general solution of (6.2) with the independent parameters Suppose now that we seek a solution of period

or a so-called harmonic solution.

We must have then

=y(O).

x(2n) =x(O),

(7.1)

Upon substituting the solution (6.3) in (7. 1) and setting for simplicity

=

=

'i)'

n> 0,

one obtains a system B1

H(E, ,j,

(7 2)

ij)

= C1

+ ... = 0,

+ 4u B2

+

C2

'j)

+ ... = 0.

This is a so-called "algebroid" system and there is no difficulty in discussing its solution completely (see VIII, § 8). However, this would scarcely be profitable here and we shall confine our attention to the following special case :—The algebraic curves C1 plane intersect in points such B1 'i) = 0 of the that if '1° is one of them 8 (Fl, K)

(7 3)

8

— 0 (B1, C1) 0

o

'io)

This makes it possible to apply the implicit function theorem to

(7.2) and obtain a solution

—=

(7.4) g

h

and g(0)

h(0) =

0.

Correspondingly the general solution (6.3) of (6.2) will be periodic.

(7.5) Sub harmonics. Suppose that f has actually the period 2,r/n, n> 1. This will be the case for instance if f is a polynomial

in cos nt and sin nt. Then the same procedure that we have i.e., used will continue to yield a solution with the period n times the period of the forcing term. This solution is then in an evident sense a subharmonic.

XII.

323


8. Stability of the periodic solution. Let us consider the stability of the periodic solution corresponding to (7.4). We cannot use the method of (VIII, 11) since the matrix P there considered has pure imaginary characteristic roots. What we shall do really is to apply a variant of Poincaré's method of sections (VIII, 5). If one starts at a point P at time t = 0 near F and reaches at time t = a point F' (a', ii') then the periodic solution will be orbitally stable if the ratio of distances PF/PF

is < 1 and unstable if it is > 1. Now the coordinates of P' are x we have (8 1) 1

x

—

=—+

y

—

— — + ILC1

Since

correspond

and

y

'i) +

+

to a periodic solution

=

x

y

,u,

=

Hence

+ •.. = 0, + ... = 0,

+ + ,.i2C2

1uC1

and therefore (8. 1) may be written

+...,

(8 2)

Let

.

=

. .,

denote the partials of B1,

= b', C1E0

. .

+ ., as to i,... and set

c,

...,

Finally to indicate the fact that terms in 4u2, (J dropped we shall replace = by We then have B1

=—

ij) —B1

Recalling now that —B1

B1

+

+

—

—

O(p),

—

=— B1e0,,0

Bie* ± (fl —

B11,..

— = O(1u), we have: +

+ O(u2)) + (?7 —

+ (p7* —

are

— {B1,,0

B1,,02 + 0(u2)),

324

DIFFERENTIAL EQUATIONS: GEOMETRiC THEORY

and similarly with C1 in place of B1. Hence (8.2) yields ii, p, 2,i) —

(8 3)

r

(1 + 1ub)

+ 1zb' thj,

+ (1 +

If the roots of (8.4) /AC

,

both < 1 in absolute value the periodic solution under consideration is orbitally asymptotically stable; if unity separates are

the absolute values the periodic solution is orbitally conditionally

stable; if the absolute values are both greater than one it is unstable. For this comparison one may clearly replace, for smafl, (8.4) by (8.5)

If we set

=0.

Pc r

s then s satisfies

P

lb—s. b' ,C —S C

(8.6)

=0,

which is the characteristic equation of the Jacobian matrix (B1, C1) / 1, 2 those Let s1, s2 be the roots of (8.6) and r5 = 1 + s5, j of (8.5). Suppose first s1 and s2 real and of the same sign. Since

= — (r2 — 1), r1 and r2 are on the same side of 1. ifs1 and s2 are both negative Since ('i — + — = + 1)

s1

1)

1)

s1

[positive] and r2 are both < 1 [> 1] and the periodic solution is orbitally asymptotically stable [is unstable]. ifi. a Suppose now s1 and s2 complex: s1 = a + 1 + 2p a + 0(u2). Hence 1r212 = (1 + p a)2 + p2 fl2 Then

XII.

325


if a < 0 [> 0] we have

1 [> 1] and so the periodic

fr'21
1: Teft =

afte1

+ 5>1

Thus

,0 \Qh, bft2,

.

•,

349

APPENDIX 1

Hence B

= (bhJ) has the characteristic roots

.

is a matrix of a transformation on the

. .,

Now B

1 spanned by

By the hypothesis of the induction it may be reduced to the forip (2. la) and this implies (2. 1). e2, ..

., en.

Observe that the characteristic roots are merely the values of A which make T —2 I singular. Let in particular have the multiplicity k, i.e., it is repeated k times among the characteristic roots. (2.2) There is a direct sum

Q3ii

=

has the sole characteristic root where: (a) (b) has the remaining charac(and therefore to multiplicity k); (c)

teristic roots each taken with its

In the reduction of A to the form (2. Ia) one may suppose that the first k roots 25 are repeated k times. Thus the space spanned by e1, . .., is already such that = and repeated k times. has for characteristic roots = and take ek +1* = Let now Tek +1 = Ak +1 ek +1 + 'i, n e Now Thus ek +1* is not in et +1 +

Tek+j*=Ak+lek+1+fl± T1C 2k+1 C + n + TiC. = Ak+i Since Ak ÷ i is not a characteristic root of T1, (T1 — Ak +1 I) is non-singular. One may then choose C = —(T1 — Ak +1 1)—i

2k + 1 ek + 1*. We take ek +1* and as a consequence Tek + as new none Suppose that we have already obtained ek +1, ..., e5 is the vector space which they span, and such that if in — then T9.9, — — i. Then Tej + C + A5e5, Take e,* = e5 + 0, Thus e,* is not in Q31 1. And now

Ajej*

+ (Ti—251) 0 + +

and as before we can choose 0 such that the two middle terms the space spanned disappear. Then Te;* Ajej* + CE — 1

has its obvious meaning, then = Pursuing this we will arrive at = and behave in accordance with (2.2).

by e,*. Hence if we take e,* as new e5 and


350

denote this time the distinct characteristic roots and let kh be the multiplicity of Aft. By applying (2.2) to etc, or else by an evident induction we obtain: Let

.

. .,

A,.

s...

(2. 3) There is a direct sum decomposition = has the sole characteristic root Ah (and therefore to where Tft = 0 we have = multiplicity kh). Moreover since

As we have seen (2) if Aft is the block of Tft then the initial matrix A diag (A1, . . ., Ar). Hence to prove (1, 8.4) it is Aft, i.e., for a matrix sufficient to prove it for the triple Tft, 3.

with a single characteristic root A 0. Let then the initial matrix A already have that property. It is not ruled out that T may still be reducible. Let it have associated subspaces is with blocks A1, . .., A8 where this time irreducible. Once more (1, 8.4) will follow if we prove it for i.e., for an irreducible T.

