Theory of Limit Cycles

TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME 66 Theory of Limit Cycles YEYAN·QIAN Nanjlng University and Cal Sui·U...

Author: Yan-Qian Ye | Chi Y. Lo

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TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME

66

Theory of Limit Cycles YEYAN·QIAN Nanjlng University and

Cal Sui·Un Chen Lan-sun Huang Ke·cbeng Luo Dlng-jun

MaZbl-en Wang Er·nlan Wang Mlng-sbu YangXIn·an

Translated by

Cbl Y. Lo Mlcblgan State University

AmericanMathematicaiSociety · Providence · Rhodelsland

1965, 1984 Translation edited by S. H. Gould and J. K. Hale

1980 Mathematics Subject Qassi{ication (1985 Revision).. Primary 34C05. SUMMARY OF CONTENTS. This book belongs to the "Modern Mathematics Series." Its first edition was published in 1965. Now this is the second edition, with many substantial changes in content. It contains tremendous results obtained during the past twenty years in China, and also introduces some great achievements from abroad. This book has eighteen sections, which can be divided into three parts. Part I (§§1-8) discusses limit cycles of general plane stationary systems, including their existence, nonexistence, stability, and uniqueness. Part II (§§9-17) discusses the global topological structure of limit cycles and phase-portraits of quadratic systems. Part III (§18) collects some important results which either could not be included under the subject matter of the previous sections or appeared in the literature very recently. This book serves as a reference book for college seniors, graduate students, and researchers in mathematics and physics.

Library of Congress Cataloging-in-Publication Data Yeh, Yen-chi'ien. Theory of limit cycles. (Translations of mathematical monographs, ISSN 0065-9282; v. 66) Translation of: Chi hsien huan lun, 2nd ed. Bibliography: p. 415 1. Differential equations. 2. Curves. l. Cai, Sui-lin. II. Lo, Chi Y. (Chi Yeung) III. Title~ IV. Series. QA371.Y413 1986 515.3'5 86·14070 ISBN 0-8218-4518-7 Copyright (£)1986 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government.

Contents Preface to the second edition

vii

Abstract

ix

Introduction

1

§1. Fundamental concepts, concrete examples, and several criteria to establish existence and nonexistence of limit cycles

5

§2. Multiplicity and stability of limit cycles

25

§3. Limit cycles in rotated vector fields

41

§4. General behavior of limit cycles depending on a parameter

77

§5. Existence of a limit cycle

91

§6. Uniqueness "'fa limit -cycle

117

§7. Existence of several limit cycles

155

§8. Structural stability of different systems

175

§9. Work of M. Frommer and N. N. Bautin

191

§10. Global structural analysis of some quadratic systems without limit cycles

219

§11. General properties and relative positions of limit cycles in quadratic differential systems

245

§12. Classification of quadratic differential systems. Limit cycles of equations of Class I

261

§13. Global structure of trajectories of equations of Class II without limit cycles

281

v

vi

THEORY OF LIMIT CYCLES

§14. Relative positions of limit cycles and conditions for having at most one and two limit cycles in equations of Class II 305 §15. Several local and global properties of equations of Class III

335

§16. Method of Dulac functions in the qualitative study of quadratic systems

359

§17. Limit cycles in bounded quadratic systems

371

§18. Appendix

401

Bibliography

415

Preface to the Second Edition

It has been eighteen years since the publication of the first edition of this book in 1964. Over these eighteen years, there has been tremendous progress in the theory of limit cycles, especially in quadratic differential systems, in China as well as in other countries, and many new results have been obtained. Some work in this area which was considered correct in the past has turned out to be wrong; some work which was considered important in the past now seems to be not worthy of further development. It is worthwhile to mention that, owing to research developments from other fields such as biology and chemistry, research on limit cycles, especially in the theory of polynomial systems, has become more and more important. In China, there are at least ten mathematicians who have produced good work in this area over the past twenty-some years. In other countries, besides the Russian mathematicians who have had a traditional interest and made solid contributions in this area, more people in the United States and France have shown strong interest in the study of limit cycles of quadratic systems. For these reasons, the author felt that this book should be completely revised and its second edition should be published without delay. Not having enough time to work on this project, and hoping to collect new ideas and valuable comments from others for the better presentation of the second edition, I have omitted §7, §9 and the latter part of §10 from the first edition, and concentrated on the overall rearrangement of material, final editing of the manuscripts and unification of the use of mathematical symbols in the new edition; except for the sections dealing with my own work, I entrusted the task of revision and supplementation to some of my colleagues, as follows: Wang Er-nian supplemented part of §3 and part of §6; Ma Zhi-en supplemented §3; Huang Ke-cheng supplemented §5 and wrote the new §7; Luo Ding-jun rewrote §8 and supplemented the first half of §9; Cai Sui-lin rewrote §10; Wang Ming-shu supplemented §11 and wrote the new vii

viii


§15; Yang Xin-an supplemented §12 and wrote the new §17; and Chen Lan-sun supplemented the latter two-thirds of §14 and wrote the new §16. In my final editorial work I have made very few changes in some of their manuscripts while in other cases I have made considerable changes, but they all applied their expertise in this field and spent their time and energy to make the second edition reflect more completely the current development of the theory of limit cycles. It is obvious that I could not have taken on such a task all by myself. I therefore express my heartfelt gratitude to the above eight colleagues. Due to our limited time and ability, it is natural that there may still be some mistakes or important omissions in the new edition. We shall be grateful if our readers would kindly let us know. Ye Yan-qian Department of Mathematics Nanjing University August, 1982.

Abstract Limit cycles of plane autonomous differential systems appear in the very famous classical paper "Memoire sur les courbes definies par une equation differentielle" of H. Poincare (1881-1886). In the 1930s, van der Pol and A. A. Andronov showed that the closed orbit in the phase plane of a self-sustained oscillation occurring in a vacuum tube circuit is a limit cycle as considered by Poincare. After this observation, the existence, nonexistence, uniqueness and other properties of limit cycles have been studied extensively by mathematicians and physicists. Then, from the 1950s, very many mathematical models from physics, engineering, chemistry, biology, economics, etc., were displayed as plane autonomous systems with limit cycles. Also, due to the well-known paper of I. G. Petrovskil and E. M. Landis concerning the maximum number of limit cycles of all quadratic differential systems (the second part of Hilbert's 16th problem), the problem of limit cycles has become more and more important and has attracted the attention of many pure and applied mathematicians. The purpose of this book is to bring together in one place most of the main contributions in the theory of limit cycles. Aside from the introduction (a brief historical review), it divides into three parts. §§1-8 are concerned with limit cycles of general plane autonomous systems, and §§9--17 with limit cycles and the global topological structure of phase-portraits of quadratic systems. At the end of every section, a large number of reference papers are listed. The last part, §18, has the character of an appendix, in which we mention briefly results that either could not be included in the subject matter of the foregoing sections, or have appeared in periodicals very recently. We assume that the readers have a basic knowledge of the qualitative and stability theory of ODE. Here are the main contents of each section in the first two parts. §1 gives the fundamental concepts and examples of limit cycles, and also some criteria for the existence and nonexistence of limit cycles, including well-known ones ix

X


as well as some new ones. §2 gives criteria for the determination of the stability and multiplicity of limit cycles. Aside from the classical ones, we introduce also results of V. F. Tkachev and M. Urabe. §3 deals with the theory of rotated vector fields due to G. F. D. Duff, and also many of its extensions and applications by X. Y. Chen and Z. E. Ma. We will use this theory very often in the second part. §4 is concerned with the variation of limit cycles with the varying of a parameter in the general case; the main contents are the classical formula of H. Poincare and contributions of M. Urabe and X. Y. Chen. §5 discusses the question of the existence of limit cycles. Aside from the well-known theorems of A. F. Filippov and A. V. Dragilev, we present here also contributions ofK. Z. Hwang, Z. J. Wu, and X. W. Zheng. This section is divided into six subsections according to the methods of proof. §6 is concerned with the problem of uniqueness of limit cycles. It also divides into seven subsections, in which we introduce methods of point-transformation due to H. Poincare, A. A. Andronov, and E. A. Leontovich, and also results of G. Sansone, J. L. Massera, Z. F. Zhang, L. A. Cherkas, and G. S. Rychkov. §7 deals with the problem of the existence of any given number of limit cycles. The main results almost all belong to Chinese mathematicians, among which the contribution of Z. F. Zhang is preeminent, solving completely the problem of the number and position of limit cycles of the equation x + p, sin x + x = 0. §8 is a short introduction to the well-known necessary and sufficient conditions for the structural stability of a plane autonomous system in a bounded domain, which we will use in the second part. It contains also some new results of G. T. dos Santos and D. J. Luo about polynomial systems. §9 deals with classical results of H. Dulac and M. Frommer on necessary and sufficient conditions for a quadratic differential system to have a center, and presents the corresponding phase-portraits due to Frommer. In this section we also give a detailed proof of an important result of N. N. Bautin concerning the maximum order of fineness of a focus of any quadratic system and the maximum number of limit cycles that can be generated from this focus. In §10, we analyze the global topological structure of phase-portraits of three types of quadratic systems without limit cycle, namely, the homogeneous systems (results of L. S. Lyagina, L. Markus, et al.), the system = x + h.o.t., iJ = + h.o.t. (result of A. N. Berlenskil) and, finally, the structurally stable quadratic systems without limit cycle (results of G. T. dos Santos and S. L. Zai). §11 deals with general properties and possible relative positions of limit cycles of quadratic systems, among which results of Y. Q. Ye, C. C. Tung, Y. S. Chin, M. S. Wang, and S. L. Shi are presented. §12 introduces the classification of quadratic systems due toY. Q. Ye, and proves a theorem on the existence, nonexistence and uniqueness of limit cycles of systems of type I, due toY. Q. Ye, Y. H.

x

y

ABSTRACT

xi

Deng, D. J. Luo, L. S. Chen, and X. A. Yang. §13 investigates the global topological structure of phase-portraits of a special system of type II, which contains two parameters a and m and has no limit cycle. We obtain global bifurcation curves in the (a, m) projective plane. §14 is concerned with the relative position (especially coexistence), uniqueness and the number of limit cycles of systems of type II containing only two second-degree terms in the first equation. These results are due mainly to M. S. Wang, K. T. Lee, S. X. Yu, N. D. Zhu, K. C. Chen, L. I. Zhilevich, and L. A. Cherkas. §15 discusses various interesting global properties of systems of type ill; especially, we give the detailed proof of a theorem (concerning system IDa=o) similar to that in §12, which was conjectured and partly proved by N. D. Zhu, and later completely proved by L.A. Cherkas, L. I. Zhilevich, and G. S. Rychkov. §16 discusses the Dulac function method used frequently by Chinese mathematicians in their research work on the qualitative investigation of quadratic systems, and uses this method to prove an interesting theorem of L. S. Chen and M. S. Wang, concerning the nesting of limit cycles surrounding just one focus. §17 introduces X. A. Yang's results on the uniqueness or nonexistence of limit cycles of bounded quadratic systems. These systems were first studied by R. J. Dickson and L. M. Perko, but the limit cycle problem remained open il' their papers.

Introduction

In the qualitative theory of differential equations, research on limit cycles is an interesting and difficult part. Following the discovery of the limit cycle by Poincare in his four-part paper Integral curves defined by differential equations (1881-1886) [1], it was immediately given special consideration by this famous mathematician. In order to determine whether there existed a limit cycle for a given differential equation and to study the properties of limit cycles, he first introduced such important theoretical concepts as the method of a topographical system, the successor function, the method of small parameter (which first appeared in his book New methods in celestial mechanics) and the Annular Region Theorem, and he constructed many examples to examine the effects of these methods. At the same time, he had already noticed the close relationship between the study of limit cycles and the solutions of the global structural problems of a family of integral curves of differential equations. In 1901 the Swedish mathematician Ivar Bendixson published a very important paper using virtually the same title [2]. In this paper he gave a rigorous proof of the Annular Region Theorem and extended it to the well-known PoincareBendixson theorem on the limit set of trajectories of dynamical systems in a bounded region. Moreover, he first applied Green's formula to establish the relationship between possession of a closed trajectory and the divergence of a plane vector field, and obtained a theorem for the nonexistence of a closed trajectory. This connection has been continuously extended and strengthened, and the relation between the value of the divergence integrated once around a closed trajectory and its stability, the relation between the value of a saddle point and the stability from inside of a singular closed trajectory passing the saddle point, and so on, have been obtained. In the same year as Bendixson's paper was published, Hilbert set forth his famous list of problems [3] at the International Congress of Mathematics. The second half of the sixteenth problem was: What is the maximum number of 1

2


limit cycles for the equation dy Qn(x,y) dx = Pn(x,y)

(1)

(Pn and Qn are polynomials of real variables x, y with real coefficients and degree not higher than n), and what are their relative positions? Strangely, mathematicians have shown great interest in the other Hilbert problems, and have done intensive research on them; but very few people have studied this problem.(!) To our knowledge, in the first three decades of the twentieth century, only the French mathematician Henri Dulac produced good results on this problem. In 1923 he published a• very long paper of 140 pages [4], in which he proved that the number of limit cycles of equation (1) is finite. Moreover, he also studied the necessary and sufficient conditions for existence of a center for equation (1) when n = 2 (see [5]). It seems to us that he had already sensed a close relationship between these two problems. Dulac also produced some other fundamental results in the theory of limit cycles; readers can see these in §1. Later, the German mathematician Max Frommer published a paper in 1934 [6] giving necessary and sufficient conditions for the existence of a center for equation (1) (n = 2). At the same time, he also pointed out that the equation dy x + (1 + c)x 2 + 2xy- y2 -y + 2xy - y2 dx =

( 2)

has a limit cycle when c > 0 is sufficiently small. In fact, as we shall see later, limit cycles occur only for nonlinear equations, and they are the most common trajectories for these equations. The trend of development of mathematical theory is often guided by practical problems. For the field of differential equations, this situation is particularly clear. The driving force behind the study of the theory of limit cycles was furnished much more by practical problems than by great mathematicians. This is the situation: during the twentieth century, applied electronics had made rapid advancement; physicists invented the triode vacuum tube which was able to produce stable self-excited oscillations of constant amplitude, thus making it possible to propagate sound and pictures through electronics. However, it was not possible to describe this oscillation phenomenon by linear differential equations. In 1926, van der Pol [7) first obtained a differential equation, which was later named after him, to describe oscillations of constant amplitude of a triode vacuum tube:

x+ ~t(x 2 - I)x + x = o

(~t ~ o).

(3)

( 1 )The situation has changed since the first edition of this book appeared, and in reent years more people have become interested in this problem.

INTRODUCTION

3

After transforming this equation into an equivalent differential system in the phase plane, he used graphical methods to prove the existence of an isolated closed trajectory. Moreover, he used an averaging method (van der Pol's method), which was at that time without rigorous mathematical foundation, to obtain an approximate equation for the closed trajectory when IJ.tl was very small. It is obvious that he was not familar with the work of Poincare and Dulac on limit cycles. Three years later, the Russian theoretical physicist A. A. Andronov in a short paper [8] clarified that the isolated closed trajectory of the van der Pol equation was the limit cycle studied earlier by Poincare. Thus he established a close relationship between pure mathematical theory and electronic technology. From that time onward, the schools of Moscow and Gorki in the Soviet Union began to produce a tremendous amount of research work on electronic technology and the theory of limit cycles. In mathematical theory, they studied mainly the existence, uniqueness and stability of limit cycles and the problems of how the limit cycles were generated and disappeared. Most of their important and fundamental work can be found in the book Theory of oscillations by A. A. Andronov, A. A. Vitt, and S. E. Kh&"kin [104]. Although equations for nonlinear oscillations include time-dependent systems as well as stationary systems (or autonomous systems) which do not contain the time variable explicitly, most of the publications of Andronov and others on differential equations belong to steady-state systems. Hence we can say that their book is devoted entirely to the mathematical theory of limit cycles and its applications in physics. Naturally, its emphasis is on applications. As for other eountries, most of the research on limit cycles after van der Pol was produced after 1940, except some earlier work of the French engineer A. Lienard and the geometers E. Cartan and H. Cartan [9], [10]. Among the research done after 1940, we especially note that done by N. Levinson, G. F. D. Duff, S. P. Diliberto, G. Sansone, R. Canti, and M. Urabe. In China, since 1957, we have carried on deep and systematic research on problems of limit cycles of equations whose right sides are quadratic polynomials. (See [11]-[18]; in the earlier fifties, Chinese scholars had done some work on the existence, stability, and uniqueness of limit cycles.) The three main problems of research are as follows: 1. relative positions of limit cycles of equation (1) when n = 2, 2. quadratic algebraic curve cycles of (1) when n = 2 or 3, and 3. research on the number of limit cycles and the global structure of trajectories for a given equation (1) (n = 2). The first two problems were thought to be completely solved. However, since the conjectures of Petrovski1 and Landis [19], [20] have been disproved, the first problem is still a long way from being completely solved and research

4


on the second is also still in progress. From abroad, in addition to the famous work of N. N. Bautin in 1952 [21}, we have found out that since 1960 there has been a seminar at the Belorussian State University to study the third problem, and many papers have been published. Surveying all the international research work on the limit cycle and its academic trend, our view is that although problems of limit cycles in the qualitative theory of differential equations are of primary importance, the workers in the field of differential equations do not pay enough attention to them. As a result, there has never been a treatise on limit cycles in the theory of pure mathematics, or a comprehensive report with special topics on limit cycles in an international journal. In China there was only one report by the present author, published in 1962. Engineers and physicists are still interested in these problems, but without strong support from mathematicians. (Recently this situation has improved.) Hence, results in this area are fragmentary and unorganized. The aim of this book is to collect important results on limit cycles for the past four decades from China and abroad, and to introduce them to beginning workers in the field, and at the same time to correlate this theory with the other areas of the qualitative theory of differential equations. Besides giving an account of more important and fundamental work in the main text of this book, we also introduce at the end of each section some secondary or deeper results; moreover, we provide enough exercises to help beginners better grasp the contents and methods of each section.

§1. Fundamental Concepts, Concrete Examples, and Several Criteria to Establish Existence and Nonexistence of Limit Cycles

We are given a system of differential equations dx dt

dy dt

= P(x, y),

= Q(x, y),

(1.1)

where x, y, and t are real variables, P and Q are continuous single-valued real functions of x, y, and uniqueness of its solution is guaranteed. DEFINITION 1.1. H the solution x = 0,

x+ x = /o

when

x < 0;

(1.9)

here x represents angular displacement, and we take gfl = 1. Changing (1.9) into an equivalent system of equations, it is easy to see that the trajectory in the upper half-plane (x,y = x) is a circular arc with 0"(-/0 ,0) as center, whereas the trajectory in the lower half-plane is a circular arc with 0'{!0 , 0) as center, whereas the trajectory in the lower half-plane is a circular arc with 0' {!o, 0) as center; the clockwise direction on the trajectory is the direction of increasing t. Every point on the line segment (-/o, /o) on the x-axis is a singular point, which corresponds to the equilibrium position; since at that time the restoring force is less than the frictional force, and also x = 0, the motion should stop. From this we can see if the spring does not supply enough energy to the pendulum, then its motion will cease after a finite number of swings. Its trajectory is indicated in Figure 1.2. The energy is supplied by the spring as follows: when the pendulum reaches its lowest point (x = 0), it receives an impulse given by the spring through its trigger mechanism such that it receives constant energy. To simplify our calculations, we assume in the following that the impulse occurs at x =- /o, y > 0. Then in the phase plane we have path of motion as indicated by ABCDE as shown in Figure 1.3, where AB is a circular arc with center 0"(-/o,O) and radius equal to Ro, BC = R1- Ro, and R~- ~ = h 2 = constant; CD is a circular arc with 0" as center and with radius R 1 ; and DE is a circular arc with O'{!o, 0) as center, and with radius R2 = R1 - 2/o. Let

§1. FUNDAMENTAL CONCEPTS

13

the coordinates of A and E be (-x 0 ,0) and (-x 1 ,0) respectively. Then it is easy to calculate the relation between xo and x1: (x1

+ 3fo) 2 -

(xo- fo) 2

= h2 •

(1.10)

This is the topological map of the negative x-a.xis into itself determined by the trajectory of the motion. The fixed point xi = x0 of this map can be determined from (1.11) that is,

x(j = h 2 /Bfo - fo. Of course we should assume x0 > fo, which is equivalent to h

> 4fo.

(1.12) (1.13)

When (1.13) is satisfied, not only is the uniqueness and existence of a closed trajec~ory guaranteed, but also it is not difficult to prove that it is a stable limit cycle. In fact, from (1.11) we can get

x0 = .jh2 + (x0 - fo)2- 3fo; hence, when fo

< xo < x0, we have 2 -+--:(,....xo---J,=-o)= 2 x 0 > x1 = Jr-:-h:::-

3fo

(1.14)

> xo,

(1.15)

where the first inequality can be seen from (1.10) and (1.14) and the second inequality can easily be deduced from (1.11). Similarly, when x 0 > x0, we have Xo < X1 < Xo. (1.16) Inequalities (1.15) and {1.16) indicate that when A =f. A*, E should be between A and A*. But the value of x0 is uniquely determined from (1.12); hence when t ---> oo, all motion should approach the closed trajectory passing through A•. Finally, we introduce several rules to distinguish nonexistence of closed trajectories. THEOREM 1.9 (POINCARE METHOD OF TANGENTIAL CURVES). Let F(x, y) = C be a family of curves, where F(x, y) is continuously differentiable. If in a region G the quantity

dF = aFdx + aFdy = paF +QaF dt ax dt ay dt ax ay (which represents the rate of change of the function F with respect to t along a trajectory of system {1.1)) has constant sign, and the curve PaF +QaF =O ax ay


14

(which represents the locus of points of contact between curves in the family and the trajectories of (1.1), and is called a tangential curve) does not contain the whole trajectory of (1.1) or any closed branch, then the system (1.1) does not possess a closed trajectory which is wholly contained in G and does not have a singular closed trajectory with only one singular point on it.

PROOF. Suppose the theorem is not true. Then the system (1.1) contains a closed trajectory which is wholly contained in G, or a singular closed trajectory which contains only one singular point. Now integrate P8Ff8x + Q8Ff8y along the above trajectory r once in the direction of increasing t; we obtain

i (p~: +Q~:) i ~ dt=

dt.

Since F is single-valued, we know that the right side of the above formula is equal to zero; however, on the other hand, the integrand on the left of the inequality has a constant sign, but is not identically zero on r, and, along r' t monotonically increases; hence its value should be different from zero, a contradiction. THEOREM 1.10 (BENDIXSON). If the divergence 8Pf8x + 8Qf8y of (1.1) has constant sign in a simply connected region G, and is not identically zero on any subregion of G, then the system (1.1) does not possess any closed trajectory or singular closed trajectory which lies entirely in G. ( We assume P and Q have continuous partial derivatives.) PROOF. Suppose the theorem is not true. System (1.1) has a closed trajectory r. Since G is simply connected, rand its interior lie entirely in G. From Green's formula we have

i

Pdy- Qdx =/Is (

~: + ~~)

dxdy.

(1.17)

But Pdy = Qdx holds everywhere along r. Hence the left side of (1.17) is zero, while the integrand on the right has constant sign but is not identically zero in S; hence the double integral is not zero, a contradiction. When r is a singular closed trajectory, since dyfdx does not have a definite value at the singular point, the tangential direction of r may be discontinuous at that point, and the above proof is not rigorous. However, we can still prove the theorem after a slight modification. For this, let r have only one singular point 0 as in Figure 1.4. Use a smooth arc "Y = AB (dotted line) to replace a small section of the arc AOB which contains 0 on r and obtain a closed curve f'. Applying Green's formula to f' and its interior 8', we obtain

fr,

--

Pdy- Qdx

=

i v• ·

ds =

jfs, (~: + ~~) dxdy,

(1.18)

§1.

FUNDAMENTAL CONCEPTS

15

r 0

FIGURE 1.4 where v• represents the vector ( -Q, P) and ds represents the element of arc length of 1· We have P = Q = 0 at 0, and according to the continuity of P and Q we only have to choose 1 lying in a sufficiently small neighborhood of 0; then the absolute value of the left side of (1.18) is less than any small positive e. For the double integral on the right, its value should be very close to the value ofthe right side of (1.17); hence its absolute value cannot be less thane, a contradiction. REMARK. The proof of Theorem 1.10 mainly depends on the following fact: PROPOSITION. In the interiorS of any closed trajectory or singular closed trajectory of system (1.1), we have

/Is (~: + ~~)

dxdy = 0.

(1.19)

Readers can see easily that (1.19) has an obvious physical meaning, because the integrand represents the divergence of the vector field defined by (1.1). THEOREM 1.11 (DULAC). If there exists a continuously differentiable function B(x, y) in a simply connected region G such that 8(BP)/8x + 8(BQ)j8y has constant sign and is not identically zero in any subregion, then system (1.1) does not have any closed trajectory or singular closed trajectory lying entirely in G. PROOF. We follow the proof of Theorem 1.10, but use BP and BQ to replace P and Q respectively. We shall call B(x, y) the Dulac function, and the method of proving nonexistence of a closed or singular closed trajectory by Theorem 1.11 the method of Dulac functions. Theorems 1.10 and 1.11 can be extended to multiply connected regions as THEOREM 1. 12 (DULAC). If we change the region G in Theorem 1.10 or Theorem 1.11 to be n-multiply connected (i.e. G has one or several outer


16

boundary curves, and n- 1 inner boundary curves), then system (1.1) has at most n - 1 closed trajectories which lie entirely in G. PROOF. From the proof of Theorem 1.10, we know that if there is a closed trajectory r of system (1.1) in G, then r should contain at least one inner boundary curve C of Gin its interior. Similarly, we also know if the interior of r also contains other closed trajectories r ..... ' r k' then the region in the interior of r but in the exterior of all the trajectories r 1' ... ' r k also contains at least one inner boundary curve C. Let C correspond tor; we can see that for different r their corresponding curves C are also different. Hence if the number of closed trajectories in G is more than n - 1, then the connectivity number of G must be greater than n. The theorem is proved. For the rules to distinguish nonexistence of closed trajectories other than these classical results mentioned above, there is some recent work of V. F. Tkachev and Vl. F. Tkachev [23], Yu. S. Bogdanov [24], Chen Guang-qing [25], Chen Xiang-yan [26], and Yang Zong-pei [27]. In the following we introduce a theorem in [26] which has more generality and has obvious geometrical meaning.(5 ) THEOREM 1.13. For system (1.1), assume there exist a simply connected region G and continuously differentiable functions M(x, y) and N(x, y) such that, in G,

E(x, y) = M(x, y)P(x, y)

+ N(x, y)Q(x, y)

aM aN F(x, y) = ay - ax ~ 0

~ 0,

(1.20)

(::; 0).

(1.21)

F(x, y) ¢ 0 in any subregion of G,

(1.22)

1°. If

then (i) if E(x,y)

=0 in G,

(1.23)

then system (1.1) does not have any closed and singular closed trajectory in

G; and (ii) if E(x,y) ¢0 in G,

(1.24)

then (1.1) does not have any positively (negatively) oriented closed trajectory or singular closed trajectory in G. 2°. If (1.25) F(x,y) 0 in G,

=

( 5 )The conditions in (23) are similar to those in Theorem 1.13, but much stronger; moreover, two sets of the conditions in (23) are incorrect. The proof given here is not the same.

§1.


17

then {1.1) does not have any closed trajectory or singular closed trajectory unless it is contained entirely in the point set of F(x,y) = 0.

PROOF. Suppose (1.1) has a positively (negatively) oriented closed trajectory r. From Green's formula, we have 1M dx

Jr

+ N dy =- f f

JJintr

(1.26)

F(x,y) dxdy.

Under condition 1°, the right side of (1.26) is negative (positive). On the other hand, the left side of (1.26) is equal to

+(-)loT E(x, y) dt,

{1.27)

where x = x(t), y = y(t) represent the solution of (1.1) along r, and T > 0 is the smallest period of the solution. Thus we know that when (1.23) holds, the left side of (1.26) is zero; hence (1.26) does not hold, i.e., r does not exist. Also when (1.24) holds, the expression (1.27) is nonnegative (nonpositive), so (1.26) still does not hold. For a singular closed trajectory, we only have to change JoT to The proof for condition 2° is similar. Now we would like to explain the geometrical meaning of Theorem 1.13. The vector field orthogonal to (M, N) is ( -N, M), and its corresponding system of equations is

L":o.

dxfdt = -N(x, y),

{1.28)

dyfdt = M(x, y).

Condition (1.20) indicates that on moving the vector field (P, Q) to the vector field ( -N, M) in a counterclockwise direction, the angle O(x, y) traversed satisfies 0 ~ O(x, y) ~ 1r (length of the vector is adjustable). Conditions {1.21) and (1.22) illustrate that (1.28) satisfies the conditions of Theorem 1.10, and therefore does not have a closed trajectory or a singular closed trajectory, and its divergence always keeps positive (negative) sign. Therefore, Theorem 1.13(1 °) indicates that under the above conditions (P, Q) is obtained from (-N, M) along the counterclockwise direction through an angle O(x, y); when (J 1r, (P, Q) does not have a closed or singular closed trajectory (obvious), and when(}¢. 1r, (P, Q) does not have a positively (negatively) oriented closed or singular closed trajectory.( 6 ) Theorem 1.13(2°) indicates that when {1.28) has a first integral, or M dx + N dy = 0 is a total differential, {1.1) does not have a closed trajectory or a singular closed trajectory, unless that closed (or

=

( 6 )However, a negatively (positively) oriented closed trajectory may still exist. For example, if (M,N) = (y,-x) and (P,Q) == (M,N), then for 9 = 7r:/2, (P,Q) has a family of negatively oriented closed trl\iectories x 2 + y 2 = C with div( -N, M) 2 > 0.

=

18


singular closed) trajectory is also a closed (or singular closed) trajectory of (1.28). Readers will understand more about the above results after they become acquainted with the theory of rotated vector fields to be introduced in

§3. It is easy to see that if we take M = auI ax and N = auI ay in Theorem 1.13, we can deduce Theorem 1.9 (change F(x, y) in Theorem 1.9 to U(x, y)); and if we take M = BQ and N = -BP, we can deduce Theorem 1.12. Moreover, we still have the following corollary.

COROLLARY [25]. If there exist nonnegative C 1 functions Mo(x,y) and N 0 (x, y) and a C 1 function B(x, y) such that in any simply connected region G a a ay (MoP) - ax (NoQ)

a

a

+ ax (BP) + ay (BQ)

~0

(~ 0),

(1.29)

and the points where equality is attained do not completely fill any subregion ofG, then system (1.1) does not have any positively (negatively) oriented limit cycles in G. PROOF. This can be done by taking M = MoP+BQ and N = NoQ-BP in Theorem 1.13. Finally we note that if E(x, y) 0 and F(x, y) 0 in Theorem 1.13, then we cannot obtain any useful conclusions. At the same time we can prove

=

=

THEOREM 1.14. If P and Q of system (1.1) belong to C 1 in a simply connected region G, and aPiax + aQiay 0, i.e. the equation

=

Pdy-Qdx = 0

(1.30)

is a total differential equation, then (1.1) does not have limit cycles, and does not even have any one-sided limit cycles. PROOF. Let the general integral of (1.30) be ~(x, y) = C, which represents the equation of a family of trajectories of ( 1.1). Along each trajectory ~ takes constant value, and along different but adjacent trajectories ~ cannot take the same value. If there exists a limit cycle f of (1.1), then every trajectory in a sufficiently small neighborhood off takes r as w- or a-limit set; hence, by the continuity of ~(x, y), we know in this neighborhood ~(x, y) Cr, which is impossible. Also if f is a compound limit cycle, then in every arbitrarily small neighborhood of f there exist nonclosed trajectories which cover some open annular region near f, in which ~(x,y) should be identically equal to a constant. This is impossible. Similarly we can prove that if in some simply connected region the system (1.1) has continuous integrating factor ~-t(x,y), then (1.1) cannot have limit

=

§1.


19

cycles, foci or nodal points in this region. Conversely, if there exist limit cycles, foci or nodal points, then the integrating factor of (1.30) cannot be continuous. EXAMPLE 4. The system dyfdt = y,

dxfdt = x

takes (0, 0) as nodal point, and at the same time the equation x dy- y dx = 0 takes 1/x2 and 1/(x2 + y 2 ) as integrating factor, but these integrating factors are discontinuous at x = 0 or x = y = 0. EXAMPLE 5. The system

~;

= -y + x( x 2

+ y2 -

(1.31)

1),

+ y2 = 1, and at the same time the equation [-y + x(x 2 + y 2 - 1)] dy- [x + y(x 2 + y 2 - 1)] dx = 0

has limit cycle x 2

has integrating factor

J.L(x, y) = (x 2

y 2 ) 2 exp { -2tan- 1

:

~}

(1.32)

and general integral U(x,y) =

x2

+ y2 -

X

2

+y 2

1

{ -1 y } exp -2tan - = C, X

(1.33)

where J.L and U are discontinuous at (0, 0), i.e., they must have a point of discontinuity in any simply connected region containing x 2 + y 2 = 1. Starting from Theorem 1.10, since we want to apply Green's formula, we have to assume P(x, y) and Q(x, y) have continuous partial derivatives in the region G. N. P. Erugin [28] first pointed out that if we rewrite Green's formula as

£

P(x, y) dy- Q(x, y) dx =

JIs

dxP(x, y) dy

+ dyQ(x, y) dx,

(1.34)

where the double integral on the right is a Stieltjes integral, it could be understood as the limit of L[dxP(x, y)dy + dyQ(x, y)dx] and the requirement on P and Q could be kept the same without strengthening it. N. N. Krasovksil [29] had made use of the above rewritten Green's formula to study the global stability of the zero solution of some nonlinear secondorder differential equations. In that paper, Theorem 1.10 was extended to the case where P and Q did not necessarily have partial derivatives.


20

EXAMPLE 6. Prove that if in the system

dxfdt

= ft(x) + h(y),

dyfdt =ax+ by,

(1.35)

ft(O) = /2(0) = 0, f1(x) and h(y) are continuous nonlinear functions, a# 0, and rp(x) = ft(x) + bx is a monotone function of x, then this system does not have a closed trajectory. PROOF. Apply the transformation X1 = -ax- by, Y1 = y. The original equations become

dxtfdt = -a[ft(x) + h(y)} + bx1, Suppose there exists a closed trajectory equation, we obtain 0 = £[-a(ft(x)

r.

Applying formula {1.34) to this

+ h(y)) + bx1} dy1 + x1dx1

=lis dx 1[-a(ft(x) + h(y)) + bx1} dy1 + dy 1X1 dx1 = -a

IIs

dx1 rp (- byl:

Xl) dyt.

This is a contradiction since, according to the assumption, the right side of the above formula should not be equal to zero. In addition, Krasovski1 [30] also obtained the following theorem. THEOREM 1.15. Let P and Q of system (1.1) have continuous partial derivatives, let the origin be the unique singular point of {1.1), and let the roots A1(x,y) and A2(x,y) of the quadratic equation of A

ap _A ax aQ ax

aP ay =0 aQ -A ay

have positive real parts at the origin and negative real parts outside the circle x 2 + y 2 = R 2 ; moreover, 00 ./

m(r) dr = oo,

Then system (1.1) has at least one externally stable cycle and one internally stable cycle, and they may coincide to form a stable cycle.

Not only is this theorem interesting in itself, but also in the proof the author used equations of orthogonal trajectories of the system (1.1). This method is rarely used in plane qualitative theory.


21

Readers should note that if we want to extend the Poincare Annular Region Theorem to a multiply connected region in the plane, it is necessary to consider regions containing singular points, since according to the well-known theory of indices, if the outer and inner boundaries of any n-multiply connected region satisfy the conditions of the above theorem, then the sum of the indices of all interior singular points is 1 - n, but when n > 1, r must have singular points. Ye Yan-qian and Ma Zhi-en (31] extended the concepts of singular closed trajectories of Definition 1.1 and general concepts of singular points and then extended the Annular Region Theorem to regions which contain singular points and general n-multiply connected regions. Yu Shu-xiang (32] applied the results of (31] to study existence of closed trajectories of dynamical systems with singular points on the two-dimensional manifolds, and extended and improved the results of Sacker and Sell [33]; (32] also gave a new proof of • extended the theorem the theorem of (31]. In addition, Dong Zhen-xi (34] also of (31] to general two-dimensional manifolds. It is very interesting to consider the extension of the problems of limit cycles of the system in Example 1 on the annular surface. Tian Jing-huang [35] proved that the van der Pol equations

~: =

-siny + J.Lsin3x,

~~ = sinx

{0::::; x::::; 2rr, 0::::; y::::; 2rr)

on the annular surface have at least two limit cycles when 0 < IJ.LI « 1. As J.L -+ 0, these cycles approach two different nonzero periodic solutions of the corresponding equations. Similarly, Tian and Gao Long-chang (36] proved that the system

~:

=-siny+J.Lsin2x,

~~

=sinx

(O::::;x::::;2rr, o::::;y::::;2rr)

has at least one limit cycle when 0 < IJ.LI « 1. But in the above two papers, the properties of having at most one or two limit cycles have not been proved.

Exercises 1. Prove that the equations dxjdt = xy,

dyjdt = ao + a 1 x + a2x 2 + a3y 2

have two centers when ao > 0 and a2 < 0. 2. Prove that the equations dxjdt = x(x 2 + y2 - l)(x 2 + y 2 - 9) - y(x 2 + y 2 - 2x- 8), dyjdt = y(x 2 + y 2 - l)(x 2 + Y2 - 9) + x(x 2 + y 2 - 2x- 8)

have a unique (stable) limit cycle.


22

3. Calculate the curvature of the orthogonal trajectory of equations (1.1), and show it is equal to

:x (,(P:+Q') + ~ ( ,(P Q+Q 2

2 ).

Thus prove that if equations (1.1) have a closed trajectory

r, then [37]

·~ H(x,y)dxdy=O. !Jintr 4. Let G be a simply connected region. If there exist continuously differentiable functions B1 (x, y) and F(x, y) such that the expression.

a a [ aF aF] ax (BtP) + ay (BtQ) + Bt Pax + Q ay keeps constant sign and is not identically zero on any subregion, then system (1.1) does not have any closed trajectory or singular closed trajectory which lies entirely in G [23]. Hint: take B(x, y) = B1 (x, y)eF(x,y) in Theorem 1.11. 5. Let G be a simply connected region. If there exist continuously differentiable functions M(x, y), N(x, y), F(x, y), and B(x, y) which satisfy

M(x,y) ~ 0,

N(x,y) ~ 0,

MP 2

-

2FPQ + NQ 2 ~ 0,

a a a a ay (M P)- ax (NQ) + ax [P(B +F)]+ ay [Q(B- F)] ~ 0 (~ 0), and the points where the above equality holds do not cover completely any subregion of G, then (1.1) does not have any positively (negatively) oriented closed or singular closed trajectory [27]. 6. Prove that the van der Pol equation is equivalent to the system of equations

dzjdt

= x,

dxjdt

= -z + p,(x- x3 /3),

and use this system to construct an annular region to satisfy the corollary of Theorem 1.6 following the method of Example 2. 7. Assuming we have proved the uniqueness and existence of the limit cycle r of Example 2, prove that any trajectory starting from any point different from 0(0, 0) will take r as w-limit set as t -+ oo. 8. Suppose in a region G that IRI < h/2 holds everywhere, where R(x, y) is the radius of curvature of a trajectory of system (1.1) at (x, y) and h is a diameter of some bounded set V in G. Prove that then G does not have any closed trajectory which contains V in its interior.


23

9. Use Theorem 1.11 and the Dulac function similar to B(x, y) = xkyh to prove that when

the system

dxjdt = x(ax +by+ c),

= y(a1x + b1y + cl) when u = 0 the system has

dyjdt

does not have a closed trajectory, and integral but does not have a limit cycle. 10. Suppose in the system of equations

dxjdt

a first

= y- F(x), dyjdt = -g(x), F(O) = 0, g(x) is an odd function, xg(x) > 0 (x =I 0),

F(x) is an even function, g'(O) > 0, and F(x) and g(x) have continuous second-order derivatives. Prove that 0(0, 0) is a center [38].

§2 . Multiplicity and Stability of Limit Cycles In §1 we have defined the stability of limit cycles for a system of equations

dyfdt = Q(x, y).

dxfdt = P(x, y),

(2.1)

In applications only a stable limit cycle has practical significance, since every spiral sufficiently close to a limit cycle can approximately represent an oscillation of constant amplitude independent of initial conditions; and an unstable limit cycle fs similar to an unstable equilibrium position in mechanics, which in reality does not exist; hence, how to distinguish stability of limit cycles becomes a very important problem. First of all, if the existence of a limit cycle is determined from the fixed point of a point transformation, then no matter whether P and Q on the right sides of (2.1) are continuous or not, we can, under suitable conditions, distinguish whether the limit cycle is stable, provided that the point transformation is continuous. This is the often-used Konigs theorem in the theory of nonlinear oscillations. DEFINITION 2.1. Let 8 = f(s) be a continuous point transformation which carries some line. segment l into itself, and let s* be a fixed point of this transformation, i.e., s* = f(s*). If there exists a small neighborhood of s* (on l) such that for any point s inside it the sequence of points St

= f(s), 82 = /(81}, ... , 8n+l = f(sn), ...

always converges to 8*, then 8* is stable under this point transformation. Conversely, if in any small neighborhood of 8* we can find a point 8 such that the above sequence of points do not converge to 8*, then 8* is called unstable. (1) ( 1 )Note that under this definition, when the right sides of (2.1) are continuous and the uniqueness of its solution is assured, an unstable fixed point may correspond to a one-sided compound limit cycle.

2S


26

THEOREM 2.1 (KONIGS). Lets= f(s) be a continuous point transformation of a line segment l into itself, and let s = 0 be a fixed point of this transformation. If the section of the arc near the origin in the curves= f(s) in the (s, s) plane lies in the angular region ls/sl :51- E (~ 1 +E), E > 0,

(2.2)

then the fixed point s = 0 is stable (unstable). PROOF. Suppose ls/sl :5 1- E = 6 < 1 near the origin, i.e., there exists a small neighborhood lsi :5 11 of s = 0 such that for all points s :f 0 we have lsi :5 6lsl < lsi. Then, provided that lsi :5 17, the sequence of numbers lsl,lsll,ls21, ... will satisfy the inequality lsnl :5 lsl6n. Hence lsnl- 0 as n-co, i.e., s = 0 is stable. > 1, i.e., Conversely, if for every lsi :5 11 we have ls/sl ~ 1 + E = lsi ~ elsl > lsi, then it is easy to see any sequence of points s, SI, 82, ... cannot converge to 0; hence s = 0 is unstable.

e

COROLLARY. I/ the functions= f(s) has a derivative at s = 0, then when lds/dsls=O

< 1 (> 1)

(2.3)

s = 0 is stable (unstable). From this we can see that for some system whose right side may not be continuous, if there exists a point transformation on some transversal of the closed trajectory r which satisfies condition (2.2) or (2.3) and the solution of the equation depends continuously on the initial condition in some small neighborhood of r' then r is a stable or unstable limit cycle. EXAMPLE 1. Consider the point transformation obtained in Example 3 of §1 on the negative x-axis x1

= Jh 2 + (xo- /o)2 -

which has a fixed point x(j = h2 /8/o we verify condition (2.3):

< (dx1)

3/o,

- / 0 , and we know that

x(j

> f 0 • Now

x(j - fo < 1. ..jh2 + (x(j - /o)2 Moreover, although every trajectory in the vicinity of the closed trajectory r overlaps r on the line x = -/0 , it is obvious that the solution varies continuously with respect to its initial condition. Thus, from the corollary of Theorem 2.1, we can prover is a stable limit cycle. If condition (2.2) or (2.3) is not satisfied (for example, if the point transformations= f(s) satisfies lds/dsls=O = 1), then there is no way to distinguish the stability of the corresponding limit cycle. 0

dXQ

=

:ro=Xo

§2 . MULTIPLICITY AND STABILITY

27

In order to further study this problem, we shall assume P and Q on the right sides of (2.1) have continuous partial derivatives of any required order. Suppose system (2.1) has a closed trajectory r, negatively oriented, whose equations are y = g(t), (2.4) X= f(t), where f and g are periodic functions of t with period T. Now introduce curvilinear coordinates (8, n) in a sufficiently small neighborhood of r, where 8 denotes arc length on r' measured from a fixed point on r' positive in the clockwise direction, i.e., the direction of increasing 8 is the same as the direction of increasing t, and where n is the length of the normal of r, whose outward direction is taken as positive. Suppose the equations of r with s as parameter are

x = rp(8),

(2.5)

For a point A in the vicinity of r, assume that A lies on the normal line of a point Bon r (Figure 2.1). Let the rectangular coordinates of B be (rp(s),.,P(s)). Then the formula connecting the rectangular coordinates (x, y) of A and its curvilinear coordinates (s,n) is(2)

x = rp(8)- nt/11 (8),

y = '1/1(8)

+ nrp'(s),

(2.6)

where

rp'( 8) = dxl = Po '1/1'(8) = dyl = Qo d8 8 JPJ+Qf d8 IJ JPJ +Qf Po and Qo being the values of P and Qat B, i.e. Po= P(rp(8),'1/1(8)),

(2.7)

Qo = Q(rp(8),'1/1(8)).

Substituting (2.6) into (2.1), we obtain

dy '1/1 1(8) + rp 1 (8)(dn/d8) + nrp11 (8) dx = -'-rp..,.-7'(-78)----''1/1":-:'-7-(87)(7-:d----'n/,. . ,d;- - ;8):----_----'n.,P'-:- :,: -(--7-8) 7, Q(rp(8)- nt/1 1 (8),'1/1(8) +nrp1 (8)) = P(rp(8)- nt/1 1 (8), '1/1(8) + nrp 1 (8)) · (2)We can see from a(x,y) a(s,n)

=

ltp'-m/l' '1/J'+ntp"

that, if we require P and Q to have continuous first partial derivatives, then tp1 , '1/J', tp 11 , and f/1 11 are continuous; hence, provided In I is sufficiently small, we have a(x, y)ja(s, n) > 0,

i.e., in a sufficiently small neighborhood of r, (2.6) represents a coordinate transformation which preserves its orientation. If we choose n as the arc length of the orthogonal trajectory of (2.1), measured from the point on r instead, then the coordinate transformation can even be global; but this fact is not needed here.

28


FIGURE 2.1

From this we can solve the equation

dn ds

=

Qcp'- Pt/J'- n(Pcp" + Qt/J") Pep'+ Qt/J'

(2.8)

= F(s, n). Now we shall use equation (2.8) to study the stability of r.