Suppose then that T is irreducible with sole characteristic 0. Let us write eft' = Teft. We may select a base such that e1' = 2e1. Then for h> 1:

root 2

eh=ahlel+ ... the ajg cannot all be zero since T would then be reducible. Let us range e2, e3, ..., in such order that a21 0. Upon reand

placing e2 by e2/a21 the situation will be the same save that a21

1.

e2 the situation will still Replacing then ek, k> 2, by ek— be the same save that every akl = 0 for k> 2. Thus the matrix A assumes now the form

20

30

1, a22

(3.1)

One

0,a32

may consider the matrix

j, k = 2, .. ., n spanned by as matrix of a transformation T* of the space Since its characteristic roots are also characteristic A*

= (ajft);

351

APPENDIX i

roots of (3.1) it has 2 as only such root. One may now choose a base e,, . . ., for such that a22 = 2, a2h = 0 for It> 2. Repeating the same reasoning we may choose the base so that a32 = 1, a3h = 0 for It> 3, etc. In the last analysis a base will be obtained such that (3.1) assumes the normal form

/2, o (3.2)

C(2)==(

:

:

\.

.

:

:

: .

.

To complete the proof of theorem (I, 8.4) when aji the charac-

teristic roots are non-zero there remains to show that the direct sum decomposition (1.3) is unique. Suppose that Then

=

fl

fl

is irreducible one of the terms at the right is and the others are zero. If say = .... 0, n s,', n then = This together with an evident induction proves that the decomposition is unique and that (I, 8:4) holds. Suppose now that zero is a characteristic root of A. Take = T —21, where 2 is not a characteristic root of A. The matrix of T1 is A1 = A —2 E whose characteristic roots —2 are all 0. Hence there is a choice of base e such that C1 = is in normal form. The corresponding matrix C for T is C = C1 +2 E = P (A1 + 2 E) P' = PAP-' A and it is also in Since

normal form. Thus (I, 8.4) holds also in the present case, and its proof is completed. (3.3) Remark. Let a be any number 0. Then one may modify the reduction process so as to replace the block C(2) by a block

2,0

C(2,a)=

:

:

:

:

,a,2 All that is necessary for instance at the beginning is to replace


352 e2

by ae2/a21. The remaining modifications required are obvious.

Evidently also a may vary from block to block. Thus when A is real for blocks corresponding to pairs of conjugate roots A, I It is simpler, however, one may choose conjugate pairs a, 0, the same for all the blocks, and to take just one real a this will suffice for the applications in the text.

§ 2. Normal Form for Real Matrices 4. We proceed now to the proof of the complement (1, 9. 1) for real transformations of real spaces. Thus there is a real base e

relative to which the transformation T has a real matrix A

and Te5 =

api,

ek.

It follows that real vectors have real transforms. However a complex vector

f=

x5 complex,

has a conjugate and now

Tf —

x,ajkek;

Tf

xjajkek.

It follows that if instead of e we have a complex base e* then Tej* =

bp

Tèj*

=

Thus the matrix of transformation of

èk*.

is the conjugate of that

5. For convenience let the distinct characteristic roots A5 be and that ranged in such an order that A2 k +2 = 12 +1, k A2

p +1, •. ., Aq are

real. Thus the first

are the p

pairs of

complex conjugate roots. Take also the decomposition

= where

...

is associated with Ah. Observe that the

+5 are all

real. For in the argument leading to (2. la) if one makes "real

853

APPENDIX 1

transformation" part of the induction process, when is real then the associated space is also real. Let Ah be the matrix of When is real the ehi and {ehl, . . ., eh(jj a base for may then likewise be chosen real. We may assume also that A = diag (A1, . .., At,) where A is the matrix of T. has its obvious Notice now that the matrix Ak — meaning), in normal form and hence in any form, is nilpotent when and only when 2 2h and non-singular otherwise. Hence (5.1) for every v.

(T —

Let us set

=

'84 (fle'??

-.-

CV)

4'

CV)

is a base for that there is a non-trivial relation

Observe that {e2k —

(5.3) E

i

;}, k

e2k_1,J +

192k—1,J e2k-_1,J +

Suppose now e2,+h,J=0.

Let also

... Owing to (5.1), (5.2) upon applying for some v to (5.3) the first two sums are annulled while if the third is non-trivial it is

transformed into a non-trivial relation between the e2, + h, j. Since they are linearly independent every + h.j 0. Applying now f'(x) for some v the first sum is annulled and if the second sum is non-trivial it becomes a non-trivial relation 2k — 1, 1 e2 k — 1,

1 = 0.

This implies a non-trivial relation EÔ2

e2k.....1,5

0,

which is ruled out since the e2 k ...i, form a base for *Y. Hence


=

Thus (5.3) reduces to its first sum and so by the remark JUSt made every 0. Thus (5.3) is 1, 5 every flz k

0.

trivial. Since the total number of elements ehk in the three sums in (5.3) is the dimension of these elements make up a base for We have then a direct sum decomposition =-

into irreducible subspaces

Consider now a decomposition of

=

..

. .

may suppose the e2 Ic — 1, j so chosen that they consist of The (h fixed) span the irrea set of bases {ehj'} for the and we have of ducible subspace One

pairs together with the real irremake up a decomducible subspaces of a decomposition of position of whose subspaces behave in accordance with (1,9.1). 2h' consist of conjugate of Since the bases {ehj}, elements, referring to (4) the associated blocks in the final transformation matrix C are likewise conjugate. Since those associated are manifestly real, the with the irreducible subspaces of proof 01(1, 9. 1) is completed.

Thus the

§

3.

Normal Form of the Inverse of a Matrix

6. As an application of the preceding considerations let us prove the following property which is required in the text: (6.1) If a non-singular matrix A diag (C1 (As), . . ., C, is of the type of (3.2),

where C,.

A'

diag (C1 (ic'),

).

Let Vh, Tft correspond as before to CA, so that both are irre-

ducible. Thus in association with T, where Th =

we

have

(6.2)

It follows that T

Hence

also T'

=

Thus the

355

APPENDIX I

direct sum decomposition (4.2) corresponds also to T1 and = Th1. From this follows that it is sufficient to prove (6.1) for r = 1, or that: (6.3) If A C(A), A 0, then A—' need to show that if A 0 is the unique characThus we A teristic root of A then of order n. We recall that RI is the R but with its diagonal of units moved j steps down. Hence = 0. Now C(A) = A E + R. Hence if

+ ... +

D(A)

then C(A) D(A) = E, and hence D(A) = C—'(A). But D(2) is merely

a matrix with 1/2 along the main diagonal and zeros above. Thus D(2) has for sole characteristic root 1/2. We already know that This it is irreducible and so D(2) = C'(A) C (2—') '.. proves (6.3) and hence also (6. 1).

DeterminatIons of log A 7. Let A be a non-singular matrix and let B be a matrix defined by A = ek If B is reduced to normal form: B diag (C .. ., C(A,.)), then. A is reduced to diag (eC(ai), . . ., e°('r)). Let n be the Thus we may suppose that B = C(A), A = §

4.

common order of the matrices A, B and let C(A) =

that sueD = 4u

=

I

0.

+

Thus if 2 = D

log

then A =

+ ... + (

A — E is known and is nilpotent:

eD

AE

=

+ D so eD =

Now

the matrix F =

=

The relation

0.

F=D/1!+... may be inverted as if F, D were analytical variables and it yields

Fe—'

F2

Since 2=log1uwehave F2

logA=(log1u + 2kni)E+F_T+...

____ DIFFERENTIAL EQUATIONS: GEOMETRIC THEORY

356

Returning to the general case if B is as above with and C(2A) is of order

and

= log

— Ek then finally

if Fh =

+

logA 2

The main observation to be made is that each corresponding where to every block of A is only known up to a multiple hA may vary from block to block, even though the corresponding IUA

remains the same.

A Certain Matrix Equation. 8. In the text (VI, '7) the solution of a certain partial differential equation has been reduced to the solution in B of a matrix system §

5

B'=B

(8.1)

A'B+BA=eC

under the following assumptions: = ± 1, sA is a known stable matrix of order n, and C> 0 (notations of VI, 7). It is asserted that under these conditions (8.1) has a unique solution B and B> 0. This is what we propose &to prove. Two preliminary properties are required. I. A transformation of coordinates does not affect the form

(8.1).

Let P be the (non-singular) matrix of the

transformation. Using the vector and quadratic form notations of (VI, 7), a quadratic form x'Dx will become, under x Px, a quadratic form x'P'DPx, so that the effect of the transformation on the quadratic form is expressed by D —* P'DP. On the other hand the effect on the basic differential equation dx

is B*

by A

P'AP. Hence let A

= P'BP, = P'CP. Upon multiplying both sides of (8.1) by P' to the left and by P to the right we obtain

357

APPENDIX I

P'A'BP + P'BAP =

eP'CP

=

eC*

or

P'A'(P-')'P'BP + P'BPP-'AP which is

A*'B* + B*A* = sC*. That is, the form of (8. 1) has not been affected by the transformation of coordinates. Property II. The characteristic roots of —A are the negatives of those of A. In fact if jA — =1(2), then 1—A — 2E1 = from which our assertion follows. — IA—(—2E)I —f(----,2.) 9. Returning now to our problem we must first distinguish two cases: I. The normal form of the matrix A is diagonal. Thus if A. are Choose the characteristic roots then A diag ., coordinates such that A is normal. Upon writing (8.1) as a system of linear equations between the terms of the matrices we obtain . .

(9.1)

+

=

ecu.

Since the have real parts of the sign of e, the 2h + A, are all 0 and so (8.2), hence also (9.1), has a solution and it is unique

in the case under consideration. II. The normal form of the matrix A is not diagonal. Referring to (3.3) we may assume that A is in a normal form with the A, in the main diagonal and underneath it a subdiagonal consisting (in fixed places) of a number a 0 and of zeros. Upon writing

again the analogue of (8.2) for the b2, there results a linear system with a determinant which is a polynomial LI (a) in a. Now LI (0) corresponds to the system with A = d.iag (2k,. . ., 0. If we take 0. Hence 4 (a) By the preceding result LI (0)

for a some value not a root of the polynomial we will have 0. Hence the system (8.1) will again have a unique LI (a) solution. Observe that if a0 is a root of LI (a), then LI (a0)

0, and so, seemingly, we have a system with a non-unique solution. This contradiction proves that LI (a) has no roots, i.e. LI (a) = 4 (0)

is just a constant

0.

358


10. There remains to show that the unique matrix solution B of (8.1) is> 0. Suppose first that A is stable. Then the unique

matrix solution of (8.1) is given by B

(10. 1)

=,ff

CeAtdt.

One must first prove that B exists. Since A is stable so is A'. Now e" being a solution of dxfdt

—

Ax

its terms and those of the integrand are finite sums of expressions of the form H (t) = (cos fit + I sin fit) g (i), where g is a polynomial, and a + ifi is a characteristic root of A. Hence

f H (1) exists and so does (10.1).

We must now prove that B satisfies (5.1). We have (by

integration by parts) (10.2)

A'B =

f

= —c — BA,

which is (8.1) in the present instance. Let now —A be stable. Then and this is (8.1) for the present case. Thus the property asserted in (8) is now completely proved.

§ 6. Another Matrix Problem. 11. We propose to prove two properties of real matrices utilized in (VI, 7) and to be described presently. Let be

. .

.,

the characteristic roots of a real non-singular matrix A.

Suppose that the real parts of Ar,. ., Z, are positive while those of ., are negative. On the strength of the .

reduction to a normal form we have A

d.iag (A1, A2) where

359

APPENDIX I

A has for characteristic roots, Ar,. ., A, and A2, the characterWe wish to prove: istic roots . ., (a) One may choose A, and A2 real. (b) The transformation matrix P from A to diag (A,, A 2) may also be chosen real. is likewise among the first is complex, then If 2h' h p characteristic roots. Assuming first A in normal form and writing A for 2h' if the block C (A) is in A1 so is C (X). Let be the corresponding conjugate vector subspaces such as in (1), and let eh be two conjugate elements of the associated .

= I — e4) are real and bases. Then éh = ek + and we may replace in the joint bases for the element pair by the pair e"A. Proceeding thus, one will arrive at a real base for the space and hence this space is real. Upon treating all the blocks for A, in the same manner, we will have, say A1

diag (B,,

..

., B1)

where the B, are all real. A similar treatment applies of course to A2. Hence (a) follows. Regarding (b) it is sufficient to prove (c) If A B and both matrices are real there is a real transformation matrix P such Mat B = PAP-'. as unknown. The relation BP = PA is Consider P = equivalent to a real linear homogeneous system S of relations

between the p,,. Now all the solutions of S are linear combinations of a finite real system of solutions ., P'. Take a Its determinant D (c) is a form general solution P (c) = 0 of degree is in the ch and this form 0 since there is a P (c)

I

for some values of the c,. If we apply a real linear transformation U to c1,. . ., the result is the same as changing the One may choose in particular set of independent matrices a transformation U such that in the new D (c) the coefficient 0. For in the contrary of the term of highest degree in c, be case D (c) 0 for all real ch and therefore D (c) = 0, which is not the case. Suppose then that the highest coefficient in c, 0. As a consequence D (1, 0,. . ., 0) 0 alone in D (c) be and so there exists a real non-singular matrix P. This proves (c) and hence also (b).

APPENDIX II

Some Topological Complements

In the present appendix we shall give more complete topological

details regarding a number of questions considered in the text, more particularly in Chs. IX and XI. We shall especially develop the properties of the index.

§ 1. The Index In the Plane 1. For a proper development of the theory of the index one requires the freedom offered by a deformation of a Jordan curve. This is best provided through the concept of circuit. A circuit fin a space 91 is a pair (f, J) consisting of an oriented Jordan curve J and of a mapping f: J -÷ 81. One agrees that if J' is a second Jordan curve and a topological orientation preserving mapping J' -÷ J then the circuit (f J') is still r. Notice that J is the circuit (1, J). We also denote by f for short

the set fJ. Let three arcs 2, ,u, v form a 0 curve (curve lUce a 0) with common end points a, b and let them be oriented from a to b. ThusJ1 = 2—v,J2 = oriented + v,J = Jordan curves. Let f map the figure into 81, producing circuits 1'l, j'2,

I. Then one defines f =

+ j'2,

and similarly

be two circuits in 91. Consider the cylinder J x 1, where 1 is the segment 0 u 1. [1 there exists a mapping Jxl-÷81 such that then in 91 and is a The and f1 are said to be J x 1) is the cylinder and if M€J then 0(1 x M) pair is the jath of M in the homotopy. If I'0 = (1, J) then the homoLet 1's,