Since cp and 1/J are periodic functions, we known (2.8) is a nonlinear equation with periodic coefficients, taking n as unknown function and s as independent variable. From (2.7), we can see that (2.8) has a zero solution, n = 0, which corresponds to the periodic trajectory r of (2.5). When P and Q have continuous partial derivatives of first order, F(s, n) also has first-order continuous partial derivatives with respect to n; hence (2.8) can be written as

dn/ds

= F~(s, n)in=O · n + o(n).

(2.9)

In order to compute F~(s, n)in=O• we note that

cp"(s) = '1/J"(s) =

(PJ

~oQ 5 )2 [P~Qxo + PoQo(Q11o- Pxo)- Q~P11 o],

(P~ :oQ5)2 [P~Qxo + PoQo(Q11o- Pxo)- Q~P11 o]

(2.10)

(where P 11o, Pxo, Q11o, and Qxo denote the values of the partial derivatives of P and Qat n = 0), and we know that P0 cp" + Q0 1/J" = 0. Noting again that P = Po and Q = Qo when n = 0, and using the above relation formula, we can easily calculate

F' ( n

s, n

)I _ n-0

= PJQ 11o- PoQo(P11 o + Qxo) (PJ + Q5)3/2

+ Q5Pxo

= H( )· 8 '

(2.1l)

here H(s) is the curvature of the orthogonal trajectory of system (2.1) at the point B. Hence the linear approximation equation is dn/ds = H(s)n,

(2.12)


29

whose solution is

n = noexp

(lol

(no= n(O)).

H(s')ds')

(2.13)

From this we get THEOREM 2. 2. Let the arc length of a closed trajectory r of system (2.1) bel. Then, when

fo' H(s) ds < 0 r is a stable (unstable)

(> 0),

{2.14)

limit cycle.

PROOF. When (2.14) holds, we have from (2.13)

ln(l)l < lnol (> lnol). This shows that the zero solution of {2.12) is asymptotically stable {negatively oriented, asymptotically stable). From the well-known theorem of Lyapunov we know that the zero solution of {2.9) is also asymptotically stable (negatively oriented, asymptotically stable) and it is obvious that r is a stable (unstable) limit cycle. COROLLARY (DILIBERTO). If H(s) < 0 (> 0) holds everywhere along a closed trajectory r then r is a stable (unstable) limit cycle [39]. Now we rewrite (2.14) in a familiar form. Substituting ds = into (2.14), we ge~

.jPJ + Q~ dt

{T P. 2 1 Q2 [PJQyo- PoQo(Pyo + Qxo) + Q~Pxo] dt fo o + o = {T + Q _ PJPxo + PoQo(Pyo + Qxo) + Q~Qyo] dt fo xo. yO PJ + Q~ {T 1 j d(P.2 + Q2)

f' H(s) ds =

fo

[p

=

lo

(Pxa

+ Qyo) dt- 2!r

P} + Q5o

=loT (Pxo + Qyo) dt. Thus we obtain the well-known THEOREM 2.3. !fin a closed trajectory

iT (~= then

r is a stable (unstable)

+ ~~)

limit cycle.

r

dt < 0

of system (2.1}

(> 0),

(2.15)


30

From the above two theorems, we know that when r is a periodic cycle, semistable cycle or compound limit cycle, we must have

{T

Jo

(8P8x + 8Q) {1 8y dt Jo H(s)ds =

= 0.

(2.16)

However, when r is an ordinary stable or unstable limit cycle, (2.16) may also hold, since (2.14) and (2.15) are only sufficient conditions, not necessary conditions. DEFINITION 2.2. When condition (2.14) or (2.15) holds, r is a single cycle or coarse cycle:(*) when condition (2.16) holds, r is a multiple cycle or

noncoarse cycle. In order to study stability of multiple cycles, we integrate (2.8) from s = 0 to s = l, and obtain w(no) = n(l,no)- no=

fa' F(s,n(s,no)) ds.

(2.17)

\ll(no) is called a successor function which denotes, when a point (0, no) in a neighborhood of r moves one round in the direction of increasing t (i.e., s) and arrives at the point

(l, n(l, no)) = (0, n(l, no)), the difference between the n-coordinates of these two points. Thus it is obvious \ll(no) = 0 is a necessary and sufficient condition for any trajectory passing through (0, n 0 ) to be a closed trajectory; noW(no)

< 0 (> 0)

r

is a necessary and sufficient condition for sufficiently small lnoli and w(no)

to be stable (unstable) for all

< 0 (> 0)

is a necessary and sufficient condition for r to be externally stable but internally unstable (externally unstable but internally stable) for all sufficiently smalllnol· Since we know that \11(0) = 0, we have THEOREM 2.4. If

w'(o) < o (> o) then

r

(2.18)

is a stable (unstable) limit cycle; if

w'(O)

=o

but

w"(o)

# o,

(2.19)

then r is a semistable limit cycle. (•)Translator's note: A single or coarse cycle is referred to in English journals as hyperbolic.


31

Using a well-known theorem in differential calculus, we can in general establish THEOREM 2. 5. If, for a given closed trajectory

w'(O) = w"(O) = ... = wCk- 1)(0) = o,

but

r,

wCk)(O) < o (> o), (2.20)

when k is odd, then r is a stable (unstable) limit cycle. If

w'(o) = w"(o) = ... = wCk- 1)(0) = o, where k is even, then

r

but

wCk)(o) -1 o,

(2.21)

is a semistable limit cycle.

REMARK. If P and Q (hence 'IJ(no)) are analytic functions, then (2.20) or (2.21) is also a necessary condition for being stable (unstable) or semistable. DEFINITION 2.3. Any r satisfying condition (2.20) or (2.21) is called a k- fold limit cycle. From this definition it is easy to see if r is a k-fold limit cycle, then no = 0 is a k-multiple root of the equation 'IJ(n0 ) = 0. If we draw the linen= no and the curve n = n(l, no) in the (n 0 , n) plane, then the origin is a k-fold point of intersection of these two curves. In particular, 'if( no), which corresponds to the periodic cycle, is identically zero for all sufficiently small values of lnol· In the following we shall prove that for k = 1 Definition 2.3 and Definition 2.2 are the same, i.e. the conditions

w'(O) < 0 (> 0) and

fa' H(s) ds
0)

are equivalent. F6r this, we differentiate both sides of (2.17) with respect to

no, and obtain w'(no)

In order to compute equation

=fa' F~(s,n(s,no))n~0 (s,no)ds. n~ 0 (s,

! n~0 (s,

and the initial condition

no), we note that it satisfies the variational

no)=

F~(s, n(s, no))· n~ 0 (s, no)

n~ 0 (0, no)

= 1; then we obtain

1 ( ) ];" F~(s,n(.s,no)) ds nno s, no = e o ,

and from this ,y,/( '!I!' no

)-!o'F'( n s, n (s, no ))J.•·F~(r,n(r,no))drds e 0

0

J.'F~(s,n(s,no))ds - . l =eo [ J. ·F~(r,n(r,n 0 ))dr] =eo 1

0

32


Note that n(8, 0)

= 0, and we know that

"\II'(O)

= ef~ F~(s,n)ln=ods _ 1 = ef~ H(s)ds -1.

(2.22 )

From this it is obvious "111'(0) < 0 (> 0) is equivalent to f~ H(8) d8 < 0 (> 0). In general, w(k)(O) can be obtained from the above method, but its representation formula becomes more complicated ask increases (see [40]). For example, "111 11 (0) = "111 111 (0)

fol F::2ef: F~(T,n)dT d8ln=O,

when "111'(0) = 0,

= fol F::~e 2 f: F~(T,n)dT d8ln=O,

w< 4>(o) =

when "111'(0) = "111"(0)

= 0,

fol F~!)e3 I: F~(T,n)dT dsln=O +2

1 1

F"' nse 2 J." o F~(T,n) dT

0

(2.23)

1 8

F"n2efo F~(T,n) dT dt d 8 In==O,

0

when "Ill., (0) = "Ill" (0) = "Ill"' (0)

= 0,

etc.

From the above formulas we see that in order to compute "111 11 (0), "111 111 (0), ... we first have to compute 2 (8, 0), F::~ (8, 0), ... , i.e., the coefficients of the Taylor series expansion of F(8, n) at n = 0. In the following, we introduce a method of the Japanese mathematician Minoru Urabe [41]; using this method we can also obtain, when r is a periodic cycle, the approximate representation formula for the period of a closed trajectory in the vicinity of r if we know that the period of r is T. Now we use the transformation (2.6), but still use t to denote the independent variable of the parametric equations of r j let T (time) be the independent variable of the parametric equations of a trajectory in the vicinity of r, with r(to) =to. Let x1(r),y1(T) be the point coordinates of this trajectory. Thus, referring to (2.4) and (2.6), we have

F::

r(to) =to,

x1(r) = f(t)

+ p(t)l(t),

y 1(r) =

g(t) + p(t)m(t),

(2.24)

where

= n(s,no), m(t) = 0, ao:

and if, moreover, there exists a positive function k(x, y) such that, for all (x, y), T > 0 is the smallest positive number for which

P(x, y, o: + T) = -kP(x, y, o:), Q(x, y, o: + T) = -kQ(x, y, o:),

(3.4)

then we say that F(o:) (for all o:, 0 :5 o: < T) forms a complete family of rotated vector fields. Let 0 be the angle which the vector (P, Q) makes with the x-axis. From (3.3) and (3.4) it is easy to see that for any 0:1 and 0:2 with lo:1- o:2l < T the following inequality holds: 0

< IO(x, y, o:t)- O(x, y, o:2)l < 1r.

(3.5)

Now we shall explain the geometrical significance of the above definition. First, from (3.3) it is easy to see that

80 ao:

= _i_ tan-1 Q = ao:

P

p

p2

1

+ Q2

Q

8P

8Q > 0

ao:

ao:

(3.6)

at all regular points. This shows that the vector at any regular point of the plane vector field defined by (3.1) rotates in a counterclockwise direction as o:

§3.

ROTATED VECTOR FIELDS

43

increases. From (3.4) and (3.5) we know that O(x, y, o: + T) = O(x, y, o:) + 71'j moreover, when the parameter varies from o: to o: + T, its corresponding vector completes exactly half a rotation, but does not necessarily keep the same length; and when o: increases to 2T, the vector completes exactly one round and returns to its original direction. This is the meaning of the S ao and a 1 - ao is sufficiently small, then for the a between ao and a1 the positive semitrajectory -y+(A, a) of F(a) starting from A will meet AB at a point Bo. near B; moreover, AB is a transversal with respect to F(a), i.e., all the trajectories of F(a), after crossing AB, will enter the above-mentioned simply connected region. On the other hand, 0 is also an unstable focus of F(a); hence the proof of this theorem is obtained by the Annular Region Theorem. REMARK. For Theorem 3. 7, conditions of rotated vector fields only guarantee that, as a varies from 0 toT, the elementary singular point whose index is +1 must change its stability; hence, for values of a in some interval, there exist limit cycles in a neighborhood of this singular point, all of which cover a region. If as a varies the system (3.1} does not form a family of rotated vector fields, but only its elementary singular point 0 can change its stability (naturally still passing the fine focus), for example, when a = ao, then from the proof of Theorem 3.7 we know(2) that if 0 is a stable focus for a= ao, then, for those a (in the vicinity of ao) which change 0 to an unstable focus, in a neighborhood of 0 the system (3.1) must have a stable cycle; conversely, if 0 is an unstable focus for a = o:o, then for those a (in the vicinity of ao) which make 0 become a stable focus, (3.1) must have an unstable cycle in a neighborhood of 0. Next we shall discuss the behavior of the separatrices of saddle points and the separatrix cycles formed by separatrices in the rotated vector fields. In ( 2 )Here

we use only the part asserting that 0 changes its stability at oo.


54

0

0

FIGURE 3.5

FIGURE 3.6

this area, except for the work of Duff, Andronov, and Leontovich in the sixties, we rely on recent work of Ma Zhi-en [51]. In addition, we shall introduce some conclusions which do not require F(o:) to be a family of rotated vector fields. THEOREM 3.8 [291]. Let F(o:) be a complete family of rotated vector fields, 0 :5 o: < T, and let the sector AOB be an arbitrary hyperbolic region of F(o:o) at the singular point 0 (Figure 3.5). Then as o: increases from o:o, the two sides 6A and OB of this hyperbolic region will continuously rotate around 0 in the counterclockwise direction, and when o: changes to o:o + T, OB will turn into the position of OA. PROOF. With 0 as center, draw a sufficiently small circular arc Jn, which forms from the hyperbolic region a bounded region G whose interior does not contain singular points of F(o:) and is covered completely by the hyperbolic trajectories of F(o:o). Because of the rotational property of vector fields, when o: > o:o the trajectories of F(o:), when meeting the two sides OA and OB of G, always enter the region G from the positive direction. Let M be an arbitrary point on Then 1+(M, o:) (o: > o:o) after entering C can neither run out of G from AO, nor enter the point 0, for otherwise it would touch tangentially some trajectory of F(o:o) in G, which is impossible. Hence 1+(M,o:) must run out of the region G from some point N on As the point M moves continuously to 0 and approaches 0, the point N will move along BA in the direction of A, but it cannot cross the point A; hence there must exist a limit point N. It is easy to prove that any negative semitrajectory of F( o:) through N must enter the singular point 0. Now we prove that as o:---+ o:o, ON will come close to and coincide with OB. For this we choose a monotonically decreasing sequence of parameters {O:n} which approaches o:o. Let a negatively oriented trajectory of the vector field

- -

OB.

AB.

-

-

-

55

§3. ROTATED VECTOR FIELDS

-

F( an) in G entering 0 meet the circular arc AB at N n (if there is more than ~ one such trajectory, then we choose the one closest to OB as ON n)· From (3.3) and (3.5}, it is easy to prove that, as n --+ oo, Nn will monotonically approach B along the arc AB, but it cannot pass B; hence Nn must have a limit position, which is denoted by N•. We now prove that N" = B. Suppose not. Then N• should be on the right of B. At the same time, the negative semitrajectory 1-(B, ao) of F(ao) which passes through a point B between the point B on BA and N" should arrive at a point C in the vicinity of Ao (Figure 3.6). By the continuity of a solution with respect to its parameter a, when n is sufficiently large (that is, lan- aol « 1), 1-(B, o:n) will also reach the vicinity of the point C. But this is impossible since F(an) already has a negatively oriented trajectory, which passes through N n and enters 0, to separate B from C. Hence we must have N" = B. Again we note AB is an arbitrary circular arc in a neighborhood of 0 which intersects the two sides of the hyperbolic region, and we know OB will rotate in the counterclockwise direction with respect to a. Since F(ao + T) and F(ao} possess similar trajectories, but with opposite directions, and since (3.5} holds when ao :::; a< ao + T, it follows that OB rotates to the position of 6A if and only if a = ao + T. The theorem is completely proved.(3}

-

-

COROLLARY. Let F(a) be a complete family of rotated vector fields, 0:::; o: < T. Then, as a varies, the four separatrices of the saddle point 0 all rotate around 0 in the counterclockwise direction.( 4 ) When a varies from 0 toT, the separatrix leaving (entering) the point 0 just turns into the position of the separatrix entering (leaving) 0.

From this corollary, we immediately know that, in the family of rotated vector fields F(o:}, when a varies from a 0 , the separatrix cycle of F(a 0 ) passing through a saddle point must split.( 3 ) When the separatrix cycle splits, a limit cycle may appear in its vicinity. To study this question, we first introduce a more classical theorem to distinguish the stability of a separatrix cycle, but use the method of proof in [51]. THEOREM 3.9. Let the system (2.1) have a separatrix cycle r 0 passing through its unique saddle point N, and let the other two separatrices passing (3)Refer to footnote 4. it is required that the rotated vector fields in a bounded interval of the parameters form a complete family, and that they rotate everywhere in a neighborhood of 0; otherwise this assertion does not hold. For an example, consider the system dx/dt = y, dy/dt = x + ax 2 y, which for 0 :5 a< oo forms a complete family, but whose vector fields on the two lines x = 0 and y = 0 do not rotate. At the same time, each trajectory entering (leaving) 0 has fixed slope ±1, no matter what value a takes. ( 4 )Here


56

through N lie in the exterior of r 0 • If P and Q are continuously differentiable, and at the point N

aP ~ aQ < 0 (> 0),

(3.20)

ax 8y then fo is internally stable (unstable).( 5 )

PROOF. In the following we consider only the case (Px + Qy)N < 0. We may as well assume fo is negatively oriented. Construct a complete family of rotated vector fields

F(o:):

dxfdt

= P(x, y, o:),

dyfdt

= "Q(x, y, o:),

such that (2.1) becomes F(O). For example, we can construct a family of uniform rotated vector fields starting from (2.1). Let 8() f8o: > 0. Since (P x(x, y, 0) + "Qy(x, y, O))N < 0, from the continuity property we know there exist a a-neighborhood of N and T/ > 0 such that, when (x, y) is in this neighborhood and lo:l < TJ,

P x(x, y, o:) + "Qy(x, y, o:) :::; -a < 0.

(3.21)

We first prove that in a sufficiently small inner neighborhood of r 0 there does not exist a closed trajectory of F(O). Suppose not. On the two separatrices of fo in a a-neighborhood of N we take regular points At and A 2 respectively (Figure 3. 7). The time corresponding to the section of the trajecis finite, say To, whereas the time corresponding to the sections tory NAt and A;N of the trajectory is infinite. Since Px + Qy is continuous, it is bounded in the interior of f 0 • Let Px + Qy < M. By virtue of the continuity of a solution with respect to its initial value, we can take very small neighborhoods f3t of At and fJ2 of A 2 such that the time from any point A~ on the closed trajectory of F(O) in fJ2 to any point A~ in f3t is greater than 2MTo/a. Moreover, let r meet the boundary of a a-neighborhood of Nat Bt and B2 respectively. Thus

A'JZA2

j(Px+Qy)dt= f_..._ (Px+Qy)dt+ f-..(Px+Qy)dt

1B1BB~

fr

+

f,__(Px

1A;A;

1.B~A~

+ Qy)dt +

f_..._(Px

1A;Bl

+ Qy)dt.

We note that the second and fourth integrals on the right side are negative, and, from the above estimate, we get 2MTo (3.22) (Px + Qy)dt < MTo- a· - - < 0. r a

i

( 5 )This theorem has been generalized to the case of a simple closed trajectory passing through more than one saddle point [162].


FIGURE 3.7

57

FIGURE 3.8

If in an arbitrary inner neighborhood of fo there exists a closed trajectory r of F(O), then r is a periodic cycle or one of the sequence of limit cycles. In the former case, we should have

i

(Px

+ Qy)dt =

0,

which contradicts (3.22). In the latter case, any limit cycle adjacent tor does not satisfy (3.22); hence in a sufficiently small inner neighborhood of fo there does not exist a closed trajectory of F(O). Now we prove Theorem 3.9. From the above analysis we know that fo is either internally stable or internally unstable. Supposing r 0 is internally unstable, we shall obtain a contradiction. Draw a sufficiently short transversal l of F(O) at an arbitrary regular point A of fo such that a positive semitrajectory of F(O) passing thrugh a point Bon l will meet l again at C, where B lies between A and C. When 0 < a « 1, l is also a transversal of F(a), and a positive semitrajectory of F(a) passing through B will meet l again at a point D between Band C (Figure 3.8). On the other hand, by Theorem 3.8 we know that the separatrix of F(a) leaving N intersects lata pointE in the exterior of fo. Hence the trajectories NE, fiF, and ED and sections of the transversals DB and FE form an annular region which satisfies the corollary of Theorem 1.6; hence in this region there exist one externally unstable cycle and one internally unstable cycle of F(a) (they may coincide). It is easy to see


58

that they should lie in the interior of r 0 . Hence, along this cycle, we should have

f

(P x(x, y, o:) + Qy(x, y, o:)) dt

~ 0.

On the other hand, by virtue of the continuity of a solution with respect to its parameter and its initial value, as long as B and A are sufficiently close, and !o:l is sufficiently small, the above integral should be less than zero, a contradiction. Thus, r 0 must be internally stable. The proof is completed. REMARK. The proof of this theorem does not use all the properties of saddle points. In fact, as long as (3.20) holds at the unique singular point N, and ro is a singular closed trajectory passing through N, with no other trajectories in the interior of ro passing through N (that is, F(O) in the interior of ro is a hyperbolic region), then the conclusion of Theorem 3.9 is established. We should point out if Px + Qy = 0 holds at N, then the stability of any separatrix cycle r o through N cannot be determined. EXAMPLE 3. Consider the system of equations dx

dt

= 2y = P(x, y),

It has a separatrix cycle 0(0, 0); here

~; = 2x- 3x2 + o:y(x 3 -

r 0 : x3 -

x 2 + y2

Px(O,O)

x2 + y2 )

= Q(x, y).

(3.23)

= 0 passing through the saddle point

+ Qy(O,O) =

0.