1'1

360

APPENDIX II

361

topy is known as a deformation. If is is a point one says: Thus: homotopic to or cieformable into a point in (1.1) A Jordan curve in a 2-cell E2 is deformable into a point of E2. W e have an e-homotopy or s-deformation whenever every path

i s of diameter <e. Evidently: (1. 2) Take two circuits in

=

(fo, j), I',

= (f1, J). Suppose may be joined by a segment in 1)1 varying continuously and which reduces to the points them-

that w hatever MEJ the points

selves when they coincide. Then F0 and F1 are homotopic in

2. The preceding con siderations may be applied to the orientation of an Eucidean p lane H and of its Jordan curves. Generally 11 is oriented by assig ning a concordant sense to all its circles. Let J be a Jordan cur ye in El and Q its interior. Take a ference C contained in Si and draw the Jordan curve H sufficiently clearly indicated by th e construction of Fig. 1. One orients now J so that in the resulti ng orientation for C this crcumference is described negatively. T his process yields a unique positive orien-

tation for J. Convers ely given a preassigned orientation of J specified as positive t he process yields a positive orientation for C and hence for 17. The orienting curve J is called an indicatrix of the plane. The process is also applicable: (a)To a 2-cell One maps E2 topologically Ofl ITo riented and assigns to the Jordan curves of

Fig. 1

362


E2 the orientation specified by that of their images in E. (b) To a differentiable orientable manifold M2. One orients directly the Jordan curves in one cell of the basic covering of M2 and then step by step those of the other cells, operating from cell to overlapping cell.

defined at all points of a circuit 3. Consider now a field J) in the Eudidean plane 11 and free from critical points P = (f, on r. As MEJ describes J once the angle of the vector V(M)

at the point fM with a fixed direction varies by 'ii.

The

number m is the index of P relative to written Index (F, or merely Index P. In particular one may define Index (J, (3.1) Given two fields

defined

and free from critical points

then on r, if their vectors V(M), V'(M) are never in = Index (1', a'). Index (F, of both vectors. Since Let a be the greatest angle, a their angular variations differ by less than 2x. Since it is aJ m = 0, hence (3.1) follows. On the other hand evidently: in

(3.2) If V(M), V'(M) are always in opposition then Index (1', Index (F,

=

The central property of the index of circuits is: (3.3) Let be a field defined in a plane region £2 and free from critical points in £2. If two circuits F0, F1 are homotopic in £2 then they have the same index to

Let 0, u, 1, J have the same meaning as in (1), where now 0(1 x J) cQ. Let the segment 0u 1 be decomposed into n equal parts by the subdivision points 1/n, 2/n Let denote the circuit (0, h/n x J). If M e J let be the field defined on fh by transferring to h/n x M the vector of at (Ii + 1/n) x M. = Index (J'h +1, However, owing to Clearly Index (I'h, the continuity of 0 and of the field on the compact set 0(1 x J), and since the latter is free from critical points, for n sufficiently

large, the vectors of

at h/n x M and (h + 1/n) x M for all

MEJ and all h, may be made to make an arbitrarily small largest will never be in opposition on any angle. Hence and

Applying now (3.1) we have Index (Ph, Hence for any h: Index (1", Index (I'd, which proves (3.3).

=

Index

(P1k +

=

1,

Index (Ph,

and therefore Index (F1,

363

APPENDIX ii

Applications. (3.4) If the field

is defined over a closed twoand is free from critical points in £2, then the index of every cell circuit r in £2 is zero.

This follows at once from the fact that 1' is homotopic to a point in Q. (3.5) THE BROUWER FIXED POINT THEOREM. Every

of a closed two-cell into itself has at least one fixed point. Take as the closed two-cell a closed circular region Let C be the boundary circumference of and 0 the center of C. If and M' = M then the directed is the mapping, M a point of

segment V(M) = MM' defines a vector distribution on whose critical points are the fixed points of Suppose that q has no fixed point on C. Then V(M)

0 on C. Let it be replaced

on C by V'(M) = MO, thus resulting in a new field

on C.

Since V(M) and V'(M) are never in opposition on C, Index (C, = Index (C, W) = 1• Hence must have a critical point (3.4) and 9? must have a fixed point.

As a last property of the index of circuits we have evidently: where the circuits are in 11. Let the field (3.6) Let = I'1 + be defined and be free from critical points on and Then it behaves likewise relative to I' and Index

=

Index I',

+ Index

Let a field be defined in a plane region £2 and let A be be a closed 2-cell in £2 containing A and a point of £2. Let suppose that in the field has no other critical point than possibly A. Draw in a positive convex curve C surrounding A. if C' is another such curve then C and C' are homotopic in — A. Thus Index (C, Index (C', Hence Index (C, is the same for all convex curves such as C surrounding A in its value, clearly independent of is by definition the index of the point A relative to the field written Index (A, or merely Index A. Let J be a positively oriented Jordan curve in Its interior is then in Let A€1R1 and let C be a circumference of center A contained in Let H be the composite Jordan curve in 4.

heavy lines in Fig. 1. The interior of H is in — A. Hence Index (H, = 0. The construction can be so carried out that Index H — Index C + Index is arbitrarily small. Since the expression is an integer it is zero. Hence Index J = Index C

364


Index A. Thus the latter may be obtained from any positively oriented Jordan curve surrounding A and no other critical point, as previously from a small convex curve surrounding the point. Notice that if one reverses the orientation of the plane ft one must also reverse that of J, and so its index, hence that of A, is unchanged. Thus: (4.1) The index of a point is independent of the orientation of the plane. (4.2) If £2 is a differentiable cell one may treat it in the same

to H and again its orientation (by manner without means of its Jordan curves) does not affect the indices of the points. Referring to (2.4) we also have at once: (4.3) The index of a non-critical point is zero. The central property of the indices of points is the following, which recalls the classical residue theorem of complex variables: (4.4) THEOREM. Let Q be the closed interior of a Jordan curve in the plane H oriented concordantly with H and a field defined over Q, free from critical points on J and having at most a finite number A1, . . ., in Q. Then

Index j

E Index

Make the construction indicated in Fig. 2. As above Index H = One may carry out the scheme so that

llndex J — Index H — E Index A1f < 1/2.

Fig. 2

0.

APPENDIX II

365

Since this expression must be an integer, it is zero and this yields (4.4). (4.5) Remark. Although our treatment of the index is tinged with considerable topological flavor, it does not follow that the concept as defmed is topological. Indeed this cannot be the case, since vectors, a non-topological notion, have played an essential role in the definitions. The following mode of defining say the index of a plane Jordan

curve J is more topological and may also be readily extended to higher dimensions: Take a circumference C in the plane of

J and from the center 0 of C draw a ray M' parallel to and sensed like the vector V(M) at M€J. Then M —+ M' defines a mapping J —+ C whose degree, in the sense of Brouwer (see The identification with the Lefschetz [1], p. 124) is Index (J, previous definition is immediate. Since the degree can be defined for any dimension, this new formulation carries over to higher

dimensions. In particular by means of a "higher dimensional index" one may prove the Brouwer fixed point theorem for any closed cell.

5. The following proposition has been proved by Jaime Lifshitz [1]:

(5.1) Let J be a planar positively oriented Jordan curve. Let T be a topological transformation of J into itself which is free from fixed points. If M is a point of J then M' = TM M and so the vector distribution given by V(M) = MM' is defined and free

from critical /'oints on J. The assertion is that the index of this distribution is unity. It is clear that if J is an oriented closed path for a differential system such as (VIII, 1. 1) then as limiting case (5. 1) yields the Poincaré index theorem (VIII, 11.1), and indeed this is the main

merit of (5.1). In outline the proof runs as follows. A polygon K1 with sufficiently small sides is first inscribed in J. It may have multiple crossings but these are readily suppressed reducing K1 to a simple closed polygon K. The polygon is so constructed that there is a point P interior to both J and K. An e-homotopy J —+ K

in 17— P is established and a topological transformation T1: K K is defined with vector distribution analogous to One shows that Index (J, = Index (K, The second index

366


is, however, readily computed and found to be unity, which proves (5.1). §

2.

The Index of a Surface

6. We have had occasion to utilize the index of a sphere or a projective plane in (IX). Both are given by a general theorem on the index of a surface due to Poincaré ([2], p. 125) and which we shall prove in the present section. His proof based on his definition of the index (see IX, 7.4) was only given for orientable surfaces but the extension to any surface offers no difficulty. Let us state that the term "surface" in the present context will stand for "compact differentiable two-dimensional manifold." Let 1) be a surface. There is no difficulty in defining a field over 0 (See I, 15.4). Let us suppose that has at most a finite number of critical points A1, . ., A1. Let {Uh) be the finite open covering by differentiable 2-cells which serves to define 0 and let U5 contain Ah. If x, y are the parameters serving to define U5 then determines a field y) over the region Uk of the Euclidean plane of the coordinates x, y. This field has Ak as an isolated critical point and it has then an associated index: .

Index (Ak, The argument of (4) shows that if Ak

Uk and one defines the

index through the medium of Uk, its value is the same as before. This value is now taken as Index (Ak, and we define = E Index (Ak, Index (0, (6.1) P0INcARE's INDEX THEOREM. The index of a surface relative to any field with at most a finite number of critical points, is independent of the field and equal to the Euler-P oincaré characteristic x(0) of the surface.

7. We will first treat the case where 0 is a sphere. The tetrahedron is a suitable triangulation in which there are a0 =

4

vertices, a1 =

6

Hence z(0) = a0 — a1 + a2 =

edges, a2 = 2.

(7.1) The index of a vector field points, on the sphere, is two.

4

triangles.

We thus have to prove with a finite number of critical

APPENDIX Ii

367

Fig. 3

Let us suppose first that the sphere

is Euclidean. We follow

now Gomory. Take a non-critical point A on

and draw a

circle C of center A so small that within C the angular variation is very small. In a certain circular region of center A containing C we may modify the vectors of and make them parallel. As a consequence they will be tangent to the circle at two points F, Q. We now join P to Q by two arcs without contact 2, 1u in C disposed as in the figure. Let J = Then Index =

Index J relative to the outside of J, i.e., to the component of S — J which does not contain the point A. This outside com-

ponent 4 together with its boundary has the appearance of

Fig. 4. Join P to Q by an arc v of great circle outside C and hence in A and complete the distribution as indicated in Fig. 4. There

Fig. 4

368


result two 2-cells with vectors pointing inward on one and outward on the other, and so each with index one. Hence, the sum is two and this is Index 0 as asserted. If the sphere 0 is merely differentiable the reasoning is the same save that PQ is merely a differentiable arc orthogonal to the vectors at P and Q and that the vectors alongv are normal to v.

Let now 0 be a pro jective plane. Let I? be the closed circular

region such that by identifying the diametral points of the boundary circumference I' one obtains 0. Let Q be considered as one half of a two sided disk A whose two faces Q and Q1 are matched along 1'. If is a field on 0 one obtains by natural on A which has twice the number of extension across r a field critical points of

each

going into two points of the same nature

on 4. Therefore as A is a sphere Index 0 = On the other hand x (0)

1/2

(4 — r) + x(I')

1/2

Index 4 =

1.

= 1.

Hence (6. 1) holds also for the projective plane.

8. To complete the proof of the Poincaré theorem we must consider surfaces other than the sphere and the projective plane.

In this connection we will liberalize the term "surface" by omitting the condition of differentiability in the definition. This

is merely done to "speed" up the treatment that follows and that is based on a most ingenious argtrment due to Gomory. It may be observed that only spheres and projective planes occur

in the text: the proof for other surfaces is thus a topological luxury.

Suppose first that 0 is an orientable surface of genus

0

with at most a finite number of critical points. Take as canonical model of 0 a planar double faced disk with carrying a field

p holes. (See Lefschetz [1] p. 83). Let J be the Jordan curve separating the two faces of the disk and ..., Jp the borders

of the holes. Thus, in the plane, J surrounds the and the latter are mutually exterior to one another (Fig. 5). Let us now make for each hole two incisions such as shown in the figure, taking care that between the two incisions for each hole there

369

APPENDIX II

5

Let the borders of all the be the surface outside incisions all be contracted to points. Let

are no critical points of the field

is still a surface, and is in effect a sphere since it is a double faced disk bounded by a single Jordan curve. Let be the field on On the part common to The two outside coincides with and incision borders say for the k - tb hole yield two critical points Bk', Bk" of The two inside incision borders and the part which has two and between make up a sphere 5,., with field only two critical points C,.,' corresponding to the two incisions. Evidently

of all incisions, it is readily shown that

Index (Bk',

= Index (Ck",

Index (BA,',

Index (CA,',

Hence Index (BA,',

+

Index (B,.,",

= 2.

Thus

2 = Index

+ and therefore

(Index (BA,',

= Index (.P,

+ Index (BA,',

370


(See Lefschetz [1], p. 82). This proves the theorem for an orientable surface. Let now P be a non-orientable surface carrying the same type of field as before. Choose for 0 the canonical model consisting of a sphere S with q holes along whose borders one has matched the boundaries of q Möbius strips W1, ..., Let the median

line Lk of the strip Wk be drawn in such a way that it passes through no critical point of the field Since one may replace Wk by an arbitrary narrow neighborhood of Lk, one may assume

We will now make that Wk contains no critical point of incisions along the borders rk of the Wk and reduce these to

points. The remaining surface is again seen to be a sphere. will coincide with on on the portion The new field and to 0, and in place of I'k it will acquire a common to critical point Bk. The Möbius strip Wk will have become a closed = 1. Hence surface Wk and a ready calculation shows that on Wk is a projective plane. The new field !Pk has a single critical point C corresponding to Bk and with equal index. Thus

1 = Index (Ck, 1 = Index (0k,

=

Index (0,

+

Index (Bk,

Therefore

Index (0, F) = 1 — q and this completes the proof of Poincaré's theorem. A

§

Property of Planar Jordan Curves

9. In the present section we propose to prove the following property utilized in the text (XI, 15.1): be Jordan curves in the Euclidean + intersect for be the interior of Ji, If

(9.1) THEOREM. Let J1,.

fi and let

. .

1=1,2, .. .,n— 1, then the infinite comf.onent V of 17—(J1u U

U...UJn.

J contained in J1

The basic part of the proof will follow the one contributed

by Floyd to Cartwright [1] (p. 175) but some preliminaries are required.

APPENDIX II

371

V

Fig. 6 10. (10. 1) Let J be a Jordan curve in the j.lane 17, U the interior

of J, A an arc joining in U two distinct points P, Q of J. Then A divides U into two distinct regions with the common boundary A in U.