But take V(x, y) = x 3 -x 2 +y 2 • It is easy to calculate that along the trajectory of the equation dV fdt = 2o:y2 (x3 - x2 + y2 ). We can see that ro is internally stable when o: < 0, and ro is internally unstable when o: > 0. In the following we discuss the problem of generating limit cycles from separatrix cycles passing through saddle points. First we consider the case when F( o:) is a family of rotated vector fields. THEOREM 3 .1 0 [52]. Suppose the system (3.1) forms a complete family of rotated vector fields with respect too:, andro is a separatrix cycle of F(o:o) passing through a saddle point N. If at N

oP(x, y, o:o)

ax

+

oQ(x, y, o:o)

oy

'

(3.24)

then when o: varies from o:o in a suitable sense, a unique stable (unstable) limit cycle of F(o:) will be generated close to the inside of ro, and when o:


59

FIGURE 3.9

varies in the opposite sense, cycle in its neighborhood.

r o disappears

and there does not exist a limit

PROOF. We only prove the case in which ro is internally stable. As in Figure 3.9, let ro be positively oriented. Take any point A on r 0, and draw a normal line segment through A; when a point B on l is sufficiently close to A, the positive.semitrajectory "f+(B, ao) through B will meet l again at a point C between A and B. Let a increase from ao to at, lat - aol « 1, so that the positive semitrajectory "f+(B,at) of F(at) through B meets l again at a point D between B and C, and the line segment B D on l still remains as transversal. On the other hand, F( at) does not have any separatrix cycle, the separatrix r1 of F(at) leaving N always stays in the interior of ro, and the separatrix r2 entering N always stays in the exterior of r 0. Let them meet l at E and F respectively, and let EF remain as transversal to F(at). Similarly to the proof of the preceding theorem, we obtain an annular region in which there exist at least one externally stable closed trajectory and one internally stable closed trajectory (they may coincide). In fact, similarly to the proof of Theorem 3.9, we can show that the one generated is a unique stable limit cycle. According to Theorem 3.4, we know that, following the increase in a, the stable cycle generated by the separatrix cycle r 0 will monotonically contract inwards and completely cover an inner neighborhood of ro. Hence, by Theorem 3.2, we also know that as a decreases from ao, there does not exist any


60

closed trajectory of F(o:) in this neighborhood. The theorem is completely proved. REMARK. If at the singular point N 8P(x, y, o:o) 8x

+

8Q(x, y, o:o) _ 0 8y - '

(3.25)

then even if r 0 is internally stable (internally unstable) in a family of rotated vector fields, the uniqueness of a limit cycle of F(o:) (o: 'I o:o} generated by r 0 may not be guaranteed. EXAMPLE 4. Multiply the right side of each equation in (3.14) by a factor [(x- r 0 ) 2 + y 2 ]. The equations thus obtained have one more higher-order singular point (ro, 0); therefore r = ro also changes into a separatrix cycle. It is easy to see that condition (3.25} holds at (ro, 0). Outside the circle r = ro, both the new and the old equations have the same trajectories. Hence, according to the analysis of Example 2, we know that when vector fields rotate uniformly, the separatrix cycle at o: = 0 can generate more than one limit cycle when o: -:J 0. Similarly to Theorem 3. 7 on generation of limit cycles by a fine focus, we can give a sufficient condition for the generation of separatrix cycles passing through a fine saddle point; namely, that at this point Px + Q 11 = 0 (the condition is not confined to rotated vector fields). THEOREM 3.11 [51]. Let there be given equations containing a parameter o:

F(o:):

dx

dt = P(x, y, o:),

dy dt = Q(x, y, o:),

(3.1)

where P and Q are continuously differentiable with respect to x, y, and o:. Let F(o:) have a saddle point N which does not vary with o:. Moreover, when lo:- o:ol « 1, let F(o:) always have a separatrix cycle r a passing through N (its position can vary with o:). If

[Px(x, y, o:o) + Q 11 (x, y, o:o)]N = 0 and

r ao

is an internally stable (unstable) separatrix cycle of F(o:o), and for

0 < lo:- o:ol

«

1 lPx(x, y, o:)

+ Q11 (x, y, o:)]N > 0

( < 0),

(3.26)


61

FIGURE 3.10

then in an inner neighborhood of r 0 there must exist an externally stable (unstable) limit cycle and an internally stable (unstable) limit cycle of F( a) (they may coincide). PROOF. Here we only prove the case when r 00 is internally stable and the divergence Px + Q11 is greater than zero at N. We may as well assume fo is negatively oriented. At any regular point A on r 00 construct a transversal I of F(ao). From the internal stability of r ao and the continuity of F(a) with respect to a, we know as before that when 0 < a - a 0 « 1, the positive semitrajectory "Y+(B, a) of F(a) through B (which lies on l insider ao and is sufficiently close to A) will meet l again at a point D between A and B. It is easy to see that r a must lie in the exterior of the region surrounded by -y+(B, a) and DB (Figure 3.10). Let the point of intersection of r a and l beR. On the other hand, since for 0 < Ia- aol « 1 the inequality > holds in (3.26), r a is internally unstable, and then in a sufficiently small inner neighborhood of r a the positive semitrajectory "Y+(E, a) of F(a) passing through a point E on l sufficiently close to R will meet l again at a point F between E and D. Thus we apply the Annular Region Theorem in the annular region formed by "Y+(B, a), "Y+(E, a) and transversal segments BD and EF, and obtain the conclusion of this theorem. REMARK. The proofs of Theorems 3.11 and 3.10 are very similar, but now the problem is whether all the assumed conditions of Theorem 3.11 can be


62

established. This does not seem as obvious as for Theorem 3.10 or Theorem 3.7. The following is a concrete example mentioned in [51]. EXAMPLE 5. Investigate the equations dx dt

)

(l

= - 5ax+

17a- 5 2 2(a- 1) 2 6a y 3a2 x y

+ 36a2xy2 -

6ay4

-

192ax3 y + 192x5 + 24x2y3

= P(x,y,a), dy dt

l)

(

(3.27)

= 2 3a - y + = Q(x,y,a),

16(1- 4a) 2 2(a- 1) 2 3a x 3a2 xy

containing a parameter a. It is easy to verify that when a > 1/2 the origin 0 is a saddle point of (3.27). Let F(x, y, a) = 8x3 + y 3 - 6axy; the total derivative of F along the trajectory of (3.27) is

: = (8x 3 + y 3 -

6axy) [(a -1) ( 1- : 2 xy)

+ (24x2 -

6ay) 2].

It is easy to see that when a > 1/2, (3.27) always has a separatrix cycle passing through the saddle point 0,

r a: 8x3 + y 3 -

6axy = 0.

(3.28)

Since Px(O,O,a) +Q11 (0,0, a)= a-1, we know from Theorem 3.9 that r"' is internally stable for 1/2 1, and the critical case appears for a = 1. At the same time,

+ y 3 - 6xy)(24x 2 - 6y) 2; interior of r 1· It is not difficult to see here that r 1

dF /dt = (8x 3

thus dF I dt < 0 in the is internally unstable. Hence, according to Theorem 3.11, we know that as a decreases from 1, an unstable limit cycle of F(a) will be generated in an inner neighborhood of r a• In fact, when a = 1, equations (3.27) can be changed to dxfdt

= -4x + 2y2 + 6(8x3 + y 3 -

6xy)(4x 2

-

y),

dyfdt = 4(y- 4x 2 ).

These equations, besides the saddle point 0, have a singular point M(1/2, 1), which lies in the interior of the separatrix cycle r 1 , and is also a stable coarse focus. Hence, when lo:- 11 « 1, the interior of r a has only a unique singular point, which is still a stable coarse focus. But when o: < 1, since r a is


63

internally stable, there must exist an unstable cycle r a in the interior of r a, which contains the singular point Min its interior. When a increases to 1, a changes into r 1, and the stability of r 1 is changed. We should note that in the family ofvector fields F(a), as a varies, besides the limit cycles which may be generated by singular points and separatrix cycles, limit cycles may also be suddenly produced by the coalescing of trajectories. This fact can be seen from Theorem 3.5. Since a semistable cycle r ao may disappear as a varies in a fixed direction, if a varies in the opposite direction this is naturally the situation; originally F(a) does not have a limit cycle, and when a = ao a semistable cycle r ao suddenly appears; then r ao splits into at least two limit cycles. In the next section we shall prove that this suddenly generated limit cycle must be a multiple cycle; when F(a) does not form a family of rotated vector fields, this multiple cycle is not necessarily a semistable cycle. How to distinguish whether an equation involving a parameter can generate a multiple cycle when its parameter varies (even if this equation forms a family of rotated vector fields) is a very important, but very difficult and not yet solved problem. In the following we introduce some problems on the generalization of the theory of rotated vector fields. At the beginning of this section we mentioned that several persons have weakened Duff's conditions: for example, not requiring that P and Q have partial derivatives, letting inequality (3.3) become an equality, not requiring (3.4) to hold, etc.; and hope the main conclusion of Duff, i.e., the nonintersecting theorem (Theorem 3.2), can still be established. Chen Xiang-yan uged a more geometrical approach to consider this problem. In order to guarantee that Theorem 3.2 can be established, the key problem is to prove the following lemma, which seems obvious from direct observation, but is yet quite difficult to prove rigorously. Since he finally obtained a proof of the lemma, his generalization of Duff's theory seems better than others.

r

LEMMA (50]. Let Lo be a piecewise smooth simple closed curve whose parametric equations are x = rp(t) andy = .,P(t). At the points where rp' and 1/J' are defined, let

9(t) = rp'(t)Q(rp(t), .,P(t))- .,P'(t)P(rp(t), 1/J(t)). Suppose that when t increases Lo is positively oriented, and 9(t) ~ 0. Then there do not exist points on Lo where the trajectory of the equations dxfdt = P(x,y),

dyfdt = Q(x, y)

(3.29)

runs into the exterior of Lo when t increases, and there do not exist points on Lo where the trajectory of (3.29) enters the interior of Lo when t decreases. If


64

we change Lo to be negatively oriented, or change 8(t) ::::; 0, then the conclusion is just the opposite. As usual, we assume P and Q are continuous and the uniqueness of a solution of (3.29) is guaranteed. PROOF. The condition 8(t) 2:: 0 is equivalent to the folowing: any trajectory of equations (3.29) passing through a smooth point of L 0 when t increases either enters the interior of Lo or touches Lo tangentially at that point. This property is obviously independent of the parameter of £ 0 . Hence without loss of generality we can assume that the points where rp'(t) and 1/J'(t) are defined always satisfy r.p'2 + 1/1'2 'f. 0; for example, we can take t as the arc length of the curve. Let the period of r.p(t) and 1/J(t) be T; moreover, except at the n points t = ti (i = 0, 1, ... , n- 1), to < t1 < t2 < .. · < tn = to+ T, the functions r.p'(t) and 1/J'(t) exist and are continuous everywhere. We now prove that for any point t E (ti-1 1 ti) the solution (xo(t), y0 (t)) of (3.29) which passes through (r.p{t), ,P(i)) when t = i cannot leave Lo and run to the exterior of L 0 as t increases. To get a contradiction, suppose there exists a Ot > 0 such that when t E {t, t + Ot), (xo(t), Yo(t)) lies in the exterior of £ 0 • We now get a contradiction as follows. Consider an auxiliary system of equations containing a parameter a:

dxfdt

= P(x,y)- a,P'(t),

dyfdt = Q(x, y) + ar.p'(t),

t E (ti-l! ti), a > 0.

(3.30)a

Assume its solution is x = xa(t), y = Ya(t) when it satisfies x = r.p(i) and y = ,P(i) at t = t. Since r.p12 (i) + ,P12 (i) > 0, there must exist a 62 > 0, 82 < min(ti - t, t- ti-t), such that for t', t" E [t- 62, t + 82] we always have

r.p' (t')r.p' (t")

+ 1/1 t'),P' (t") > 0. 1(

Moreover, since the functions on the right sides of (3.30)a are continuous, there exists a 83 > 0 such that, when 0 L2, and let the parametric equations of L, be x = cp,(t), y = 1/J,(t); ( 6 )Ifwe consider separately the (closed) point set consisting of all the points on Lo which make 9(t) = 0, and all the comer points of Lo, and the (open) point set of all the points on Lo which make 9(t) > 0, and the property of a trajectory of (3.29) passing through these Points, then the proof of this lemma can be simplified, and there is no need to introduce equations (3.30) ...


66

suppose there are no singular points of (3.29) on Lt and L2 and in the annular region R bounded by them. Assume, moreover, at the points where IPi and tPi have derivatives, the inequalities cp~ (t)Q(cpt (t), tPt (t))

- '1/J~ (t)P(cpt (t), tPt (t)) ;::: 0 (~ 0),

cp~(t)Q(cp2(t), 'I/J2(t))- .,P~(t)P(cp2(t), 'I/J2(t)) ~ 0

(;::: 0)

always hold. Then when Lt and L2 are positively oriented there exist stable (unstable) limit cycles of (3.29) in R, and when Lt and L2 are negatively oriented there exist unstable (stable) limit cycles in R. This result is better than that in [53]. Next, Theorem 3.2 in this section can be generalized to THEOREM 3.13. If for the two systems of equations

dxjdt = Pi(x, y),

dyjdt = Qi(x, y)

(i = 1,2)

(3.31)

we have Pt(x,y)Q2(x,y)- P2(x,y)Qt(x,y);::: 0

(~ 0),

then their trajectories either coincide or do not intersect. PROOF. Let Li be a closed trajectory of (3.31)i, whose equations are x = IPi(t) andy= tPi(t) (i = 1, 2). If Lt and L2 intersect but do not coincide, then there are two possibilities: 1) Lt and L2 are tangent to each other externally or internally; 2) L2 has points in the exterior of Lt and points in the interior of Lt. In the former case, at their point of contact (if their points of contact cover a section of arc, i.e., if they have a common section of arc, then this arc will be understood as one point), as t increases or decreases L2 will enter respectively the interior or exterior of Lt. In the latter case, there must exist points of L2 on Lt entering the interior of Lt as t increases, and also points of L 2 on Lt running to the exterior of Lt as t increases. But cp~(t)Q2(1Pt(t),'I/Jt(t))-

.,pat)P2(1Pt(t),'I/Jt(t)}

= [PtQ2- QtP2]x= 0,

ayo we know that x 0 and y0 have continuous first-order partial derivatives with respect to x and y. Hence ho = h(xo, Yo), as a composite function of x and y, also has continuous first-order partial derivatives with respect to x and y. Finally, we prove that dhfdt has constant sign. To do this, we merely note that the trajectories of (3.37) are generated by rotated vector fields (3.36), but when a varies, the rotated vector fields (3.36) will strictly rotate, and hence the trajectory of (3.29) cannot be tangent to the closed trajectory of (3.37); that is, the closed trajectories of (3.37) are the arcs without contact of the rotated vector fields. Hence, along the trajectory of (3.29) dh ah ah dt = axP+ ayQ

> 0 (< O).

In the above formula, whether > or < holds depends on whether 0 is an unstable or stable singular point of (3.29). The theorem is completely proved. Before we conclude our discussion of rotated vector fields, it is interesting to study some properties of rotated vector fields in which the singular point can move. To make things clear, we now assume the singular point of a vector field moves following the variation of the parameter a, and can disappear, or split. But we require that the number of singular points after splitting be at most finite, and that this singular point not coincide with the original one. THEOREM 3. 16. A singular point which can move or disappear in rotated vector fields can only be one whose index is equal to zero and whose neighborhood does not contain an elliptic region. [291] PROOF. Suppose that for a1 ::f:. ao the singular point 0 of F(ao) moves to become a singular point 0' of F(al). Since 0 is a regular point with respect to F(at), there exists a sufficiently small circle C with 0 as center such that

72


there are no singular points of F(at) on C and in its interior, and there is a unique singular point 0 of F(ao) in it. It is obvious that the index of C relative to F(at) is zero, but the angle between the vector of F(at) and the vector of F(ao) is less than 1r. Hence the index of C relative to F(ao) is also zero, i.e. the index of 0 is zero. Similarly we can prove that the index of 0' in F( at) is also zero. Now suppose a neighborhood of the singular point 0 contains an elliptic region. For any arbitrary a1 #- a 0 , let 0 move to become a singular point 0' of F(at). Then we can always find a sufficiently small elliptic region G that does not contain any singular point of F(at). Since the boundary trajectory of G is a trajectory starting from 0 and returning to 0, the trajectory of F(at) passing through any point P on the boundary of G after entering G (when t increases or decreases) can only get out of G from 0. Since P is arbitrary, 0 is a singular point of F(at), which contradicts the assumption. COROLLARY. An elementary singular point in rotated vector fields cannot move following the variation of parameter. REMARK. If a family of rotated vector fields is not defined in the whole plane, then the elementary singular point on the boundary of the domain of definition can move following the variation of parameter. EXAMPLE 6. For the equations with 6 as parameter, dxjdt =ox- y + mxy- y 2 ,

it is easy to calculate that p

Q

8P/88 8Q/88 =-x 2 ( 1 +ax). Hence the equations on the two sides of the line 1+ax = 0 form two families of rotated vector fields with different directions of rotation. It is not difficult to verify that the positions of the two elementary singular points (whose indices are +1 and -1 respectively) on 1 + ax = 0 will move up and down following the variation of 8. Similarly to Theorem 3.2, the following still holds: THEOREM 3 .1 7. In the family of rotated vector fields whose singular point can move following the variation of the parameter a, let ri be the singular closed trajectory or closed trajectory of F(ai) (i = 1, 2). Then, when a1 #- a2, r 1 and r 2 do not intersect.

73


Corresponding to Theorems 3.3 and 3.4, we have THEOREM 3.18. In a family of rotated vector fields, let the singular point

0 move continuously following the variation of parameter a, but let it not split into several. Let r (ao) be an externally stable (externally unstable) singular closed trajectory of F(ao). Then when a varies from ao in a suitable sense, in the outer neighborhood of r 0 there exists at least one closed or singular closed trajectory of F(a) which contains fo in its interior; moreover, if the singular point of F(a) does not move out of the exterior of fo, then the outside of fo must have a closed trajectory of F(a).

The proof is omitted. See [291]. On the theory of rotated vector fields, a very important question, worthy of study, is: what kind of additional condition is required in order to guarantee that for every F(a) there exists a unique limit cycle? In this area Chen Xiangyan [50] obtained some results, but the conditions were rather too strong, and the scope of application was too small. After reading through Theorem 3.9, people naturally wonder: if there are several saddle points on a singular closed trajectory, and the divergence at these points can be positive or negative, then how are we going to distinguish the stability of this simple closed trajectory? In this area, L. A. Cherkas did some work in [54]. His main result was as (ollows: Let r be a positively oriented singular trajectory, and let A{ (j = 1, 2; i = 1, ... , n) be two characteristic roots of the i th saddle point on r, in which At > 0 > A~; let Ai = -AU At. Then for ll~=l Ai > 1 (< 1), r is stable (unstable).

Concerning limit cycles generated by the singular point in plane vector fields containing several parameters, there is some recent work by Takens [56] and others.

Exercises 1. Use the Annular Region Theorem to prove Theorem 3.2. 2. Prove that formula (3.12) and the first formula of (2.26) in §2 are the same when k* and k are the same function. 3. Starting from the equations dr 2 dt = -(r- 1)(r- 3) ,

dcp

di =

{ (r- l)(r- 2)(r- 3) seclr- 2111/2 when 0 < r < 4, but r 4/7r when r = 1, - 4/7r when r = 3,

f= 1, 3,


74

form a complete family F(o:) of uniformly rotated vector fields. Study the behavior of limit cycles following the variation of parameter o:, and indicate the region covered by the limit cycles. 4. Compare the behavior of the variation of limit cycles in rotated vector fields and the behavior of a change of real roots following the variation of the parameter C in the equation (~)

where the equation aoxn + a 1 xn-l +···+an = 0 does not have real roots and K is a constant. Describe what kind of behavior of rotated vector fields corresponds to the equation(~) at K = 0. 5. Prove that the point transformation x1 = f(x, y), Yl = g(x, y) with positive functional determinant o(xl, yt)fo(x, y) will transform a complete family ofrotated vector fields into a complete family [46]. 6. Let F1 (P, Q) and F2 ( P*, Q*) be vector fields which satisfy condition 1 (i.e., formula (3.2)) and condition 3 at the beginning of this section, and let them have common singular points. Then a necessary and sufficient condition for existence of a complete family F(o:) such that F(O) = F1 and F(7r/2) = F2 is that the inequality PQ*- QP* > 0 holds at all regular points [46]. 7. Let G be an open annular region which is completely covered by all the trajectories of system (3.1) (for all o:) containing the singular point. (0 is excluded in the region G.) In G define a function o:*(x, y) as in the proof of Theorem 3.16. Prove that if P and Q have continuous first-order partial derivatives, then o:*(x,y) also has continuous first-order partial derivatives in G, and satisfies a first-order quasilinear equation [55]:

oa• Q(x, y, o:*) oy

+ P(x, y, a*)

oa• ox = 0.

8. Use the fact that 0 is the center of

dxfdt

= P(x,y,o:*(x,y)),

dyfdt

= Q(x, y, o:*(x, y))

to prove that if r is a closed trajectory of system (3.1), then

i

H(s)ds = -

i ~: 0~ 0

ds,

where H (s) is the curvature of the orthogonal trajectory of (3.1) and oo.* I on represents the directional derivative of o.*(x, y) along the direction of the outward normal of r [37]. 9. Prove that if the equations

dxfdt

= P(>..x, y),

dyjdt = >..Q(>..x, y)

75


or

dyfdt = Q(x, )..y)

dxfdt = )..P(x, )..y),

form generalized rotated vector fields with respect to ).. E (0, oo), then the system (3.29) does not have a closed trajectory. 10. Use the following equations to verify the conclusion of Theorem 3.18

[51]: dxfdt = {[y + x(l- x2 - y 2)] cos a- [-x + y(l- x2 - y 2)] sin a} . [(x- g(a))2 + y2], dyfdt = {[y + x(l- x 2 - y 2)] sin a+ [-x + y(1- x 2 - y2 )] cos a} . [(x- g(a))2 + y2], where i) g(a) = v'l +tan a; ii) g(a) = v'l +a; iii) g(a) = v'1 + tan2a. 11. Use the viewpoint ofrotated vector fields to realize the significance of Theorem 1.13. 12. Prove that the two separatrices on the left side of x = 1/2 which pass through the saddle points (1/2, Yl > 0) of

dxfdt

= -y +ex+ xy- y 2 ,

dyfdt

= x(l- 2x).

(6 > 0)

must surround the origin 0. Suppose we kriow when D = 0, 0 is a stable focus. Prove for some D > 0 the neighborhood of 0 should have at least two limit cycles. 13. Apply Exercise 10 in §1 and the theory of rotated vector fields to prove the following criterion for nonexistence of limit cycles [38): Suppose in the equations dxfdt = y- F(x)- Q(x), dyfdt = g(x), where F(x) and g(x) are the same as in Exercise 10 of §1, Q(x) is odd and xQ(x) ~ 0 (or~ 0). Then there does not exist a closed trajectory of the equations. 14. Let the singular closed trajectory r have two saddle points Nt and N2, and let div Ni be the value of the divergence at the point Ni. Prove that when div N1

div N2

~>~ 1

r

2

( ) or
( 0, i.e.,

r

expands as a increases. Conversely, if r is a single unstable cycle, then

dnofdalno=O < 0, i.e., r contracts as a increases. These conclusions conform with those provided by the table following Theorem 3.4. If r is a multiple cycle, then

dnofdalno=O = oo, which also suitably reflects the results in Theorem 3.5. Extending the conditions of Theorem 4.2, we can get THEOREM 4.3. If, for every so, w~(O,so,O) # 0 and keeps constant sign, then in a sufficiently small neighborhood of r, closed trajectories of (4.1) corresponding to different a must not intersect. PROOF. From the conditions of the theorem, we know that for every so (0 ~so~ l), on the normal line segment off passing through (so,O) equation (4.5} defines a as a single-valued function of no whose domain is a nonzero interval and which satisfies a(s 0 , 0) = 0. Hence, according to the Weierstrass theorem, a( so, no) has a minimum interval of no which is not equal to zero and is suitable for all s0 • This shows that there exists a neighborhood of r which is filled with the closed trajectories of (4.1). These closed trajectories do not intersect, and this situation has been explained in Theorem 4.2.(2) (2)Rigorously speaking, from the conditions of the theorem we know that when lnol and lal are sufficiently small, for every so we have lll~(no,so,a) > 0 (< 0). Now suppose fa 1 and fa~ intersect at (s 0,n0). Then llf(n 0,s0,al) = llf(n0,s0,a2) = 0. Thus there exists i'i between a1 and a2 such that llf~(n0 ,s 0 ,a) = 0, which contradicts the previous formula.

§4. GENERAL BEHAVIOR

81

However, we should note that, although o: is a single-valued function of n 0 , its inverse function need not be single-valued. In particular, if for some so the range of values of o: degenerates to a single point o: = 0, then obviously for every s 0 the range of values of o: is o: = 0. At the same time, r is one of the system of periodic cycles of {4.1)a=O• and, by the property of nonintersection, system {4.1)a=O does not have a closed trajectory in a small neighborhood of r. We should also note that the nonintersecting property assured by Theorem 4.3 only applies to o: with very small absolute value; its corresponding trajectory not only lies in a small neighborhood of r, but also takes r as its limit position as o: -+ 0. We call this kind of closed trajectory the first type. It is highly possible that system (4.1) has a closed trajectory f* for some o: with rather large absolute value which intersects r, but as o:-+ 0 the limit position off* is another closed trajectory rl of (4.1)a=O· Then Theorem 4.3 is not applicable to both r• and r. We call r• (with respect to r) a closed trajectory of the second type. IT we require lo:l to be sufficiently small, then in a suitably small neighborhood of r there do not exist closed trajectories of the second type-otherwise it would contradict the continuity of solution with respect to its parameter. In order to guarantee that the conditions of Theorem 4.3 can be established, aside from assuming that system (4.1) forms a family ofrotated vector fields with respect to o:, there are some other methods. For this we first prove: LEMMA 4.1. Iff is not a single cycle, then, for 0 has the same sign. PROOF. Let so+ s =

~so~

l,

\11~(0, s 0 ,0)

s' in (4.7), and note that J~ H(s) ds = 0. We obtain

1s' H(s) ds) -80(s') ds' 8so so 89(s') = exp (Joro H(s) d1i) 1so+l so exp (- Jor' H(s) d1i) fia ds'.