Let "2 be the two arcs of J with the common end-points P, Q and let J1 U 1, J2 = U I Denote also by U1, V1 the interior and exterior of J itself. Since V is unaffected by A, a point R of can be joined to infinity

by an arc ji, which does not meet J U A. Similarly if R is any point of V. Hence V c V1. Moreover since R is not in J1, c V1 also. Since there are points of U2 in any neighborhood of R, U2 has points in V1. Since it is connected U2 which is in 17— J1, cannot meet U1, for otherwise the two components U1, V1 of would meet the same connected set in 17— Thus U1 and U2 are disjoint. Since V c V1 necessarily U1 c U — A and similarly U2 c U —2. Suppose now that, in addition to U1 and U2, U —2 contains

some open set W disjoint from both. Let M e W and let L be a line through M which does not contain the points P, Q. Let the extension of L to one side of M meet W in a last point N. This point cannot be in since then intervals of L ending in N would belong to U2 or V and not to W. Similarly N is not in A!. Since N is neither P nor Q and is not in V, N U. Since N is in one of U1, U2, necessarily N e A. Thus a full neighborhood of N in 11— J1 consists of U1 and V1. But the only points of V1 near N are those of U2. Hence U1 U U2 is a full neighborhood of N in U —2 and so N is not in W. This contradiction proves the non-existence of W. Thus U — A = U1 U U2.

372


Hence U1 and U2 are the components of U — A and this proves (10.1). (10.2) A plane curve in the form of 0 divides the plane into three regions U1, U2, U3 where if A1, A2, A3 are the three arcs of the curve then = A5 U I j k.

This is reaUy the same as (10. 1). All that is necessary for the A the region V. identification is to label

11. Proof of (9. 1). We first establish it for n = 2. Thus J1 and 12 in II are such that their interiors U1 and U2 intersect. c t72 we are through. Otherwise then 12 Now if J2 c U1 or contained in J2. enters and leaves U1. Let x be a point of

Fig. 7

Thus x is in an arc yxz of 12 with end-points y, z in Jj, where the arc y = yxz c V. Of the two arcs of J1, ending at y and z, only one ô

ywz bounds with y a region W exterior to U1 (10.2).

The set of all the y's is countable. Let it be Va' •.., and let W1, be the W, ô corresponding to yj. Notice that and yg, I j, are disjoint, for otherwise the corresponding regions W1 is one-one. would overlap and hence coincide. Thus -÷ such that the Define now a topological mapping ri: ôi common end-points remain fixed. Let t be a mapping V -÷Ji which is the identity on the points of V in 11 and coincides with on It is clear that i is topological. I = 1, 2 Hence V is a Jordan curve. As it is manifestly contained in J1 U J2, (9.1) is proved for n = 2. The proof for n> 2 follows now by an obvious induction.

PROBLEMS

1. Peano's existence theorem. Consider the vector equation dxliii = X (x; I) and let X be merely continuous in a region D c Then x through every point P (xe; of D there passes at least one trajectory of the system. Show by an example that there may even pass an infinity

of trajectories through P.

(Method: subdivide the segment t0 I 11 into q equal parts; approxi-

mate linearly by replacing the differential equation by a difference equation in each interval, then pass to the limit as q -÷+ 00. Kamke [1] and Bellman [2]).

See

notably

2. The equation being the same as in the preceding problem and real or complex, if X is holomorphic in D, prove by exhibiting the power series x (I) that there is a unique solution holomorphic in t at

that

and such

= x0, ( (x0, I,,) e D).

3. Theorem of Cauchy-Kovalevsky. Consider the real or complex system

of partial differential equations in the p-vector u and q-vector x: øu/øt = 4(u) where A is a

x

q

matrix holomorphic at u,. Let

be a given p-vector

= u0. Then the system has a

bolomorphic at x• and such that

unique solution u (x; I) holomorphic at (x0; I,) and such that u (x; t0) = 4. Derive the analogue of theorem (VI, 21.2) for X(x; t) periodic with period w in I and show that the associated function V(x; I) may be chosen with period w in I (Massera [2]). 5. Obtain the analogue of theorem (VI, 21.2) for a merely continuous function X(x; I). Show also that one may then choose V(x; I) of any

class Ck (Kurzweil [1]). 6. Prove the converse of

instability theorem (Krassovskii [1)).

7. Prove the expansion theorem (V, 9.1) when the characteristic numbers are replaced by the negatives of the Liapunov numbers and the system is regular in the sense of Liapunov (see his paper [1] p. 32). 373

374


8. Consider an analytical autonomous system

dxlii — Pz + q(x), q =

where P diag ..., Xe), the having only negative real parts. There is a regular transformation x = f(y) which reduces the system to the form

dyj/dl =

'%iyt

+ gi (Yi'

where is a polynomial containing such that there exists a relation

= "'i

Y' —

i)

solely terms in Yi" ••.

+... +

—

-'

—

(Dulac [1]).

9. Consider the two systems (a)

dxlii

Px;

dx/dI = {P + Q(i)}

where P is a constant matrix and f tions of (a) are bounded as 1

+ I

IQ(l) II

dl converges. If all the solu-

00 so are those of (a).

If

and the solutions of (a) all -+ 0 as I -÷+ °° (i.e., the characteristic roots of P all have negative real parts) so do the solutions of (a). I

(Bellman [3], Ch. 4).

10. Consider the system

where P is a constant matrix and q is continuous in flxfl 4, 1 and satisfies a Lipschitz condition in that set. Suppose also that: (a) all solutions of the first approximation are bounded; (b) II q (x; ')II for I

I

A and f +

g(t)

a g(t) I IzI I

iSis convergent. Under these conditions the

origin is stable (Bellman [3], Ch. 4).

11. Prove that the origin is unstable for the system in two variables dx/dt = —ax dy/di

=

{sin log I + cos log I — 2a} y + x1

but that it is stable for the first approximation. (Perron [3]). 12. Investigate the stability of an autonomous system with two purely conjugate complex characteristic roots, the rest having negative real parts. (Liapunov [1), Maikin [1]).

375

PROBLEMS

13. Discuss the complete phase-portrait of a system dx/dt = P (x, y), dy/di = Q (x, y) where P and Q are relatively prime real polynomials of degree two. (Büchel [1)).

14. Discuss the periodic solutions (harmonic and subharmonic) of d2x/d12 +

where p, v,

a

(x2 —

1)

dx/dt + x = v x +

dx/dt + a cos cot

are all small parameters. (Obi f 1)).

15. Discuss the complete behavior in the projective plane of the system with constant coeftlcients dxjdt = a x + by, dy/dt = cx + dy, ad — bc 0. 16. Duffing's equation. Investigate the possible values of w for which cos d2xf&2 + x + x3 (See Stoker [1]). has solutions of period

cot

17. Prove the theorem on structural stability stated in (X, 29. 1). 18. Consider the system of the third order

dx/dt =f(x)—(1 + a)x—z dy/dC = — fi (f(x) — x —

z)

dz/dt = —y(y + z), a,

fi, y>

0

which arises in a vacuum tube circuit problem. Prove that with suitable values of the constants a, fi. y the system is self-oscillatory. (L. L. Ranch [1]. See also K. 0. Friedrichs [1]).

19. Derive a more accurate estimate for the period of oscillations of van der Pol's equation for 4u large than that given in (XH, 18). (Mary Cartwright [1], Lasalle [1], Haag {fl).

BIBLIOGRAPHy

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376

BIBLIOGRAPHY

377

[2] Lectures on Di//erential Equations. Princeton, 1947 (typewritten).

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Erich

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1923.

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Lefschetz, S., and Lasalle, J. See LasaUe and Lefschetz. Leontovic, E., and Mayer, A. [1] Sur les trajectoires qui déterminent la structure qualitative de Ia division de Ia sphere en trajectoires. Dokiady A had. Nauk 14: 251—254 (1937).

Levinson, N. [1] On the existence of periodic solutions for second order differential

equations with a forcing term. Journal of Math. and Phys. 22: 41—48 (1943).

BIBLIOGRAPHY

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[2] Transformation theory of nonlinear differential equations of the second order. Annals of Math. (2) 45: 723—737 (1944).

[3] On stability of non-linear systems of differential equations. Colloquium Maihemalicum (Wroclaw) 2: 40—45 (1949).

Levinson, N., and Codthngton, E. A. see Coddington and Levinson.

Levinson, N., and Smith, 0. K. [1] A general equation for relaxation oscillations. Duhe Math. Jour. 9: 382—403 (1942).

Liapunov, A. M. [1] ProbLème Géniral de La Stabilité du Mouvement. Princeton Uni-

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1892).

Liénard, A. [1] Etude des oscillations entretenues. Revue Générale de l'Electricité 23: 901—912, 946—954 (1928).

Lifshitz, Jaime [1] Un teoreina sobre transformaciones de curvas cerradas sobre si mismas. Boletin de La Sociedad Mat. Mex. 3: 21—25 (1946).

Littlewood, J. E. See Cartwright and Littlewood. Malkin, I. G.

[1] Certain questions on the theory of stability of motion in the sense of Liapunov (in Russiaq). Sbornih Nauchnykh Trudov Kazanshovo Aviatsionnovo Instituta, no. 7, 1937, 103 p. (Amer. Math. Soc. Translation no. 20). On stability in the first approximation (in Russian). Ibid. No. 3 (1935). [2] On the stability of motion in the sense of Liapunov (in Russian). Mat. Sbornih n.s. 3: 47—100 (1938). (Amer. Math. Soc. Translation no. 41).

[3) Some basic theorems of the theory of stability of motion in critical cases (in Russian). Priki. Mat. i Mekh. 6: 411—448 (1942) (Amer. Math. Soc. Translation no. 38). [4) Oscillations of systems with one degree of freedom (in Russian).

Priki. Mat. i Mekh. 12: 561—596 (1948) (Amer. Math. Soc.

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[7] On a method for solving the stability problem in the critical case of a pair of pure complex roots (in Russian). Priki. Mat. i Mekh. 15: 473—484 (1951). [8] Some Problems in the Theory of Nonlinear Oscillations (in Russian). Moscow, Gos. Izd. Tekh.—Teor. Lit., 1956.

[9] Theory of Stability of Motion (in Russian). Moscow, Gos. lzd. Tekh.—Teor. Lit., 1952.

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[2] On Liapounoff's conditions of stability. Annals of Math. 50: 705—721 (1949).

[3] Contributions to stability theory. Annals of Math. 64: 182—206 (1956).

Mayer, A. See

and Mayer.

McLachlan, N. W. [1] Theory and Application of Mathieu Functions. Oxford, Clarendon Press, 1947.

[2] Ordinary Non-linear Diflerential Equations in Engineering and Physical Sciences. Oxford, Clarendon Press, 1950. Minorsky, N. [1] Introduction to Non-linear Mechanics. Ann Arbor, J. W. Edwards, 1947.

[2] Parametric excitation. Jour. Appi. Phys. 22: 49—54 (1951). [3] Stationary solutions of certain nonlinear differential equations. Jour. Franklin Inst. 254: 21—42 (1952). [4] On interaction of non-linear oscillations. Jour. Franklin Inst. 256: 147—165 (1953).

Mitropolskii, lu. A. [1] Non-stationary Processes in Nonlinear Oscillating Systems (in Russian). Kiev, Izdatelstvo Akad. Nauk. Ukrainskoi SSR., 1955. Mitropolskii, Ju. A. See Bogoliubov and Mitropolskii. Mizohata, Sigeru, and Yamaguti, Masaya [1] On the existence of periodic solutions of the non-linear differential Memoirs College of Science, equation + a (x) z + v(z) = Univ. of Kyoto, Ser. A Mathematics 27: 109—113 (1952).

BIBLIOGRAPHY

383

Niemytski, V. V., and Stepanov, V. V. [1) Qualitative Theory of Differential Equations (in Russian), 2d ed. Moscow, Gos. Izd. Tekb.—Teor. Lit., 1949.

Obi, Chike

[I] Subharmonic solutions of non-linear d.e. of the second order. Jour. London Math. Soc. 25: 217—226 (1960); Penodic solutions of non-linear d.e. of the second order. Proc. Cambridge Phil. Soc. 47: 741—75], 752—755 (1951); Periodic solutions of nonlinear d.e. of order 2n. Jour. London Math. Soc. 28: 163—171 (1953). Resear-

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+ (e1 + e2z)k + x + e3vz'== 0. Proc.

Cambridge Phil. Soc. 50: 26—32 (1954).

Osgood, W. F. [1) Lehrbuch der Funktionentheorie, 2d ed. 2 vol. Leipzig, Teubner, 1912—32.

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384

[2] Mémoire sur les courbes dé6iiies par une equation différentielle. Jour. Math. Pures et Af4'l. (3) 7: 375-422 (1881); 8: 251—296 (1882); (4) 1: 167—244 (1885); 2: 151—217 (1886). t.1., p. 3—84; 90—161; 167—221. le problème des trois corps

[3] Sur

1—270 Ada Mathematwa 13: [4)

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second order. II. Journal London Math. Soc. 27: 48-58 (1952). Sansone, G. [1] Equazioni

Diflerenziali nel Campo Reals, 2d ed. vol. 1—2. Bologna,

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BIBLIOGRAPHY

Stoker,

385

J. J.

Cli Nonlinear Vibrations in Mechanical and Electrical Systems. New York, Interscience, 1950.

Turritin, H. L. [1) Asymptotic expansions of solutions of systems of ordinary linear differential equations containing a parameter, 81—115 in Contri&utions to the Theory of Nonlinear Oscillations, vol. 2, Princeton University Press, 1952. (Annals of Mathematics Studies, No. 29).

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Wedderburn, J. H. M. [1] Lectures on Matrices, American Mathematical Society Coil. Pubi. 17, 1934. Witt, A. See Andronov and Witt. Yamaguti, Masaya. See Mizohata and Yamaguti.

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LiST OF PRINCIPAL SYMBOLS

is a member of; U union; n intersection; c is contained in; contains; p-cell; A closure of A; boundary of A; A x B product implies; £

n dimensional vector space referred to coordinates of the spaces A, B; Kronecker deltas (= 1 for j = h, = 0 for j direct sum; Mxli norm of vector x; (a,k) matrix with term a;k iuj — throw and k — ih column; x1, x2, ..., coordinates of vector x. [f A = (aik) then A' is its transpose (akj) and A its complex conjugate; Ct

ft

ajk(t) dt). If y(x) = ajk(t) then dA/di (daik/di) and J A dt ( Jto Jto is a vector function of a vector then its Jacobian matrix is øy/8x = have the same dimension then the Jacobian determinant and if is written The Euler-Poincaré characteristic of a two dimensional manifold M2 triangulated into a2 triangles with a1 sides and a0 vertices is J

x (Mt)

a0 —

a1

+ a2.