\11~(0, s 0 , 0)

= 1so+l exp (-

0:

(4.13)

It is easy to see that 89(s')/8o: has period l with respect to s', and from the equality

lo s' +I H(s) d1i = is' H(s) d1i + ls' +I H(s) d1i

1

os'

0

=

;~

H(s) d1i + lo H(s) d1i

r'

= lo

H(s) ds,


82

we know that exp(- J;' H(s) as) also has period l with respect to (4.13) can be rewritten as

111~(0, s0 , 0) = exp = exp

(loro H(s) d8) lo{ exp 1

(fo

80

H(s)

(

-

s'.

Thus,

89(s') ds' lot' H(s) d8) ---a;;-

d8) 111~(0, 0, 0).

The proof is completed. According to this lemma, from Theorem 4.3 we can immediately obtain THEOREM 4.4. Ifr is not a single or periodic cycle, and 111~(0, so, 0) =f. 0 for some so, then, when a varies slightly to one side or to both sides, system (4.1) has a closed trajectory in a neighborhood of r; moreover, closed trajectories corresponding to different a do not intersect. COROLLARY. If for a= 0 the system (4.1) has a system of closed trajectories which completely fill a small neighborhood U of r, then, for a =f. 0 (but lal very smalQ, a necessary and sufficient condition to have a closed trajectory in U is that 111~(0, so, 0) 0.

=

This corollary is very useful in studying the third problem. Corresponding to Theorem 4.3, we have the well-known THEOREM 4.5. Ifr is a single cycle (i.e., 111~ 0 (0,so,O) =f. 0), then when is sufficiently small, system (4.1) has, in a neighborhood of r, a unique single cycler Q of the first type, which has the same stability as r.

lal

PROOF. From 111~ 0 (0, so, 0) =f. 0, we know that when Ia I is sufficiently small (but does not degenerate to a point o: = 0), equation (4.5) defines no to be a single-valued continuously differentiable function of o:, satisfying the condition n 0 (0, so)= 0; hence the existence and uniqueness of r a is proved. Moreover, since the value of H(s, a) ds varies continuously with o: and the position of r a, if Jr. H(s, o:) ds =f. 0, then, when lal is very small, H(s, a) ds has the same sign with it, i.e., r a is also a single cycle and also has the same stability as r. The proof is completed. Theorem 4.5 shows that when a varies slightly from 0 on any side a single cycle cannot disappear, and can keep its stability. But Theorem 4.4 shows that when r is a multiple cycle (but not a periodic cycle) and 111~(0, s0 , 0) =f. 0 for some a 0 , at least when o: varies slightly in one direction, the limit cycle does not disappear, but cannot keep its multiplicity, and its uniqueness is not assured. As an example, a semistable cycle in the rotated vector fields splits into two.

J

Jr ..

§4. GENERAL BEHAVIOR

83

THEOREM 4.6. If w~(O,so,O) ::/: 0 hold3 for some so, then a necessary and sufficient condition for r to be a semistable cycle is that a( no, so) takes a strict extremal value at no = 0; that is, when a varies from 0 to some direction, there appear limit cycles simultaneously on both sides of r for system (4.1), and when a varies in the other direction, these limit cycles disappear. PROOF. We may as well assume that W'~(O,s 0 ,0) > 0 at s0. Thus there exists o > 0 such that when lal < and lnol 0 for all no ::/: 0. Now we take 11 so small that 11 < o, and a( no, s 0) < o when lnol ~ 11· Thus, if no E (0, 17), then w~(n 0 , s 0, a( no, s0)) > 0.

o

But we already know that W'(no, s0, a( no, s0))

= 0; hence

W'(no,s 0,0) < 0;

that is, r is externally stable. Similarly, if no E ( -17, 0) we can still derive llf(no, s 0,0) < 0, that is, r is internally unstable. Combining all this, we see that r is semistable. Conversely, if r is externally stable but internally unstable, then, for all no ¥- 0 in a sufficiently small interval (-17, 17],

W'(no, s0,0) < 0,

w~(no, s;), a)

> 0,

\II'( no, s0, a( no, s0))

= 0.

Altogether we can see that a( no, s 0) > 0 for no ::/: 0; that is, a takes its minimum value at no = 0. Since a semistable cycle is a multiple cycle, under the condition of Theorem 4.6, if r is semistable, then not only does a take its extreme value at n 0 = 0, but also the closed trajectories which fill completely a neighborhood of r do not intersect. It is obvious that this theorem is a generalization of Theorem 3.5. We can also generalize Theorem 4.4 to THEOREM 4. 7. Let f be a k-fold cycle of equations (4.1} 0 we can find a1 < 0 < a 2 , lo:il < c, such that ll'(O, so, o:t) and ll'(O, so, o:2) have opposite signs. Thus,

84


there also exists 6 > 0 such that for all n 0, lnol < 6, 111(n0, so, at) and 'll(no, so, a2) have opposite signs. Hence for any no which satisfies lnol < 6, we can at least find an a( no), between at and a2, to make 111(n0, s0 , a( no)) = 0. This shows that system (4.1)a(no) has a closed trajectory passing through the point (so,no). If all the a(no) =: 0, then f is a periodic cycle of (4.1)a=O• which contradicts the hypothesis. Hence the range of a(no) is a nonzero interval which may have a = 0 as an interior point or an end-point. Note that a(n0 ) is not necessarily a single-valued function; that is, limit cycles corresponding to different a may intersect. Next we assume there exists a sequence of numbers ai - 0 such that the system (4.1)a, always has k + 1 or more limit cycles of the first type in a neighborhood of r. Then for each ai there exist at least k + 1 numbers n~ (j = 1, ... , k + 1) such that

(j = 1, 2, ... , k + 1; i = 1, 2, ' .. ). Thus from Rolle's theorem we know there must exist nb such that (i=1,2, ... ),

where nb is among the n~. Let i - oo in the above formula; then ai - 0 and nb - 0. Hence we obtain w(h) = 0,

=

J£,.

where Gh is the interior region of rh. Thus, Theorem 4.10 and its corollary become THEOREM 4.13. If 4>(h*) = 0 and 4>'(h*) =I 0, then when lal is very small, system (4.26) has a unique limit cycle in the vicinity offh". Moreover, if

then system (4.26) cannot have more thank limit cycles in the vicinity ofrh".

The first half of this theorem is a result of Pontryagin [58], and the second half is the result of Zhang Zhi-fen. The problem of whether the k limit cycles mentioned in Theorem 4. 7 can be realized has been studied in detail by Andronov and Leontovich [59]. Since their results are more complicated, we do not discuss them here. Please refer to the original paper or Qin Yuan-xun's book [67], Part II, Chapter 4. Exercises 1. Prove, under the condition of the corollary of Theorem 4.11, that if rh expands ash increases, then for an A~(h*) < 0 (> 0) the unique limit cycle generated by rh" is a stable (unstable) cycle. 2. Prove that ifr is a single cycle and "llf~(O, s 0 , 0) 0, then 80f8ala=O 0 (along r). 3. Explain why the coupling theorem obtained by exchanging the positions of a and no in Theorem 4.6 is obvious, and meaningless. 4. Prove, under the condition of Theorem 4.7, that if k is odd, then in a neighborhood of r there exists at least one limit cycle of system (4.1) 0 which has the same stability as r. 5. What is the coupling problem of problem III (with the same meaning as in Exercise 3)? What is the corresponding condition of (4.18)?

=

=

§5. Existence of a Limit Cycle Since van der Pol studied the equation

X+ J.t(x 2 - 1):.i: +X= 0

(J.t > 0)

(5.1)

in 1926, and proved that the system of first-order differential equations in the corresponding phase plane

dxjdt = y,

dyjdt = -x + J.t(1- x 2 )y

(5.2)

had limit cycles, his results have been extended by many workers in physics and mathematics. Up to now, people not only have studied the problems of limit cycles of systems of second-order nonlinear equations more general than

(5.1):

x + f(x):.i: + g(x) =

0

(5.3)

and

x + f(x, :.i:):.i: + g(x) = 0,

(5.4)

but also have extended (5.2), and studied problems of limit cycles of systems of equations which cannot necessarily be transformed to second-order nonlinear equations, such as systems of the type

dxjdt = h(x) + h(y),

dyjdt = g1(x) + Y2(Y)

(5.5)

dyjdt = Q(x, y),

(5.6)

and the most general equations

dxjdt = P(x, y),

among which there is a considerable amount of research for the case where P and Q are quadratic or cubic polynomials. We shall give a deep analysis and detailed study of (5.6) in the second half of this book. In this section we shall mainly introduce methods of proving the existence of limit cycles for equations of the types (5.3), (5.4), and (5.5). As to this problem, although in essence we use the Annular Region Theorem to ascertain the existence of a limit cycle, we still apply different techniques to different types of equations. We shall 91

92


mainly introduce several typical theorems for different types of equations and different techniques. Then in the end we enumerate some recent articles for the convenience of our readers' reference. I. Equations (5.4) and (5.3) have obvious physical significance in the theory of nonlinear oscillations. Hence, other than the hypotheses which guarantee the existence and uniqueness of their solutions, such as f and g being continuous (these will not be mentioned again), we shall assume that xg(x)

> 0 when x =F 0

(5.7)

(hence necessarily g(O) = 0) and

(5.8) /(0) < 0 or /(0, 0) < 0. (5.7) means that there exists a unique equilibrium position x = :i: = 0, and the restoring force is opposite to the direction of displacement; (5.8) means that the oscillation system has negative damping near the equilibrium position. Under these conditions, (5.3) is generally called the Lienard equation, the extension of equation (5.1) which was first studied by Lienard [9]. In order to study equation (5.3), in general, let G(x)

=fox g(~) d~,

F(x)

=fox/(~) d~,

(5.9)

and proceed to study the system of equations(!) dxfdt

= y- F(x),

dyfdt

= -g(x),

(5.10)

which is equivalent to (5.3). THEOREM 5 . 1 (A. V. D RAGILEV) . Let the following conditions hold:

1) xg(x) > 0 when x =F 0, and G(±oo) = +oo. 2) xF(x) < 0 when x =F 0 and lxl is sufficiently small. 3) There exist constants M > 0 and K > K' such that F(x) ~ K when x

>M

and F(x) ~ K' when x < -M.

Then system (5.10) has stable limit cycles. PROOF.(2) Let

A(x, y) = y /2 + G(x), A1 (x, y) = (y- K) /2 + G(x), -(1)Here we let y = :i; + f(x), which is not the same as taking y = :i; in (5.2). 2

2

This transformation is called the Liinard transformation; the (x, y)-plane is called the Liinard plane. The advantages are that two nonlinear functions are put separately into two different equations, and, moreover, we get to replace /(x) by its integral F(x), which has better smoothness properties. (2)Here we adopt a simpler method of proof proposed by Huang Ke-cheng [61] which makes the estimate of the interval of existence of limit cycles much easier.

§5. EXISTENCE

93

A2(x, y) = (y- K') 2/2 + G(x).

We compute their total derivatives with respect to t separately, and obtain

dA 8Adx 8Ady dt = ax dt + ay dt = y( -g(x)) + g(x)(y- F(x)) = -g(x)F(x), dAt dt

= g(x)(K- F(x )) ,

dA2 dt

= g(x) ( K

I

(5.11)

- F ( x )) .

From condition 1) we know that A(x,y) = C (C > 0) is a family of closed curves containing the origin. We call the curve the equivalent energy curve, and A(x, y) the energy function. From condition 2) and (5.11) we know that dA/ dt > 0 in the vicinity of the origin. Hence, for every closed curve A(x, y) = C in the vicinity of the origin, every trajectory intersecting it always crosses it in the interior-to-exterior direction. In order to apply the Annular Region Theorem, we only have to construct an outer boundary curve r of the annular region such that every trajectory intersecting r always crosses it in the exterior-to-interior direction. First, tiUce Yt > 0 such that Yt > K, -y1 < K', and for all the points in the region R+ = {- M :::; x :::; M, y ~ Yt} we have

y- F(x) > 0,

-g(x) K -K' y _ F(x) < 2M ,

and for all the points in the region R_

y- F(x) < 0,

(5.12)

= {-M:::; x:::; M, y:::; -yt} we have

-g(x) K- K' y-F(x) < 2M .

(5.13)

Let l = max((y1 - K') 2 /2, (-y1 - K) 2 /2]. We inspect the closed curve r = A"BCDA (Figure 5.1 corresponds to (-y1- K) 2 /2 > (y1- K') 2 /2), where BC is a closed curve At (x, y) = l + G(M) lying to the right of the line x = M, AD is a closed curve A2(x, y) = l + G( -M) lying to the left of the line x = -M, and AB and C15 are line segments. Let the point of intersection of the lines x = M and y = K be Ot and the point of intersection of x = - M and y = K' be 02. From the construction of r it is easy to prove that Yo :::; -y.,

YA ~ Yt,

l"lhVI = I02DI,

IOtBI = I02AI.

II DC and AB and DC lie in R+ and R_ respectively, with slope (K- K')/2M. From conditions 1) and 3) and (5.11)-(5.13), we know

Thus AB

/3

=

that all the trajectories intersecting r must cross it in the exterior-to-interior direction. The theorem is completely proved. The above method of proof can be extended to obtain sufficient conditions for existence of limit cycles of equations more general than (5.10):

dx/dt = h(y) - F(x),

dyjdt

=

-g(x),

(5.14)

94

THEORY OF LIMIT CYCLES '!/

D

o:--M

FIGURE 5.1

where g(x) does not necessarily have only one zero, and yh(y) > 0 when y # 0, h( +oo) = +oo, h( -oo) = -oo. For simplicity, we only consider here the case where h(y) y to prove one theorem. As to (5.14), we refer the readers to

[61].

=

THEOREM 5. 2. Let xF(x) < 0 (or> 0) when lxl # 0 .and lxl is sufficiently small, and let there exist constants M > 0, Xt > 0, and X2 > 0 such that 1) xg(x) > 0 when -x2 < x < 0 and 0 < x < Xti 2) F(x) ~ -M (or F(x) ~ M) when 0 < x < Xt, and F(xt) ~ M + ..,fii (or F(x) ~ -M- v'2l); 3) F(x) ~ M (or F(x) ~ -M) when -x2 < x < 0, and F(-x2) ~ -M- v'2l (or F(-x 2 ) ~ M + J2I), where l = max[G(xt),G(-x2)]. Then system (5.10) has stable (unstable) limit cycles. PROOF. Let

= !(Y + M) 2 + G(x), .X2(x,y) = !(y- M) 2 +G(x).

.X1(x,y) Thus it is easy to compute

= -g(x)[M + F(x)], d.X 2 jdt = g(x)[M- F(x)].

d.Xtfdt

(5.15)

The construction of the inner boundary curve of the annular region is the same as in Theorem 5.1. Now we are going to construct the outer boundary curve. In conditions 2) and 3) we only consider the case outside the

§5. EXISTENCE

0

95

s

FIGURE 5.2

parentheses. Inspect the closed curve r = ABCDEFA (Figure 5.2 corresponds to the case G(xt) > G( -x2)) in which AB is part of the curve ..\1(x,y) =(2M+ v'2i) 2/2; taking A(O,M +..til) as left end-point, it is easy to see that YB > M; AF is part of the curve ...\2(x, y) = l, and YF ~ M. DE is part of the curve ...\2(x, y) = (2M+ v'2i) 2/2, taking D(O, -M- ..til) as right end-point, and it is easy to see that YE < -M; fiG is part of the curve ..\1(x, y) = l, and Yo ~ -M; 1JlJ and EF are line segments. It is obvious that YB < YA = M +..til, YE > YD = -M- ..til. From the conditions of the theorem and (5.15) we know that all the trajectories of system (5.10) intersecting r always cross it in the exterior-to-interior direction. The theorem is completely proved. Since all the additional conditions on F(x) and g(x) in Theorem 5.2 are local, when the singular point of system (5.10) is not unique, i.e., g(x) has more than one zero, or the conditions G(±oo) = +oo do not hold, Theorem 5.2 is still applicable. EXAMPLE 1. The system of equations dxfdt = y- x 3

+ 3x,

dyfdt = x 2

-

6x

{5.16)

has limit cycles. (This system has singular points (0, 0) and (6, 198).) PROOF. Taking M = 2 and x 1 = x2 = 3, we can verify that the conditions of Theorem 5.2 hold. Applying the technique in Theorem 5.2 several times, we can give sufficient conditions for system (5.10) to have at least n limit cycles. For example, we


96

can prove that the systems of equations dxjdt

= y + x 2 sin x,

dyjdt = -x,

and dxjdt = y + x 2 sinx,

all have at least n limit cycles in the strip system of equations, 2x g(x) = (1 + x2)2,

lxl

~

mr + 1rj2.

x2

G(x)

= 1 + x2,

G(±oo)

For the second

= 1.

II. The second method for proving the existence of limit cycles of system (5.10) is quite different from the methods used in the previous two theorems. We now introduce this method, using a method of A. F. Filippov [62] as its representative. THEOREM 5.3. Suppose that in (5.10) g(x) satisfies

xg(x)

>0

when xi= 0,

G(±oo) = +oo;

(5.17)

and suppose that after a change of variables

fox g(~)d~=z1 (x), fox f(~)d~=F(x)=Fl(zl)

whenx>O,

(5.18)

fox g(~) d~ = z2(x), fox f(~) d~ = F(x) = F2(z2)

when X< 0,

(5.19)

the functions F 1(z) and F2(z) satisfy the following conditions: 1) For small z (0 < z < o) we have F1(z) ~ F2(z) but F1(z) ¢. F2(z); and F 1 (z) < a.fi and F2(z) > -a.fi, where 0 1 + We know that the integral curve passing through a point Bon the isocline y = F(z) lies to the left of the line z = zo. It has negative slope above y = F(z), and positive slope below y = F(z). (Figure 5.3.) But both its ends must intersect the y-axis; let the points of intersection be A and A'. It is easy to see that A = A' is not possible. PROOF. max[o,zo]

IF(z)l.

LEMMA 5.2. Let F(z) be continuous, F(O) = 0, and F(z) < aJZ (0 < a < v's) when z > zo. Then the integral curve of (5.22) starting from any point K(O, YK) on the negative y-axis must return and cut the y-axis at a point L(O, YL) with YL 2': 0. (Similarly, if F(z) > -aJZ, then the integral curve starting from any point M(O, YM) on the positive y-axis must return and cut they-axis at a point N(O,yN) with YN ~ 0.) PROOF. If the integral curve of (5.22) starting from K does not intersect the line z = zo, then the proof is the same as Lemma 5.1. Now let the integral

98


,

FIGURE 5.3

curve intersect z = zo at a point P, which lies below the isocline y == F(z). In the half-plane z > zo, we compare the integral curves of (5.22) and of the equation

dwfdy

= a.fW- y

(5.23)

passing through the same point P. Since (5.23) has the general integral

2w-ay.fW+y2 = Cexp{ h t a n - 1

[a~y]},

any integral curve of (5.23) has a point of intersection with the positive and also with the negative y-axis. We know that for the integral curve z = z(y) and the integral curve of (5.23)( 3 ) w = w(y) passing through the same point P, we should have z(y) < w(y) when y > yp; hence z(y) must intersect y = F(z) and then return to intersect they-axis at a point L. It is clear that YL ~ 0. REMARK. We can see from the proof of Lemma 5.2 that if in Lemma 5.1 we let F(z) > -a..(Z, 0 < a < v'B, then we must have YA > 0; if we let F(z) < a..(Z, then YA' < 0. Hence if under the condition of Lemma 5.2 we also have F(z) > -avz (or F(z) < a..(Z) when 0 < z < 6, then we can prove that YL > 0 (or YN < 0). ( 3 )1£ we let .,fiii = u, then (5.23) becomes du/dy = (au- y)/2u, when a 2 equation takes the origin as focus, and all the integral curves are spirals.

YA ;:::: 0 and Yc < YD ~ 0. The figure in the (x, y)-plane corresponding to Figure 5.4 is shown in Figure 5.5. The points with the same letter in the two figures correspond to each other. Since dxfdt > 0 on the line segment A'G' on the positive y-axis and dxfdt < 0 on the line segment C' D' on the negative y-axis, we know that B' C' D' E' G' A' B' forms the inner boundary curve of the required annular region.( 4 ) Next, according to the conditions of Theorem 5.3 and the remark after Lemma 5.2 we know that the integral curve of equation (5.21)* starting from the point K on the negative y-axis must return to intersect the y-axis at L, YL > 0, and YL increases when YK decreases. Let limyK-+-oo YL = YM; then either YM = +oo, or 0 < YM < +oo.

=

( 4 )If D 0, then D' = 0. Then we use the trajectory_...£assing C' on the left half-plane and the vertical line passing E' to replace the segment 0 E'.

100


z

N'

FIGURE 5.6

FIGURE 5.7

We first consider the case YM < +oo. In this case the integral curve of equation (5.20)* starting from M should return to intersect the negative yaxis at a point N, YN < 0; and the integral curve of (5.21)* starting from N should return to intersect the positive y-axis at a point P below M (Figure 5.6). Turning to the (x, y)-plane, we immediately see that the trajectory M'ii'P' and the line segment M' P' form the outer boundary curve of the annular region, because the trajectory when intersecting M' P' must cross it from left to right. This completes the proof of existence of stable limit cycles (Figure 5.7). When YM = +oo, the above proof is not suitable. We still need the following lemmas. LEMMA 5.3. Let Yt(z) and y2(z) be solutions of equations (5.20)* and (5.21)* respectively, and let them satisfy the same initial conditions Yt(O) = y 2 (0) = YO· For zo in condition 2) of the theorem and for any e > 0, we can take IYol sufficiently large that Yt(z) and Y2(z) are defined on the interval [0, zj, and satisfy the inequalities IYt(z)- Yol < e and IY2(z)- Yo I < e when 0 ~ z ~ Zo·

PROOF. Let m = max[o,zoJfiFt(z)l, IF2(z)j]. If IYol ;;::: m + zo/e + e, it is easy to prove Lemma 5.3. The readers can supply the details for themselves.

§5.

101

EXISTENCE

From Lemma 5.3 we can immediately obtain LEMMA 5.4. AsiYol-+ oo, 2

Yo -+ 1 (Ft(z)- Y1(z))(F2(z)- Y2(z)) uniformly in [0, zo]. LEMMA 5.5. There exists a positive number p such that Yt(zo) for IYol > p.

< Y2(zo)

PROOF. From y 1 (0) = y2 (0) and (5.20)* and (5.21)*, it is easy to establish that

y~(Y2(zo)- Yt(zo)) _

{ "0

- lo

(y2(z)- Yt(z))y~ dz {"0 (F1(z) - Y1 (z))(F2(z) - Y2(z)) + lo (Ft (z)- F2(z)) dz

~ + lor~( (Ft(z)- Y1(z))(F2(z)Y2(z)) -

)

1 (Ft(z)- F2(z)) dz

= /1 +/2 +/3. As IYol -+ +oo, we know that It -+ 0 by Lemmas 5.3 and 5.4. Moreover, since F1 (z) - F2(z) is bounded on [0, zo], I a -+ 0. Besides, according to condition 2) we know that /2 > 0. Hence, as IYol-+ +oo, the right side of the above formula approaches a positive limit; and when IYo I is sufficiently large, Y2(zo)- Yt(zo) > 0. The lemma is proved. We now prove Theorem 5.3 when YM = +oo. Take an integral curve KQRL of equation (5.21)* such that YK < -p and YL > p. Through L and K construct integral curves LST and KV of (5.20)*. From Lemma 5.5 we know that Ys < YR and Yv < YQ· From condition 2) and the comparison theorem we know that §T cannot intersect RQ and must return to meet z = zo at a point U above Q (Figure 5.8). Returning to the (x, y)-plane, we know that the section of the trajectory V'K'Q'R'L'S'T'U' and the line segment U'V' together form the outer boundary curve of the required annular region (Figure 5.9). The theorem is completely proved. REMARK 1. H for every z in (0, t5) we have

---

-av'z < F2(z)

= Ft(z) < a..[Z,

then it is easy to see that system (5.10) takes the origin as its center. This is the extension of the symmetry principle in §1. REMARK 2. It is easy to prove that Theorem 5.1 is a special example of Theorem 5.3. Hence in Theorem 5.1 we only need to assume f(x) and g(x) are continuous; then we can guarantee the existence and uniqueness of the solution. In fact, even the theorem of de Figueiredo [63], which was published


102

" ::

FIGURE 5.8

FIGURE 5.9

a few years later than the results of Filippov, is also a special example of this theorem. Besides, from condition 1) of Theorem 5.3 we can establish a rule to determine nonexistence of closed trajectories. THEOREM

5.4. If we have F1(z) $ F2(z) for allz > 0, andF1(z) ¢ F2(z) > 0, then system (5.10) does not have a closed trajectory.

in (0, 6) for any t5

COROLLARY. If g(x) = x, F(x) = s(x) + r(x), where s(x) is an even polynomial, s(O) = 0, r(x) is an odd polynomial, and xr(x) ~ 0, then system (5.10) does not have a closed trajectory.( 5 )

m. In the following we tum to study equation (5.4). Note that we can only use general methods to transform this equation into a system of firstorder equations; we cannot use the Lienard transformation. The problem of limit cycles of this equation was first studied by Norman Levinson and Oliver K. Smith, but here we introduce the result of Dragilev [65] in which the hypotheses are weaker than the conditions of Levinson and Smith, and the method of proof has been improved, and is different from the two methods introduced in the previous subsections. THEOREM 5.5. Inequation(5.4) letf(x,v) andg(x) satisfythefollowing

conditions: 1) xg(x)

>0

( 5 )Compare

when

lxl > 0,

Exercise 13 in §3.

and G(±oo)

= +oo.

§5. EXISTENCE

103

FIGURE 5.10

2) f(O, 0) < 0. 3) There-exists xo > 0 such that f(x, v) ~ 0 when lxl ~ xo, and there exists M > 0 such that f(x, v) > -M when lxl ~ xo. 4) There exists Xt > xo such that for any decreasing positive function v(x)

1

2:1

f(x, v) dx ~ 4Mxo +a

(a> 0).

(5.24)

dvjdt =- f(x, v)v- g(x),

(5.25)

zo

Then equation (5.4) has stable limit cycles. PROOF. Consider the system of equations

dxjdt = v,

which is equivalent to (5.4), and, as in Theorem 5.1, let

From condition 1) we know that ..\ (x, v) = C represents a system of simple closed curves containing the origin; from condition 2) we know that in the vicinity of the origin we have d..\ dv dx 2 dt =vdt +g(x)dt =-v f(x,v)~O.

Hence we can take a very small C1 > 0 such that d>./dt ~ 0 everywhere on .\(x, v) =Ct. This closed curve can be taken as the inner boundary curve of the annular region (Figure 5.10).

104


Now we study the trajectory 1 starting from a point Ao(xo, a) (a> 0) on the line x = xo. From condition 3) we know that d>.jdt ~ 0 to the right of x = x0 • However, the singular point is unique; hence 1 must intersect the x-axis and then return to intersect x = x0 at a point Aa. Since we shall take a very large later, we may as well assume 1 also meets x = x1 at two points A1 and A2. When lxl ~ xo we know from condition 3) that

dv = _ f(x, v) _ g(x) ~ M + maxlxi<xo ig(x)i < M dx v ivi

+1

'

when lvl > maxlxl~xo lg(x)l. On the other hand, from the inner boundary curve of the annular region constructed before, and the fact that dvjdtiv=O = -g(x), we know that 1 after Aa must intersect the negative v-axis and enter the left half-plane. After that, it may cross the negative x-axis from the right of x = -xo and enter the second quadrant, or it may cross x = -xo through A4. But on the left of x = -xo we have d>.jdt ~ 0; hence, later on, 1 should still intersect the negative x-axis and then turn back to intersect x = -xo again at As Using the same proof in the lower half-plane, we know 1 will cross the positive v-axis, and then intersect x = x 0 again at a point A6 • Now we shall prove that if a > 0 is taken sufficiently large, we can make A6 lie below Ao. At the same time, the section of the arc from Ao to A6 on 1 and the line segment AQ'A6 can form the outer boundary curve of the required annular region. We first prove that, for any given positive number N, we can always choose the ordinate a of Ao greater than N and so large that the arc AoA1 · · · A6 can intersect x = XI. and its part inside the strip -xo ~ x ~ X1 must lie above v = N or below v = -N. In fact, if 1 after Ao does not intersect x = x1. then we can choose instead the trajectory 1' starting from Ai (xi. a'), a' > a, to replace 1, so that when t increases 1' can still move around the origin as 1 does. It would be better if 1' can intersect x = xo as t decreases; otherwise, the negative semitrajectory of 1' must approach oo from the right of x = xo (since dxjdt > 0 and dv/dt < 0), while at the same time we can take A~ = oo, and A~ must be below A~. So the proof of the theorem is complete. Similarly, for example, if A 3 lies above v = -N, then we can take the trajectory 1" starting from A~(xo, -N- 1) to replace 1, etc. Let Vi denote the ordinate of Ai. Now we estimate v5. Use (5.24) to obtain

Vt-Vo = -

l."'' xo

1

f(x,v)dx-

[."' x0

g(x) dx V

Xl

.(x,v) decre~Ules Bll t incre~Ules, but G(x) hBll the same value at A1 and A2; hence (5.27) Along A--;A;, >.(x, v) still continues to decreBlle, and so !v~ G(xl); thus

+ G(x0 ) ~ !v~ +

I 3 I I 2I < 2[G(xl)- G(xo)] G(xl) - G(xo) v - v lv31 + lv2l < N ·

(5.28)

For the arc A~, we have V4

-v3 =- 1

-xo xo

f(x,v)dx-

1-xo ( ) xo

~dx v

f(x,v)dx+ { 0 g(x) dx+ ro g(x) dx. 1o V 1-xo 1-xo V Using condition 3), we can get = r

lv41-lv31 < Along

ro 1o

g~) dx +

roM dx = 2Mxo + 1-xo

G~o).

(5.29)

....G"k we have d.\jdt ~ 0; hence vs -

lv41 ~ 0.

(5.30)

Finally, for A~ and A~, similar estimates lead to

v6- Vs < 2Mxo + G(-xo)/N.

(5.31)

Adding (5.26)-(5.31), we get

v6- vo 0 and M > 0 such that

fox(Cg(x) -F(x)sgnx)dx < M

(x > 0).

Then system (5.25) has stable limit cycles. PROOF. As in Theorem 5.5, it is not difficult to construct an inner boundary curve. Now let us construct an outer boundary curve. First we prove that if any trajectory 1 from any point (xo, vo) in the first quadrant does not intersect the x-axis as t increases, then x(t) must go to +oo. Suppose not. Since dxjdt = v > 0, 1 must remain in the region xo :5; x :5; x1 < oo, v > 0. Since 0(0, 0) is the unique singular point of system (5.25), v(t) must be unbounded; that is, lim v(t) = +oo, t-+T"

where T* is an upper bound of t which can be continued along the positive semitrajectory of 1, and may be +oo. But when T < t < T* (x(T) = x0 , v(T) = vo), we have

v(t) = v(T) -1x f(x, v) dx -1x z(T)

:5; v(T) -1x

g(x) dx

:z:(T)

V

F(x) dx;

:z:(T)

here the right side is bounded as t -+ T*, and the left side is unbounded. This is impossible. We next prove that 1 cannot remain in the first quadrant and still make x(t) -+ +oo. Since

v :5; vo-

1z

F(x) dx-

:z:o

1z zo

-g(x) dx < v0 v

-

1x

we must have lim

fx F(x) dx < +oo,

z-++oo }zo

zo

F(x) dx,

(5.32)

107

§5. EXISTENCE

for otherwise the left side of the above formula cannot always be greater than zero. Then from condition 3), we get that

fo+oo g(x) dx =

+oo,

and so by the boundedness of v the three terms in the middle of (5.32) can take negative value. This is also impossible. In short, the trajectory"' starting from any point (xo, vo) in the first quadrant must later intersect the x-axis and enter the fourth quadrant. We note that in the fourth quadrant after"' crosses v = -6, we have v ~ -6 < 0 along 1 (refer to the proof of the corresponding part in Theorem 5.5) and dxfdt < 0; that is, x remains bounded. Hence from

v-vo

~-

l:r::or: F(x)dx- l:r::r:o -g(x)d x

(x < xo)

V

we know that v has a lower bound; thus "' must intersect the negative v-axis. Since condition 3) of the theorem is independent of the sign of x, the behavior of 1 in the third quadrant is similar to its behavior in the first quadrant, and the behavior in the second quadrant is similar to its behavior in the fourth quadrant. This means "' must later go around the origin once and return to the first quadrant. We have thus proved under conditions 1), 2), and 3) that the polar angle of the point along the trajectory "' must increase without bound as t increases. Now we prove J;here exists a negative semitrajectory,(6 ) which goes to infinity by condition 4). Consider the trajectory "' passing through the point (O,vo) at t =to, where vo = -M -2/C. Let the maximum time during which "(can be continued in the negative direction beT < t ~ t 0 • Now we prove that in this interval we must have v(t) < -1/C. First, this inequality holds near t =to. If there exists t1 > T such that v(t) < -1/C when h < t ~to and v(tl) = -1/C, then in the interval h < t ~to we have

r

r

2 v(t) 0.

In fact, using (6.15) it is easy to compute that at (s', 8 1 = s), we have ds - dst = e2h2r2+"11rt) 0. ds' ds' rp(r2, -12) Hence we must have 0

< dstfdsip < 1.

{6.17)

If these two curves have a second point of intersection Q, then at Q we should have ds1 > ds ds' - ds'

0 > ' which implies that

~:

L

2::: 1.

{6.18)

But this is impossible because we can see from {6.16) that as 8 increases ft increases whereas f2 decreases, and so we would have dstfdslq < dstfdslp, which, together with {6.17), indicates that (6.18) cannot hold. This proves the uniqueness of the fixed point; that is, system {6.5) has a unique limit cycle. From (6.17) we can see (according to Theorem 2.1 in §2) that it should be a stable cycle. Case 2. 0 < h1 :::::; h2 < 1. In this case if the two curves s = f(s') and s1 = g(s') intersect, then at the first point of intersection we would have dstfds = e2h2r2-"''t1't) :::::; 1, but this is impossible since f2 > 1r >ft. and 12 2::: 11 when h2 2::: h1. We can prove that when h1 2::: 1 > h2 > 0, system (6.5) also has a unique stable limit cycle, while (0, 0) is a stable nodal point for the trajectory in region (II). Hence {6.6) cannot be used again. The proof is similar to the first case, hence it is omitted here. For general nonlinear equations, since there is no way to give a clear analytic expression for the point transformation, we cannot prove the uniqueness of the limit cycle in the same way as we have done before. However, according


124

• FIGURE 6.8 to the idea of the point transformation, we can use the following method to prove the uniqueness. (a) We first use the qualitative method to prove that any trajectory starting from any point A on a half-ray l passing through the singular point 0 must return and meet l again at B (Figure 6.8). (b) Next we prove that IDAI: lOBI (or the difference fOAI-1 0 B I) increases (decreases) monotonically as IDAI increases. Then IOAI : lOBI can equal 1 at most once (I'OAI-IOBI can at most equal 0 once); this shows that the limit cycle, if it exists, must be unique. The authors of [63] and [64] used this method to study the uniqueness of a limit cycle, and obtained better results. II. The Poincare method. This method was given in Poincare's classical paper Integral curves defined by differential equations [1]. We first prove LEMMA. Let the differential equation in polar coordinates

dp/ dw = cp(p, w)

(6.19)

be given, where the origin 0 is a singular point. Let 1/J(p) be any single-valued continuous function of p. If equation (6.19) has two closed trajectories r1 and r2. then there exists a point between rl and r2 such that one of the following five relations holds: d'ljJ

-=oo, dp

cp(p,w) = oo, ocp -=oo,

ap

~"" dp2 = oo.

(6.20)

PROOF. If one of r 1 and r 2 does not contain the other, or if they do and 0 lies in the exterior of these two trajectories or between them, or if they do and 0 is in the interior of the inner trajectory but there exists a half-ray through 0 which has mor.e than one point of intersection with one of them, then for

§6. UNIQUENESS

125

all these cases it is easy to prove between rl and r2 there must exist a point such that cp(p, w) = oo-for this we should use the well-known formula dp

tanT = p/ dw' where T represents the angle between the radius vector passing through a point and the trajectory of (6.19) passing through this point. Now suppose that 0 c rl c r2 and any half-ray w = Wo starting from 0 intersects r i at only one point (Wo' p~i)) (i = 1' 2). Study the function

~(wo) = 1/J(p~2)) - 1/J(p~t>). It is easy to see that this is a continuous periodic function of wa; hence there exist an w1 at which ~ takes its maximum value. At w = Wt we have d~(w)/dwlw=w 1 = 0;

that is,

dp) - (d"' dp) ( df/J dp . dw (wi,P~2)(wi)) dp • dw (wl,p~l)(wi))

(6.21)

We now fix w11 and consider (df/l/dp)(dp/dw) as a function of p; the interval of variation of pis [p&1>(wt),p&2 >(wt)]. From (6.21) and Rolle's theorem we know there exists p*, p&1>(wt) < p* < p~2 >(wt), such that at the point (wt,p*) at least one of the following three equalities holds: df/J dp dp dw

=.00

'

!_

ap

(d"'

dp) dp dw

=

0,

!_ ap

(d"'

dp) dp dw

= oo.

Substituting dp/dw = cp(p,w) in the above formulas, we get (6.20). The lemma is completely proved.THEOREM 6.1. If for equation (6.19) we can find a function 1/J(p) such that in some simply connected region G none of the five relations in (6.20) hold, then there exists at most one closed traiectory in G, and it must be a limit cycle.

This is obvious from the lemma. COROLLARY. If 1/J(p,w) is a quotient of two continuous functions, and the denominator in the region G is not equal to zero, and along any half-ray w = wo, cp(p,w) is a monotonically increasing (decreasing) function of p, then there exists at most one closed traiectory in G. PROOF. By hypothesis there is no point in G such that !P(p,w) = oo. Hence from the proof of the lemma we can see that if there exist two closed trajectories r 1 and r 2t then they must contain the origin in their interior, and


126

any half-ray w = wo can only meet ri at one point. Since cp(p,w) is monotone with respect to p, we get immediately

0 = rf.. cp(p1(6), 0) dO > ftl ( 0.

k],

Then when k > x~ + y~, system (6.22) has a unique limit cycle, which is stable (unstable) when a< 0 (> 0); if k ~ x~ + y~, then (6.22) does not have a closed trajectory [105].( 5 ) PROOF. When k > x~ + y~, the circle F(x, y) = 0 contains the origin in its interior. Let V(x, y) = !(x2 + y 2 ). Then we can calculate

dV dx +ydt dy =ax ( 2 +y2)[( x-xo )2 + (y-yo )2 -k.] dt=xdt Take r > 0 so small that the circle Cr : x 2 + y 2 = r 2 lies entirely in the interior of the circle F(x, y) = 0; thus dvjdt and a have different signs on Cr. Similarly, takeR > 0 so large that the circle CR: x 2 + y2 = R 2 contains the circle F(x, y) = 0 in its interior; thus dvjdt and a have the same sign on CR. Hence there must exist a limit cycle between Cr and CR, which is stable (unstable) when a < 0 (> 0). To prove its uniqueness, we can transform (6.22) into polar coordinates, and obtain (6.23) dpjdw = pF(pcosw,psinw) = cp(p,w), where F(pcosw,psinw) = a[p2 Now take 1/J(p)

= 1/p.

d.,Pjdp-:; oo,

-

2p(xocosw+yosinw) +

pg- k],

pg = x& +Y&·

Then it is easy to see that between Cr and CR we have

x2. 2 Note that (6.42) and (6.36) are identical in the half-plane x ~ x2, and (6.42) forms a family of rotated vector fields with respect to 1 in the halfplane x > x 2 ; hence (6.42) forms a family of generalized rotated vector fields in the whole plane. Thus for sufficiently small 1 > 0 b < 0), (6.42) has an unstable (stable) cycle I\ on the outside of r 1 , and has a stable (unstable) cycle fi inside f1. On the other hand, the function F(x) and its derivative f(x) + "f(X- x2)g(x) still satisfy all the conditions of the theorem; hence, as before, h~ > h1 {here h~ and h1 are the values of the divergence of {6.42) integrated around r~ and I\ once respectively). This is impossible. Finally, if f(x), g(x), and cp(y) are only continuous, then we can handle this by Erugin's method at the end of §1. REMARK 1. If /3 = 0 and ft(x)fg(x) does not decrease when x increases in (-oo, 0) and (0, +oo ), we can obtain Zhang Zhi-fen's uniqueness theorem. REMARK 2. If the conditions of the theorem only hold in the region (a < x < b, -oo < y < oo}, then the conclusion also holds in this region. REMARK 3. If there is no hypothesis on the stability of the singular point, then the conclusion on the uniqueness cannot be obtained, since it is possible for h2 < 0 < hl that r 1 is unstable and r 2 is stable. EXAMPLE 5. Prove the uniqueness of the limit cycle of system (5.33) in §5 assuming B > 1 + A 2 • PROOF. Applying to system (5.35) the change of variables

_ F(x)

r,' = 'T'J,

{ F(x)

= F(x) +

e' =

!,

ds

J: "t(~- x2 )g(~) de

= A(1 + ~') 2 dt (for~' > -1,

i.e. x > 0),

we get d~' fds

= 'f/ 1 -

F(~'),

d'f/ 1 fds

= -g(e'),

where

F(e')

= ( A~'a _ B ~2A 2 e12 _ B- ~- A 2 g(e)

It is easy to see that F'(O)

e) /(1 + e')2,

= e/(1 + (} 2 •

= (1 + A2 - B)/A< 0, e'g(e') > 0 when e' :f. 0, e· G(e') = fo g(u) du---+ +oo when~'---+ -1 + 0,


136

and

d

df

(F'(e')) g(€')

=A+

2

A€'2

B- A 2

-

1

+ A(1+e)2 >O

for €' # 0, e' > -1. Hence from Theorem 6.4 we get the uniqueness of the limit cycle. There is a paper by L.l. Zhilevich [113] which used the comparison method to prove the uniqueness of a limit cycle of system (5.25) when f(x, 0) has only one zero. IV. Method of topographic systems. This method was first used in the classical paper of Poincare, but he only considered some artificially constructed special equations. For more general equations, the difficulty of using this method is that we do not know how to obtain a suitable topographical system. Later, J. L. Massera [114] obtained a better way of constructing the topographical system to enable him to improve some others' results; moreover, his method was later adopted by Zhang Zhi-fen and Chen Xiang-yan to obtain some good results. In this subsection we present Massera's method. Earlier, in 1950, G. Sansone [115] proved a uniqueness theorem for limit cycles of the Lienard equation.

=

THEOREM 6.5. Suppose that in system (6.26) g(x) x, f(x) is continuous, f(x) < 0 when (5_1 < x < 6, and f(x) > 0 when x < 6-1 < 0 and x > 61 > 0. Moreover, for all x in ( -oo, 0) let f(x) be a nonincreasing function, and for x in (0, oo) let it be a nondecreasing function; finally, let lf(x)l < 2. Then system (6.26) has a unique limit cycle, which is stable.

Later, Massera [114] gave a simpler proof of the above theorem and eliminated the condition lf(x)l < 2. Theorem 6.5 was extended by Roberto Conti [116] in 1952 to become THEOREM 6.6. In system (6.26) suppose that xg(x) > 0 when x ::1 0 (! and g are continuow), and G(±oo) = +oo. Moreover, suppose that after the transformation z = J2G(x)sgnx the function ~(z) = F(x(z)) satisfies the following conditions: 1. 4i(z)/z is a nonincreasing function when z < 0, and a nondecreasing function when z > 0. 2. lcll(z)/zl < 2. Then system (6.26) has a unique limit cycle, which is stable.

We can also eliminate condition 2) of the theorem using Massera's method. In 1954, Antonio de Castro [117] also gave the following uniqueness theorem for a limit cycle of system (5.25).

§6. UNIQUENESS

137

THEOREM 6. 7. Suppose g(x) and f(x,v) are continuous, xg(x) > 0 when x # 0, g(x) is an increasing function of x, and f(x, v) satisfies a Lipschitz condition with respect to x and v. Let f(O, 0) < 0. When lxl > a > 0, let f(x, v) > 0, and suppose there exist N > 0, a> 0 such that for any continuous function v( s) /_: f(s, v(s)) ds

~a> 0,

provided that lv(s)i > N and lxl >a. Finally, suppose that when increase, f(x, v) does not decrease. Then the system dxfdt

= v,

dvfdt

= -g(x)- f(x,v)v

(6.43)

lxl

and

lvl

(6.44)

has a unique limit cycle. Zhang Zhi-fen proved later [109] that the conclusion of the theorem is wrong. She gave a counterexample, and proved that when g(x) = x the conclusion of this theorem is correct provided the following hypothesis is added: there does not exist a closed curve such that in some neighborhood of it f(x, v) takes constant value along a half-ray passing through the origin. Condition (6.43) can then be dropped. She also used Massera's method. Since Theorem 6.5 is a special case of Theorem 6.7, we now prove Theorem 6.7 (assume g(x) x). PROOF. Transforming (6.44) into differential equations in polar coordinates, we obtain

=

dpfdt

= -psin2 Of(pcosO, psinO),

dO/dt

= -1- sinOcosOf(pcosO, psinO).

We first prove that any closed trajectory of this system, if it exists, must be a star with respect to the origin. That is to say, any half-ray starting from the origin can only intersect the closed trajectory at one point (maybe touching it tangentially). Suppose not. Let the half-ray 0 = Oo and a closed trajectory L intersect at at least three points (Oo, Pi), i = 1, 2, 3, 0 < Pl < P2 < P3· Then the sign of d(J / dt at these three points should vary alternatively. We know this is impossible from the additional condition on f(x,v) in the theorem. Now suppose that under the similarity transformation x' = kx, y' = ky (k > 0) the image of L is Lk. Since L is a star, it follows that Lk contains L fork> 1, but L contains Lk fork< 1. The family of curves {Lk} forms the topographical system we need. Note that the line tangent to Lk at the point Pk(kx, kv) is parallel to the tangent to L at P(x, v ); hence the slope of Lk at Pk is dv' dx

= Lt

x -vf(x, v),


138

FIGURE 6.12

but at Pk the slope of the trajectory of system (6.44) is

-~I

b X = --f(kx, kv) = --f(kx, kv).

kv

dx (6 . 44 )

v

From the conditions of the theorem, we know that X - - - f(x v) { V

'

~ -~- f(kx, kv), V X

2: - - - f(kx, kv), v

when k

< 1; (6.45)

when k > 1.