The block matrix of order Y j 2, 0

.

1,

.

(

is

written C,Q.). One writes generally f(x) for

.. x,). In the system of convergent power series, f(x) such that f(O) 0, called a unit, is written E(x). A series beginning with terms of degree p is denoted by [x],,. One denotes by (AB), [AB), (AB] [AR], an arc with end-points A, B open at both ends, closed at A and open at B, closed at B and open at A, closed at both ends. The half-path beginning at the point M of the path y and formed by [YMI, also M and all the following [preceding] points is written A— YM are written A1 (y), y'(M)[y—(M)]. The limiting sets of .

386

INDEX

Ad joint equation, 66 systems, 61 Analytic (analytical) arc, 24 curve, 24 functions of several variables, 21 manifold, 21 matrix, 20 p-cell, elementary, 24 vector function, 21

Analyticity properties of solutions of duff. eqs., 43

Arc, 6 analytical, 24 without contact, 211 Autonomous system, 46

Axis of path cylinder, ring, rectangle, 52, 53, 183

Base for the solutions of a linear system or cliff. eq., 58 of a single eq., 58 of a vector space, 10 Basic system, 88

Bendixson's theorem on limit sets, 230 criterium for limit-cycles, 238 Block of a matrix, 14 Box, 37 center of, 37

Cell, 5 p-cell, 5 Center, 187, 192

Characteristic exponents, 73

Characteristic polynomial, of a matrix, 13

roots, of a matrix, 13 Circuit, 360 deformation of, 361 homotopy of, 360 Class CT, function of, 25 Closed arc, 6 cell, 5

set, 4 Compact set or space, 6 Complete family of periodic solutions, 174

Component of a set, 8

ofavector, 10 Condensation point, 5 Connectedness, 7

Continuity properties of solutions of duff. eqs., 36

Continuous function, 5 matrix, 20 transformation, 5 Convergent series of matrices, 15 Convexity condition, 108 Covering of a space by open sets, 6 Critical point, 36 elementary, 182 general, 208 Curve, analytical, 24 Definite function, positive or negative, 113

Derivative of a matrix, 11 Diameter of a set, 2 387

DIFFERENTIAL EQUATIONS: GEOMETRIC THEORY Differentiability properties of solutions of duff. eqs., 40

Dimension of a vector space, 10 Direct sum of vector spaces, 10 Distance between two points, 2 from a point to a set, 2 Divergence of a vector, 157

Domain of analyticity of a duff. eq., 44

of continuity of a duff. eq., 35

Elementary critical points, 182, 188 Elliptic sector, 219 Equation of Cartwright-Littlewood, 279

of Hili, 337 of Levinson, 279 of Levinson-Smith, 279 of Liénard, 267 of MatlAieu, 337 of Van der Pol, 272

Eucidean space, 3 Euler- Poincaré characteristic of a two-dimensional manifold, 28 Existence theorem, 31 Exponential of a matrix, 18 Extension of solution of a duff. eq., 35 Fan, 219 Focus, 220

stable, unstable, 187, 192 Function continuous, 5 definite, positive or negative, 112 of a matrix, 17

of fixed sign, positive or negative, 113

General solution of a diR. eq., 39 Genus of a two-dimensional manifold, 28

Green's formula, 67

Half path, positive, negative, 228 Holomorphic function of several variables, 21 vector function, 21

Homeomorphism, 5 Homeomorphic spaces, 5 Homotopy, 360 cylinder, 360 Hyperbolic sector, 219

Implicit function theorem, 23 Index, 195, 360 of a closed path, 199 ofa curve, 195 of an elementary critical point, 196 of a point, 195 Instability, 78 Integral of a matrix, 12 of a system of duff. eqs., 54

ofavector, 11 Jordan curve, 6 Jordan-Schoenflies theorem, 9

Kronecker deltas, 10 Liapunov function, 113 expansion theorem, 96 numbers, 152 strong, 195, 225, 228 Limit-cycle, 223 Limiting sets of paths, positive, negative, 225, 228 position of limit-cycles, 342 Linear system, 55 with constant coefficients, 69 homogeneous, 55 with periodic coefficients, 73 Lipschitz condition, 30

Local phase-portrait at critical point, 218

Logarithm of a matrix, 18

Majorante, 22 Manifold differentiable, 27 Euler-Poincaré characteristic of, 28 genus of, 28 non-orientable, 27 orientable, 27 of section, 161 two-dimensional, 28

INDEX vector field on, 28 Matrix, 9 analytical, 20 derivative of, 11 exponential of, 18 function of, 17 function of scalars, 19 integral of, 12 inverse of a square, 10 limit of, 15 logarithm of, 18 nilpotent, 12 non-critical, 119 non-singular, 10 normal, 14 norm of, 11 series of, 15 absolutely convergent, 16 uniformly convergent, 16 similar, 13

stable 88 trace of, 12 transpose of, 12 triangular, 12 Metric, 2 space, 1

Metrization, 1 equivalent, 2

Neighborhood of a point or set, 4 Nilpotent matrix, 12 Node, stable, unstable, 184, 189 Non-analytical systems, 259 Non-homogeneous system, 72 Non-singular matrix, 10 solution of a djfI. eq., 62 Non-unit (as power series), 26 Normal matrix, 14 system in the sense of Liapunov, 154 Norm of a vector, 11 of a matrix, 11 w-set of a path, 228 Open set, 4

389

Path, 47 closed, 48 cylinder, 50 polygon, 230 rectangle, 52, 182 ring, 53

Perturbation method, 312 Phase-portrait, 181 local, 218

Picard's method of successive approximations, 31 Poincaré's closed path-theorem, 232 expansion theorem, 44 index theorem, 366 Product space, 2 Projective coordinates, 7 plane, 7 real, 7 space, 7 Quasi-harmonic system, 312 stable, 124 unstable, 124

Regular system in the sense of Liapunov, 154 transformation, 23

Saddle point, 186, 190 Self-excited system, 313 Separatrix, 223 Series of matrices, 15 absolutely convergent, 16 uniformly convergent, 16 of vectors, 15 Small parameters, method of, 313 Space, 1 metric, 1 Sphere, 5 p-sphere, 5 Spheroid, 2 center of, 2 radius of, 2 Stability, 76 asymptotic, 78, 83

390


conditional, 78, 83 exponentially asymptotic, 85 of mappings, 84 orbital, 83 structural, 250 of trajectories, 83 uniform, 78 in product spaces, 122 Stroboscopic method according to Minorsky 325 subharmonics, 322 Systems linear, 55 homogeneous, 55 normal, regular in the sense Liapunov, 154 quasi-harmonic, 170 quasi-linear, 170, of periodic solutions, 170

TO curve, 242 Topologically equivalent spaces, 5 Trajectory, 35 Transformation of real vector space, 14 regular, 23 topological, 5 Triangular matrix, 12

-

Uniform stability, Unit (as power series), 26 Vector space, 9 base of, 10 dimension of, 10 direct sum of, 10 real, 14

Well-behaved set, 96

Differential Equations, Geometric Theory, 2ed

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