Thus, if as lxl and Ivi increase f(x, v) strictly increases, then any Lk (k =f-1) is an arc without contact (Figure 6.12). Hence L must be a stable cycle, and thus is a unique limit cycle. If as IxI and IvI increase f (x, v) does not strictly increase, then according to the previously mentioned hypotheses the family of equations

dxfdt = v,

dvfdt =-x- f(x/k,v/k)v

(6.46)

forms a family of generalized rotated vector fields with respect to the parameter k, and, along any closed curve, when k varies slightly the vector cannot keep its direction unchanged everywhere. From the theory of §3, we know that closed trajectories of (6.46)k belonging to different vector fields do not intersect. On the other hand, it is easy to see that Lk is a closed trajectory of (6.46)ki thus it is immediately seen that (6.44) does not have any other closed trajectory except L, for otherwise it would intersect or coincide with some Lk (k =f.l), which is impossible. The theorem is completely proved.

§6.

UNIQUENESS

139

FIGURE 6.13 REMARK. In Theorems 6.5 and 6.6 of Sansone and Conti, we assume that lf(x)l < 2, which makes d8fdt =J 0 in the system after it is transformed into polar coordinates, and thus guarantees that the closed trajectory should be a star. From the proof of Theorem 6. 7, we see this can also be assured from the monotone property of f( x, v) with respect to x and v. Following the method of proof of Theorem 6.7, we can prove a more general result of Chen Xiang-yan [50]. ' THEOREM 6. 8. Let the system of equations dx/dt

be given. If for any ..\

= P(x, y),

> 1 the

dyfdt = Q(x, y),

(6.47)

inequality

P(x,y)Q(..\x,..\y)-P(..\x,..\y)Q(x,y)~O

(or ~0)

(6.48)

always holds and the points where equality obtains do not fill completely the closed trajectory of system (6.4 7), then system (6.47) has at most one limit cycle-which, if it exists, must be star shaped (with respect to the origin).

PROOF. Suppose (6.47) has a cycle whose equations are x

= rp(t)

and

y = ,P(t). Suppose r~ is a curve determined by the equations x = ..\rp(t) and

y = A'I/J(t), ..\ E (O,oo). It is easy to see that system

r.\ is a closed trajectory of the (6.49)

From (6.48) and the theory in §3, we know that r and r.\ (A =J 1) do not intersect. Also, it is easy to see that the origin does not lie on r, for otherwise f.\ (..\ =J 1) would intersect r at the origin (Figure 6.13). Moreover, if r is not a star, suppose r and some ray through the origin intersect at two points P 1 and P2. Let ..\o = IVP21/IOP11· Then r.\o and r will intersect at P2, which is impossible. Hence r is a star with respect to the origin.

140


r,

Now suppose system (6.47) has another limit cycle and suppose the points of intersection of r and with the positive x-axis are M(x 0 , 0) and M(xo, 0) respectively. Let X= xo/x0 • Then will intersect rx at M(xo, 0), which is also impossible. Hence does not exist.

r

r

r

COROLLARY. For the system

dx/dt = y- F(x),

dyjdt

= -x,

(6.50)

if, as x varies in (-oo,O) and (O,oo), F(x)/x is a nondecreasing (nonincreasing) function of lxl, and in the strip where the limit cycle exists, F(x)/x ¢. const, then system (6.50) has at most one limit cycle. When F'(x) exists, the above uniqueness condition is equivalent to

F'(x) - F(x)/x ~ 0

(or :::; 0)

for all x =F 0, and in the strip where the limit cycle exists the left side of the above formula is not identically zero. This result was obtained earlier by Luo Ding-jun using another method. Moreover, in [50] there is another uniqueness theorem, which is a corollary of the following theorem of de Figueiredo [63] using the point transformation. THEOREM 6.9. For system (6.50), if there exists 8 > 0 such that xF(x):::; 0 when lxl :::; 8, F(x) ¢. 0 in the vicinity of the origin, F(x) ~ 0 when x > 6, and F(x)/x is a nondecreasing function of lxl in ( -oo, 0) and [o, +oo), then the system has at most one limit cycle. REMARK. The conditions ((6.45) and (6.48)) for the system to form a family of rotated vector fields with parameter A of the similarity transformation are too strong; hence the scope of application of the above theorems is not very broad. In the second half of this book we shall see that the system

dx/dt

= -y +ox+ lx 2 + mxy + ny2 ,

dyjdt

=x

forms a family of generalized rotated vector fields with respect to the parameter 8, and for any 6, if the system has a limit cycle, we can prove it must be unique. However, the system does not form a family of rotated vector fields for the similarity transformation x' = kx, y' = ky. V. The Bendixson-Dulac method. By this we mean the special case of Theorem 1.12 in §1 when n = 2. That is, THEOREM 6.10. Suppose that for system {6.47) there exist an annular region G and a continuously differentiable function B(x, y) Uor simplicity, suppose P(x, y) and Q(x, y) are also continuously differentiable) such that

§6. UNIQUENESS

141

(8f8x)(BP) + (8/oy)(BQ) has a constant sign in G and is not identically zero in any subregion of G. Then (6.47) has at most one closed traiectory lying entirely in G.

We can make use of other similar theorems in §1 to obtain similar results, but the practical value of these methods is very little. For instance, for Example 5 in §1, we can take any circular annular region which contains the unit circle x 2 + y 2 = 1 as G, and take B 1. For further extension, as in Example 2 in this section, how to choose G and B(x, y) is not very clear. Since the Bendixson-Dulac method is very useful for proving the nonexistence of limit cycles in polynomial differential systems (as we shall see in the second half of this book), it seems still necessary to continue to explore and investigate the use of this method in the proof of uniqueness of limit cycles.

=

VI. The method of Levinson and Smith. In an earlier paper by Levinson and Smith [64], they first used the method of "proving that the divergence integrated along any limit cycle once must always take positive (or negative) value" to prove uniqueness. Hence it is natural to assume the equations have a -unique singular point (whose index is +1); thus every limit cycle, if it exists, should contain this singular point in its interior. Then according to the well-known fact that adjacent sides of adjacent limit cycles must possess different stability, we can deduce uniqueness of limit cycles in the above manner. However, the first theorem they obtained by this method not only had certain drawbacks, but also it is not easy to verify the conditions of the theorem; hence it. does not have much use. In 1970, because of the needs of qualitative work on quadratic differential systems (that is, systems (6.47) whose right sides P and Q are quadratic polynomials of x and y), and also adopting the advantages of Filippov's method of proving existence of limit cycles, G. S. Rychkov [118] obtained better results by using the method of Levinson and Smith to prove uniqueness of limit cycles. This method was recently improved and extended by Zeng Xian-wu [119], [120], so that it can include many known uniqueness theorems as special examples. For simplicity we shall only present the theorem from [118], but its proof is far more difficult than that of any of the other uniqueness theorems in this section. Suppose in the strip d1 < x < d 2 (d1d2 < 0) we are given the equations dxfdt = y- F(x),

dyfdt = -g(x),

or

(F(x)- y) dy = g(x) dx, (6.51) where xg(x) > 0 when x 1: 0, g(x) is continuous and F(x) is twice continuously differentiable. Introducing the transformation (5.18), (5.19) in §5, we change


142

!I

z*

FIGURE 6.14

(6.51) into the two equations

(Fi(z) - y) dy

= dz,

0 $; z < Zoi

(i = 1, 2),

(6.52)

on the right half {z, y)-plane, where Zoi = limx-+ld,l Zi(x), Zi(x) being taken from {5.18) and (5.19). Every closed trajectory L of (6.51) corresponds to a unique pair of curves L1 and L2, which are solutions of the two equations of (6.52) respectively, starting from the same point on the positive y-axis, passing through the same point on the negative y-axis, and satisfying (along the clockwise direction)

~

h

(apax + 8Q) dt 8y

= =

rf, F'(x) dy

h f

jL1

g(x)

FHz) dy-

f F~(z) dy.

(6.53)

jLl

Let Yt(z) and y2(z) denote the equations of L 1 and L2 respectively, let Yu and Y21 denote the sections of arcs of these two curves on the upper parts of the perpendicular isoclines of the corresponding equations respectively, and let Yt2 and Y22 denote the sections of arcs on the lower parts of the perpendicular isoclines respectively (Figure 6.14). Let A and B denote the points of intersection of L 1 and L2 with the two isoclines respectively. On the curve y == Ft(z) we take a point C which has the same ordinate as B; that is,

143

§6. UNIQUENESS

yc = YB = F2(zB) = Ft(zc). Construct an integral curve y = y3(z) of equation (6.52)1 passing through C, and extend it both ways, so that it intersects the line z = z• (z• being the abscissa of the point of intersection of the two isoclines y = Ft(z) andy= F2(z)). This point of intersection will later be assumed to exist and be unique. Let (i,j = 1, 2), and let Yi(z) be the inverse function of Zi(y); that is, the inverse function of Yi = Fi(z) is Zi = Fi(y). THEOREM 6 .11. Suppose there exist unique zo and z*, 0

< zo < z•
0 when z =I zo, 0 3) F~'(z) < 0 when 0 < z < z0 t-( 10 )

< z < z01 , and F1 (z*) = F2(z*).

4) ddy(Ft(y)-F2(y)) >0, i.e., (F~- Fn/F~F~

> 0 when /3 < y < Ft(z*), where /3 = 1.=1,2 lllax ( lim Fi(z)) . Z-+Zoi

Then equation (6.51) does not have more than one limit cycle; if it exists, it must be a single cycle. The proof of this theorem will be divided into nine small steps,(l 1 ) in which we have to use the inequalities ZA > z* and ZB > z*, as can be seen from Theorem 5.4 of §5, because in the strip lzl ~ z• it is impossible to have a closed trajectory. Moreover, if F(O) =I 0, then we can translate y to make F(O) = 0. ( 1) YA < YB; that is, the extreme right point of Lt is below the extreme right point of L2.

PROOF. Take a point A' on y = F2(z) such that YA' = YA, with A' on the right of A (Figure 6.15). Now we translate the integral curve y = y 12(z) of (6.52)1 and the isocline y = Ft(z) to the right to make A coincide with A' and obtain y = Y12(z-6.) andy= Ft(z-6.) (dotted line); here 6. = ZA' -ZA· At this timeD moves to D'. If Zc = Zv• and G is a point on y = Y22(z), then Yc ( 9 )This condition is equivalent ( 10 )This condition is equivalent

to f(x) < 0 for the Lit~nard equation when d 1 < x < 0. to condition 4) of Theorem 6.4, but the range of existence

is smaller. ( 11 )Please first see the remark after the proof of this theorem.

144


FIGURE 6.15

is clearly greater than y D'; that is, G is above D'. Moreover, it is easy to see the curve O'K'H'A'lies entirely above y = F2(z); that is, F1 (z-~) ~ F2(z), where z E [~, ZA' ]. From this we see that if YA' ~ YB we can apply differential inequalities to the two equations (F2 (z)- y) dy = dz thereby obtaining

YD' ~Yo,

and

(F1(z-

~)-

y) dy = dz,

which is impossible; hence

YA'

= YA < YB·

(2) Y21 < IY22I when 0 < z < ZB· The geometrical meaning of this is obvious. PROOF. From Y21 = Y21- F2 and Y22 = Y22- F2 it is easy to see that Y21 and IY22I are solutions of

dY

dz

1 + YF~(z) y

(a)

and

dY = 1- YF~(z) (b) dz Y respectively. Note that F2(z) < 0 when 0 < z < ZB, and so at the same point (z, Y) dYI - dYI = dz (a) dz (b)

-2F~(z) > o.

Now if there exists Z3, 0 < Z3 < ZB, such that Y21{z3) ~ IY22{z3)l, then from differential inequalities we have Y21 > IY22I at {z3, ZB), and so d dz(Y21(z) -IY22(z)\) = -2F~(z)

+ (Y21(z) -IY22(z)\)(IY22(z)IY21(z))- 1 > 0;

145

§6. UNIQUENESS

that is, Y21(z) -IY22(z)l increases as z increases, for z E [z3,ZB)· Taking the limit, we get

which contradicts Y21(zB) = IY22(zB)I; hence Y21(z)

< IY22(z)l when 0 < z
IY22I > Y21 > Yu and Y21 - Yn > Y22 - Y12 (12 ) when 0 < z ~ z*. PROOF. At (O,z*) we have F2(z) < F1(z); hence from differential inequalities we know that Y21 > Yll· Moreover from the definitions of Y21 and Yu, we immediately get Y21 > Yu. Similarly we can prove"that IY12I > IY22I· Using (2), we get the first half of (3). Next, we note that d 1 1

-(Y21- Yn) = ( ) dz F2 z - Y21 F1 - Yn (Y21 - Yn) + (F1 - F2) =..:..:..__....;;...,::-:-'-..,..,-;....._-___:_ Y21Yu

and so we can solve

r

r

r

r

F1(u)- F2(u) { de } Y21 - Yn = ] 0 Y21(u)Yu(u) exp lu Y21(e)Yu(e) du. Similarly,

F1(u)- F2(u) { de } Y22 - Y12 '"']0 Y22(u)Y12(u) exp lu Y22(e)Y12(e) du. Note that F1(u)- F2(u)

> 0 and

[Y22(u)Y12(u)]- 1 < [Y21(u)Yu(u)]- 1; that is, we get the second half of (3). ( 4) Y{ 1 (z)

< 0.

PROOF. We have Y{ 1 = y~ 1 - Fi = -1/(yu- Ft)- Fi(z). When 0 < z < ~ 0; hence Y{ 1 < 0. When zo < z ~ ZA, let o:(z) = F1 - (Ff)- 1 . Then

zo, Fi(z)

[F1- o:(z)]- 1 = F{ > F{

+ F{'(F{)- 2 = o:'(z).

To prove Y{ 1 < 0 is the same as proving 1 I 1 Yu + IY12I. Moreover, this inequality shows the curved angular region formed by Y21 and yu is wider than that formed by Y22 and Y12·


146

Now a' < 1/(Ft -a), by Yi 1 = 1/(F1 - Yu) and

Yn(ZA)

= F1(zA) < Ft(ZA)- [F{(zA)t 1 = a(zA);

hence from the comparison theorem we know that when 0 < z < ZA, Yu(z) must lie below a(z); that is, yu 0. PROOF. If 0

0 and the inequality in (3) we can

get

[(Yu)- 1 + IY12I- 1 l~ = Yi1 3 +

IYd- 3 + F{(Y1! 2 -IYd- 2) > 0,

and when zo < z < ZA, we have IY12I~ = -1/IYt21 (4), we get

+ F{ < 0.

From this and

[(Yu)- 1 + IYt2l- 1 ]~ = -(Yu)- 2Y{1 -IYt2l- 2 ·IY12I' > 0.

(6)

~ = -1z• F{ (~1 + l~2l) dz + 1z• F~ (~1 + IY~2l) dz > O. First we prove a lemma. LEMMA. Suppose a> b > c

> d and b + c >a+ d > 0. Then

1

1

1

1

a+ €t > li + c· PROOF. Since (a- d) 2

> (b- c) 2 and b + c >a+ d > 0, we have (a-d) 2 2(a+d)

(b-c) 2

> 2(b+c)"

Moreover, it is easy to see that

(b-c) 2

-=-=----'"'7

2(b+c)

=

(b+c) 2 -4bc (a+d)(b+c)-4bc > -'----------':....;.,-----..,:....____ 2(b+c) 2(b+c)

Combining these, we get (a-d) 2 (a+d) 2 -4ad a+d 2ad 2(a+d) = 2(a+d) = -2-- a+d (a+ d)(b +c)- 4bc a+ d 2bc > =---2(b+c) 2 b+c"

Hence

be b+c

ad a+d'

-->--·

§6. UNIQUENESS

147

that is,

b+c

T

1

a+d

1

1

1

= "b + -c < ---;;;r- =a+ "J·

PROOF OF (6). From (3) and the lemma we deduce that

Note that F~

1

1

Yu

+ IY12I >

1 Y21

1

+ IY22I.

< 0, and hence ci>

> foz• (F~- F{) ( IY:21 +

~~) dz.

Since (5) holds, we can apply the mean value theorem to the right side of the above formula, and we get

> [(Yu(z*))- 1 + IYt2(z•)l- 1]· [Ft(~)- F2(~)] since F1 > F2 at (0, z*).

~ 0,

ci>

(7)

1

y 31 (z")

F~ (z3(y))

dy ~

Yal(z•)

0 :::;

e: :; z*,

1Y11 (z")

Ff(z1 (y)) dy. Ytl(z•)

PROOF. Consider a fixed yo. Since in Figure 6.14 we have Yo= YB > YA and F{ ( z) < 0, it follows that C is on the upper left of A and the integral curve y3(z) is on the left of Yt(z); thus Z3(Yo) < Zt(Yo). Since F{'(z) < 0, we have F.l(z3(Yo)) > F.l(zt(Yo)); thus

l

!tat (z")

F.l(z3(y))dy >

Yal(z•)

1Y3t (z•)

F.l(zt(y))dy. Y32(z•)

But Yu(z") 1

F{(z 1 (y))dy=

Ytl(z•)

1y31(z")

F{(zt(y))dy+

1Ya2(z")

Y32(z•)

F.l(z1(y))dy

Yt2(z•)

Yll (z•)

+ /.

F{{zt(y)) dy,

Yat(z•)

where the values of the last two integrals on the right are negative. The proof of (7) is then obtained.

(8)

PROOF. The function y3(F1 (F2(z)) and Y2(z) are solutions of the equations

(F1

-

y)

dy = /3*(z) dz and (F2- y) dy = dz


148

respectively, where

!3*(z)

= F~(z)[F~(F1(F2(z)))]- 1 .

From condition 4) of the theorem, noting that Fi (z•, zc), we conclude that /3* (z) < 1. Moreover,

y3(F1(F2(zB))) Hence when z•

~

z
0, as the case b(O) < 0 can be discussed similarly. We introduce polar coordinates, eliminate dt, and expand the right side of this equation into a power series of r; thus we obtain

§6. UNIQUENESS

151

where R 1(0, >.) = a(>.)jb(>.), and ~(0, >.) is a polynomial in cos 0 and sin 0. Now we look for a solution of (6.60) in the form

T =ToUt (0, >.) + T~u2(0, >.) + T3ua(O, >.) + · · ·

= /(0, To,>.),

(6.61)

where To is the initial value of r; we can obtain the equations satisfied by the functions Uk (0, >.): du1 ( ) du2 2 (6.62) dO = u1R1 0, >. , dO = u2R1(0, >.) + u 1R2(0, >.), ... , and the initial conditions

(k = 2,3, ... ). (6.63) Ut(O, >.) = 1, Uk(O, >.) = 0 From (6.61) we see that the necessary and sufficient condition for T = f(O, To,>.) to be a periodic solution is /(2rr, ro, >.)-To= [ut(2rr, >.)- 1]To

+ u2(2rr, >.)T~ + ua(2rr, >.)T3 + · · · = 0. Cancel the factor To

# 0, and rewrite the above transcendental equation as

cp(>., ro) = vt(>.) + v2(>.)To + va(>.)T~ + · · · = 0. (6.64) In order to study whether a closed trajectory appears in the vicinity of the origin as >. varies, we have to study whether the equation cp(>., r 0 ) = 0 has a real root with respect to To. H we treat cp(>., To) = 0 as a curve in the (>., To)-plane, it is clear it must pass the origin. Moreover, from the equation satisfied by u2(0, >.)it is easy to see v2(0) = u2(2rr,O) = 0. Now suppose that • a'(O) # 0, va(O) # 0. (6.65) We wish to show that when>.# 0 and takes the suitable sign, system (6.59) has a unique limit cycle in the vicinity of the origin. Note that v1 (>.) = Ut (2rr, >.) - 1 = exp ( 1 (')

vl ""

= 2 b(>.)a'(>.)- a(>.)b'(>.)

b2(>.)

7r

We have vi(O) = 2rra'(O)/b(O)

# 0.

:~~~ 2rr) exp

1,

( 2 a(>.)) 7r b(>.) •

Thus

8cpj8>.l(o,o) = v~(O) # 0. Hence in the vicinity of the origin we can solve for >. from (6.64) as a singlevalued function of To, >. = >.(ro). Next,

d>.l = [- ocp ;acp] = _ v2(0) = dro (0,0) oro a>. (0,0) vi (0) ~>.I 2va(O) = _ b(O)v3(0) # 0 dr5 (O,O) vi (0) rra'(O) '

__

o,


152

(a)

(b)

FIGURE 6.16 and hence A= A(ro) takes extreme value at (0, 0). In the following we discuss four different cases, depending on the signs of a'(O) and v3(0). 1. a'(O) > 0 and v3(0) < 0. Then A(ro) has a minimum at the origin (Figure 6.16(a)). Since lPAjdr~l(o,o) > 0, dAjdro increases in some neighborhood above ro = 0; hence the function inverse to A = A(ro) is also single-valued in the vicinity of the origin in the first quadrant; that is, for every sufficiently small A > 0, there is a unique ro > 0 which satisfies rp(A, ro) = 0, i.e., system (6.59) has a unique limit cycle in the vicinity of the origin. Moreover, from a'(O) > 0 and a(O) = 0 we see that, for A < 0, we have a(A) < 0 and the origin is a stable focus; when A(O) > 0 we have a(A) > 0 and the origin is an unstable focus; hence the limit cycle is stable. 2. a'(O) > 0 and v3(0) > 0. Here A(ro) has a maximum at the origin (Figure 6.16(b)); when A> 0 the origin is an unstable focus, and when A< 0 it is a stable focus; hence the limit cycle which appears when A < 0 must be unstable. 3. a'(O) < 0 and v3(0) > 0. An unstable limit cycle appears when A> 0. 4. a'(O) < 0 and v3(0) < 0. A stable cycle appears when A< 0. Here we only discuss the conditions for system (6.59) with one parameter to have a limit cycle in the vicinity of the origin. Later, in §9, we study the same problem for systems with several parameters. Exercises 1. Prove that when ht ~ 1 > h2 > 0, system (6.5) has a unique (stable) limit cycle. 2. If, in system (6.44), g(x) = x and J(x, v) = 0 represents a real ellipse in which the distance from the center to the origin is less than the distance from the origin to the ellipse, then the system has a unique limit cycle.

§6. UNIQUENESS

153

=

3. Prove that if, in system (6.26), g(x) x and F(x) is a polynomial of degree not higher than three, then either system (6.26) does not have a limit cycle or it has a unique limit cycle. 4. Study whether Theorem 6.11 can contain some previous theorems as special examples. 5. Prove that when 0 < J.L < 1 the limit cycle of system (6.35) must intersect the lines x = ±~, and when J.L ~ 0 or J.L ~ 1 the system does not have a closed trajectory. 6. Prove that the function f(x) in system (6.29) has a positive zero and a negative zero when C > 0 is sufficiently large. 7. Prove that the equation x + hx + x = k when x > 0; = 0 when x < 0 (where h > 0 and k > 0 are constants) has a unique periodic solution, which is stable. 8. Suppose the right side 0 and IJ > 2.5, then (7.1) satisfies the condition of his previous paper [128], and so this system has exactly two limit cycles. This result can be said to be the earliest one on the existence of two and only two limit cycles. The idea of the proof is to study dh/da1 (the rate of change of h(L) with respect to the ordinate of a11 where h(L) is the integral value of the divergence along a spiral L leaving from, for example, a point a1 on the positive y-axis, going around the singular point (0, 0) and returning to another point a2 on the positive y-axis) and to prove that dhjda1 has a fixed sign, e.g. always positive. Since two adjacent sides of two adjacent cycles have different stability, we know at once that the system of equations has at most two cycles, the integral value of the divergence along the inner cycle is less than zero (or :50), and the integral value along the outer cycle is at least 0 (or > 0). In the proof the technique of proving step (8) in the proof of Theorem 6.11 in §6 is used. Since 1979 the Chinese mathematicians Huang Ke-cheng [130], Zhang Zhifen and Ke Qi-min [131], [132], Chen Xiu-dong [133], Huang Qi-chang and Yang Si-ren [134], and Ding Sun-hong [135] have given sufficient conditions for system (7.2) to have at most or at least n limit cycles; Zhang's result [131] solved a previous conjecture. In the following discussion we assume that F(x) E Cl, F(O) = 0, g(x) E C, and xg(x) > 0 when x '# 0. It is easy to see that (0, 0) is a unique singular g(e) de. We first point of (7.2). As before let F'(x) = f(x) and G(x) = present the result of [130].

J;

LEMMA 7 .1. Suppose there exist constants a, a', b, and b' (b' < a' < 0 < a 0, which is a contradiction. The lemma is proved. Corresponding to Lemma 7.1, we obviously have LEMMA 7. 2. Suppose there exist constants b' < a' < 0 < a < b such that the following conditions hold: 1) F(x) ~ F(a) when 0 ~ x ~a, and F(x) is monotonically nondecreasing on [a, b]. 2) F(x) ;::: F(a') when a'~ x ~ 0, and F(x) is monotonically nondecreasing on [b', a']. 3) F(x) ¢. 0 when a' ~ x ~a. Then in the strip b' ~ x ~ b, system (7.2) has at most one limit cycle which can intersect both the lines x = a and x = a'.

Next we prove another lemma. LEMMA 7. 3.

Suppose there exist constants N ;::: 0, a > 0, and b' < 0 such

that 1) F(x) ;::: -N when 0 ~ x ~ a, and

2) F(b') ~ -N- y'2G[cij. Then the limit cycle of (7.2) which intersects the line x the line x =a.

= b'

must intersect

PROOF. Let the trajectory starting from A( a, -N) as t increases intersect the negative y-axis at A', and let the trajectory starting from B(b', F(b')) when

§7. EXISTENCE OF SEVERAL CYCLES

159

FIGURE 7.3

t' decreases intersect the negative y-axis at B' (Figure 7.3). From condition 1) and (7.3) we know that YA' ~ -N- J2G(a) and YB' < YB = F(b'); hence

YB' < YA', i.e., B' is below A'. From this we can see at once that the lemma holds. Similarly, we can establish the following three lemmas. LEMMA 7. 4. Suppose there exist constants M ~ 0, a

> 0, and b' < 0

such that 1) F(x) ~ M when 0 ~ x ~a, and 2) F(b') ~ M + ~J2G(a). Then the limit cycle of system (7.2) which intersects the line x intersect the line x = a.

= b'

must

LEMMA 7. 5. Suppose there exist constants N ~ 0, a' < 0, and b > 0 such that 1) F(x) ~ -N when a'~ x ~ 0, and 2) F(b) ~ -NThen the limit cycle of system (7.2) which intersects the line x = b must intersect the line x = a'.

JWM.

LEMMA 7. 6. Suppose there exist constants M ~ 0, a' < 0, and b > 0 such that 1) F(x) ~ M when a' ~ x ~ 0, and 2) F(b) ~ M + y'2G(a'). Then the limit cycle of system (7.2) which intersects the line x = b must intersect the line x = a'.


160

From the above lemm88 it is not difficult to give sufficient conditions for system (7.2) to have at most n limit cycles. For example, THEOREM 7 .1. Suppose in system (7.2) F(x) and g(x) satisfy the follow-

ing conditions: 1) F( -x) = -F(x) and g( -x) = -g(x). 2) In the interval (0, b), f(x) has only n zeros 0 < 01 < 02 f - f(x) dt

181

182

( f - f(x) dt < f - f(x) dt).

181

(7.13)

182

PROOF. We consider the case outside the parentheses. From (7.11) we get

f - f(x) dt- f - f(x) dt

1s1

1s2

= =

lb [v1(x~(z))- v1(x~(z))] lb [v2 (x~(z))- v2 (x~(z))] lb [(v1(x~(z))- v2(x~(z)))- (v1(x:(z))- v2(x:(z)))] dz-

dz

dz,

and

d(v2(x)- v1(x)) = __ x_ + _x_ = x v2(x)- v1(x) > O, dx v2(x) v1(x) v1(x)v2(x) when x E ( o:, ,B). Again from (7.12) we get (7.13) at once. The proof is complete. Similarly, we have

-

LEMMA 7 .10. If the conditions of Lemma 7.8 hold, then along two arcs = v1(x) and s2: v = v2(x), v2(x) > v1(x) ~ 0 when x E [o:,,B], of any

s1: v

two trajectories of system (7.10) in the strip a

1 1

-f(x)dt

1 2

-f(x)dt).

We still need two more lemmas. LEMMA 7 .11. Suppose f(x) ~ 0 (f(x) ~ 0), and f(x) ¢ 0 when x [o:, ,8]. Then along two arcs s1: v = v1(x) and s2: v = v2(x), v2(x) > v1(x) 0 when x E [o:, ,8], of any two trajectories of system (7.10) in the strip o:

x

~

.B

we have

1 1

-f(x)dt >

1 2

-f(x)dt

(1

1

-f(x)dt
0)

1) f(x) > 0 (f(x) < 0), and 2) f(x) is monotonically nondecreasing (nonincreasing). Then along any two arcs of traiectories 1i (i = 1, 2) of system (7.10) passing through the points (.B;, 0) (i = 1, 2) and intersecting the line x = a twice, we have l

-f(x)dt "'11

where a

> l-f(x)dt "'12

(l

-f(x)dt

0. For convenience of presentation, we still let

f(x) = -p.cosx,

F(x)

= foz f(s)ds = -p.sinx.

First we prove some lemmas. LEMMA 7 .13. In the half-plane v ~ 0, along the same trajectory v = v(x) of system (7.19), we have v(x) > v( -1r-x) when x ~ 0, and v( -x) > v(1r+x) when x ~ 0. PROOF. From

(7.19) we get

v(1r+x) -v(-x)

11"+z e

= 1-z

v(e) de

1.,.+z --ed e = l -zo + 1.,. + o .,. v(e) = e[v(7r +e)- v( -e)I de

r lo

v(-e)v(1r+e)

r v(e) e de- lor

- lo

11" v(7r +e) de,

X~

0.

Differentiating both sides yields

d dx[v(1r+x) -v(-x)]

=

x[v(1r+x)-v(-x)] 1r v(-x)v(1r+x) - -,v(:-1r-+-x"7)

--dx k1r+x v( -x)

= v(-k1r- x)- v(-k1r -1r + x) > 0, 0 ~X~ 1rj2,

k;::: 0.

(7.31)

From (7.29)-(7.31) we obtain (7.27). Similarly we can prove (7.28). From Lemmas 7.7 and 7.8 we can get

do = f

f(x(t)) dt > 0,

(7.32)

][0,7r)

do = Let d-1

= iL1 = 0.

f(x(t)) dt > 0.

f

(7.33)

j[-1r,O)

From (7.27)-(7.33) we get m-1

f(x(t)) dt = ( -1)m-l

(-1)m-l {

I: (dk + dk) k=O

J[-m1r,m7r)

[(m-1)/2)

I:

= ( -1)m-l

(dm-2k-l + dm-2k-2)

k=O [(m-1}/2) + (-1)m-l

I:

(dm-2k-l +dm-2k-2)

k=O

> 0,

m;::: 1.

Lemma 7.15 is completely proved. Since the closed trajectory of system (7.19) is symmetric with respect to the origin and (-1)m-l f(x);::: 0 when m1r ~ x ~(2m+ 1)7r/2, from Lemma 7.15 we have LEMMA 7 .16. If the closed trajectory Lm of system (7.19) intersects the interval [m1r, (2m+ 1)7r/2] in the positive x-axis, then

( -1)m-l J.

h ...

f(x(t)) dt > 0,

m;:::

1.


170

Next we prove LEMMA 7.17. If system (7.19) has two closed trajectories Lt C L2, and

= ±(2m+ 1}rr/2 in the half-plane v ~ 0,

they both intersect the lines x

f

(-1)m

(f(x2(t))- /(xt(t))] dt > 0,

m

then

> 0.

1(-(2m+1)11'/2,(2m+l)11'/2]

PROOF. First, from Lemmas 7.9 and 7.10 we get

(-1)kDk = (-1)k

f

[f(x2(t))- f(xt(t))]dt

1((2k+ 1)11' /2,(2k+5)11' /2]

> 0,

k ~ 0,

(-1)k Dk = ( -1)k

(7.34)

f

[f(x2(t))- /(xt(t))] dt

1(-(2k+5)11'/2,-(2k+l)11'/2]

> 0,

k ~ 0,

f

(7.35)

[f(x2(t))- f(xt(t))] dt

< 0.

(7.36)

1(-11',11']

> 0 when

When m is odd, by (7.34)-(7.36) we have f(x) and by Lemma 7.11 we have

f

1r

~ lxl

< 3rr/2,

[j(x2(t))- /(xt(t))] dt

1[-(2m+1)11'/2,(2m+1)11'/2]

f

=

+f

1[-311'/2,-x] (m-1)/2

+

E

+f

1(-11',11']

(D2k-1

[f(x 2 (t))- f(x 1(t))] dt

1[11',311'/2]

+ D2k-d < 0.

k=1

When m is even, by (7.34) and (7.35) we have f(x) and by Lemma 7.11 we have

f

< 0 when 0 ~ lxl < rr/2,

[j(x2(t))- f(x1(t))] dt

1(-(2m+1)11'/2,(2m+1)11'/2] ·

=f

m/2-1

[f(x2(t))- f(xt(t))] dt +

1(-11'/2,11'/2]

E

(D2k

+ D2k) > 0.

k=O

Lemma 7.17 is completely proved. From Lemmas 7.12 and 7.17 we get

*

LEMMA 7.18. If system (7.19) has two closed trajectories L1 C L2 and both intersect the interval [(2m+ 1)rr /2, (m + 1)rr] in the positive x-axis, then

{-l)m

[.fr:

3

j(x2(t))dt-

1

f(x1(t))dt]

> 0,

m;:::

1.

§7. EXISTENCE

OF SEVERAL CYCLES

171

PROOF OF THEOREM 7.3. (I) We show that system (7.19) has at least n limit cycles in the strip lxl ~ (n + 1)11'. Suppose the trajectory starting from a point Pm(-m1r,O) of the negative x-axis, after passing through the half-plane v ~ 0, intersects the positive xaxis at Qm. From Lemma 7.14 we know that when m > 0 is odd we must have XQm > m1r; when m > 0 is even, XQm < m1r. Since the directional field is symmetric, we know that when the trajectory from the point Pm(m1r,O), after passing through the half-plane v ~ 0, intersects the negative x-axis at a point 'lJm, we must have XQm = -XQm· Let PmQ:n and PmlJ,;: represent the segments of the trajectories passing through the points Pm,Qm and Pm,Qm respectively; let QmPm and QmPm represent the line segments joining Qm, Pm and 7Jm, P m respectively. Set

r m = -p;;:Q;, U QmPm U PmQm U QmPm. Then in r 1, r 2 , ••• , r m, ... , r n+ 1 every pair of adjacent simple closed curves form the inner and outer boundary curves of a Poincare annular region, and there exists at least one closed trajectory between them. Hence in the strip lxl ~ (n + 1)11', there exist at least n limit cycles. (II) We show that there exist exactly n limit cycles of system (7.19) in the strip lxl ~ (n + 1)11'. We divide this into two cases. 1. There exists a closed trajectory Lm which intersects the interval [m1r, (2m+ 1)11'/2] in the x-axis. By Lemma 7.16 we know that when m is odd (even), Lm is stable (unstable) and there are no more trajectories intersecting (m1r, (2m+1)11'/2]. Now we prove there is no closed trajectory which intersects the interval ((2m+1)11'/2, (m+1)11']. Suppose the contrary. Then the trajectory L~ closest to Lm and containing it must be internally unstable (stable). Next we prove L~ cannot be semistable. In fact, suppose L~ is semistable. Consider the system of equations dxfdt

where fa(x) =

-JLCOSX

= v,

dvfdt = -x- !a(x)v,

(7.19*)

+ O')'m(x), a> 0, and when when

lxl ~(2m+ 1)11'/2, Ixi > (2m+ 1)11'/2.

System (7.19)* forms a family of generalized rotated vector fields with respect to a, and when a is very small, there exist closed trajectories L!.!> C L~) of (7.19)* which intersect ((2m+ 1)11'/2, (m + 1)11'] in the x-axis, and, moreover, (-1)mrf. h~l

fa(x)dt~O.

172


This contradicts Lemmas 7.12 and 7.17. Hence L~ must be an unstable (stable) cycle. From the proof of (I), at the same time there exists at least one one-sided, or even both-sided stable (unstable) closed trajectory outside L~ which intersects the interval ((2m+ 1}n/2, (m + 1)1r]. This contradicts Lemma 7.18. Hence there cannot be another closed trajectory which intersects

((2m+ 1)11"/2, (m + 1)1r]. 2. There is no closed trajectory which intersects [m1r, (2m+1)?r/2] on the xaxis. From the proof of (I), we know there exists at least one closed trajectory which intersects ((2m+ 1)11"/2, (m + 1)1r]. Suppose Lm is the one closest to the origin; when m is odd (even), it must be internally stable (unstable). Lm cannot be semistable, for otherwise by the proof of (I) there would exist at least one one-sided or even both-sided stable (unstable) closed trajectory outside Lm which intersects ((2m+ 1)11"/2, (m+ 1)1r]; this contradicts Lemma 7.18. Hence, Lm must be a stable (unstable) closed trajectory. Then, similarly to case 1 we can prove there is no other closed trajectory except Lm which intersects the interval ((2m+ 1)11"/2, (m + 1)11"]. From the above discussion, we know that when n is odd (even), system (7.19) has a unique stable (unstable) closed trajectory which intersects [m1r, (m + 1)1r] in the positive x-axis. Also, in the strip lxl ~ 1r, system (7.19) does not have a closed trajectory; hence in the strip lxl ~ (n + 1)11" there are exactly n limit cycles, and stable and unstable limit cycles are arranged alternately. Theorem 7.3 is completely proved. After proving this beautiful theorem, it is fitting for us to introduce the history of this problem. We know that the van der Pol equation

x + J.tf(x)

+ x = o,

(7.37)

where f(x) = -x+x3 /3,(1) has a unique limit cycle with respect to all J.t =f:. 0. Similarly, we can prove that if f(x) is a polynomial of degree 2n + 1 which contains only terms of odd degrees, then for sufficiently small IJ.tl equation {7.37) has at least n limit cycles. Eckweiler [136] first noticed that if we take f(x) = sinx, then in order to obtain a solution from the family of solutions z:l + z2 = A 2 for p = 0 that will produce a periodic solution for J.t =/:- 0, the amplitude A must satisfy the equation( 2 )

Jor1f sin~sin(Asin ~) d~ = 21rJ1 (A) = 0, =

(7.38)

( 1 )Differentiating (7.37) with respect to t, and letting :i y, we obtain the well-known form. Equation (7.37) is sometimes called the Rayleigh equation; it first appeared in Rayleigh's book Theory of aound (1894), and was much earlier than van der Pol. (2 )See (4.18) in §4.

§7. EXISTENCE OF SEVERAL CYCLES

173

where J1(A) is a first-order Bessel function. It is easy to calculate the derivative of the above formula, and get

{211:

lo

sin2 tp cos(A sin tp) dtp

= 21r[J0 (A) -

J2 (A)].

(7.39)

By the properties of Bessel functions,( 3 ) Jt(A) has an infinite number of positive zeros, and the large zeros obey the asymptotic law

An= n1r + 11'/4 + 0(1/n).

(7.40)

Under this condition, the right side of (7.39) becomes

4( -1Y'.../211'/An + 0(1/n).

(7.41)

Hence we see that this value is different from zero provided that n is sufficiently large. Thus [136] affirmed that the equation

x +~sin±+ x = 0

(7.42)

has an infinite number of limit cycles when 1~1 « 1. Later, Hochstadt and Stephan [138) pointed out that this affirmation is not rigorous because it may not be possible to find an interval for the same ~ such that the value of~ in this interval and the infinite number of limit cycles corresponding to the zeros of J1(A) can coexist. The authors of [138) applied more asymptotic properties of J 1 (z) and the method of constructing the Poincare annular region, first proving that the asymptotic equation

d1 / d8 = J.t sin 8 sin("Y sin 8) of the equation (7.42) in polar coordinates d""f _ ~sinOsin(""(sinO) dO- 1 + (~h)cos8sin("Ysin8)

(7.43)

has an infinite number of limit cycles when 1~1 > 0 is sufficiently small, and then showing that this conclusion also holds for (7.43) provided 1~1 is taken slightly smaller. Later, R. N. D'heedene [139) improved the method of constructing the annular region in [138] to obtain the above conclusion when 0 < IJ.tl < 2, but for IJ.tl ~ 2 he only proved that outside a sufficiently large neighborhood of the origin there exist an infinite number of limit cycles, and he conjectured that Theorem 7.3 should also hold. Zhang Zhi-fen in [131) completely proved his conjecture. REMARK 1. For the van der Pol equation

dxjdt = y, (3)Refer to [140].

dyjdt = -x + JJ(Y- y 3 /3),

(7.44)

174


FIGURE 7.5

if we let x = -J.Lv, then we get

dy dv

= J.L

2y

- y 3 /3 + v

,;;.____;;__"""y--'

(7.45)

and we can prove( 4 ) that as J.L ---+ +oo the limit position of this periodic solution of (7.45) is a discontinuous periodic solution, as shown by the bold line in Figure 7.5. Applying to (7.42) the transformation x = J.LV, or

dx/dt = y,

dy/dt

= -x- J.Lsiny,

we get

dy/dv = -J.L2 (v+siny)jy.

(7.46)

We conjecture that as J.L---+ +oo we can obtain some results on (7.46) similar to those for (7.45). REMARK 2. Exercise 5 of §6 of the first edition of this book (1965) (Exercise 3 in this edition) has shown that when F(x) in (7.1) is a cubic polynomial of x, the system has at most one limit cycle. This result was also obtained independently by A. Lins, W. de Melo and C. C. Pugh [141]. They also conjectured that if F(x) is a polynomial of degree 2n + 1 or 2n + 2, then (7.1) has at most n limit cycles. This conjecture has not yet been confirmed or disproved. We think it is correct.

( 4 )See

Appendix IV in (137).

§8. Structural Stability of Differential Systems Briefly speaking, a property possessed by a differential system is called stable if this property remains after a slight change of the system itself. For example, in §4 we have already seen that if an ordinary differential system (P(no),Q(no)) possesses a single limit cycle r 0 , then in the family ofrotated vector fields with a as parameter, for all a in a sufficiently small neighborhood of n 0 , the system (P(a), Q(n)) also has a unique single cycle in the vicinity of fo; hence the property of "possessing a single limit cycle" with respect to the rotation of vector fields is stable. However, if (P(nt), Q(nt)) possesses a semistable cycle f 1 , then for a close to n 1 on one side, (P(a), Q(n)) does not have a limit cycle in the vicinity of r 1 , but for a close to nt on the other side, (P(a), Q(n)) has at least two limit cycles in the vicinity of ft: this illustrates that the property of ''possessing a semistable limit cycle" is unstable with respect to the rotation of vector fields. In dynamical systems in the plane, the study of the existence and the number of limit cycles is an important part of the global qualitative theory; hence it must involve the problem of whether the global qualitative structure is stable. This is the structural stability to be discussed in this section. The importance of structural stability in practical applications is very obvious, since mathematical models abstracted from practical problems (which are in general represented by algebraic, ordinary or partial differential equations) have gone through processes of approximations and simplifications. To make sure the properties obtained from the study of mathematical systems can correctly reflect the state of the practical problems, We must demand that these models be structurally stable. In this section we first introduce a rigorous definition of structural stability of plane ordinary differential systems, and give topological characteristics of phase-portraits of structurally stable systems; then we discuss the problem of structural stability of polynomial systems. This section is preparatory for a discussion on the problems of limit cycles and global structure for quadratic differential systems in the following sections. 175


176

Let the plane system

dxjdt = P(x, y),

dyjdt

= Q(x, y)

(I)

be defined in a region B of the (x, y)-plane, let P, Q E C 1 (i.e., continuously differentiable), and let the boundary of B be a simple closed curve having no points of contact with respect to {I). For definiteness, we may as well assume the trajectory of (I) crossing the boundary of B does so from outside to inside. All the systems whose P and Q satisfy the above conditions form a set X. Now in X we introduce a metric p as follows: Let another system in X be

dx dt = P(x, y) + p(x, y),

dy dt = Q(x, y)

+ q(x, y).

(II)

Then we define the distance between (I) and (II) in X to be

p(I,II)

+

= m;x (IPI lql +~~~~+I ~:I+ ~~!I+ ~~:1).

We can prove that, under such a definition of the metric 1 X becomes a Banach space. DEFINITION 8.1. If there exists a 8 > 0 such that there is a topological map T from B into itself which can carry trajectories of {I) into trajectories of (II) provided that p(I, II) < 8, then we call system (I) a structurally stable system or coarse system in B, and system (II) a permissible perturbed system of (I); and p and q are called perturbations. The concept of coarse systems was first introduced in 1937 by A. A. Andronov and L. S. Pontryagin [142]. They assumed P and Q were analytic functions, and for the map T, besides the requirements of Definition 8.1, they added the following restriction: for any c > 0, we can make d(M, T(M)) < c(l) for any point M in B provided 8 > 0 is sufficiently small; here d( , ) represents the distance in R 2 • They have pointed out that the necessary and sufficient conditions for system (I) to be structurally stable are 1. It has only a finite number of elementary singular points, and the characteristic roots of its corresponding linear approximate system do not have zero real parts. 2. It has only a finite number of closed trajectories, which are all single limit cycles. 3. It does not have a trajectory from a saddle point to another saddle point. (1 )We shall call any map satisfying this conditione-homeomorphic.

(III)

§8. STRUCTURAL STABILITY

177

The details of their proof, however, have never been published. In 1952, H. F. De Baggis (143) weakened the requirement of analyticity of P and Q to P, Q E C 1 , and gave a detailed proof for the necessary and sufficient conditions. In 1959, M. M. Peixoto [144] further proved that Definition 8.1 and Andronov's original definition are equivalent, and generalized to n-dimensional syste1ns. In 1962, Peixoto [145) gave and proved necessary and sufficient conditions for structural stability of differential dynamical syste1ns on compact two-dimensional manifolds. His results will be mentioned at the end of this section. Since Peixoto has proved the equivalence of the two definitions of structural stability, we may as well add the requirement d(M, T(M)) < e in Definition 8.1 in order to simplify our proofs. Because we only have to consider perturbations p and q whose absolute values are very small, in order to prove the necessity of (ill) we can assume that the singular point of (I) under the map T becomes the singular point of (II) in its vicinity; the same hypothesis is also applied to the limit cycle. These hypotheses greatly shorten the proof of the following theorem. THEOREM 8. 1. The conditions (III) are necessary conditions for system

(I) to be structurally stable. The proof will be divided into the following four lemmas, in all of which we assume that (I) is structurally stable. LEMMA 8. 1. System (I) can only have a finite number of singular points and a finite numbe1> of closed trajectories. PROOF. From the Weierstrass approximation theorem, in any neighborhood of (I) there always exists a polynomial system (II) (that is, the right sides of the system are polynomials) such that the two polynomials of (II) do not have a common factor; hence the number of singular points of (II) can only be finite. Also, from Dulac's well-known theorem [146], [147], we know that (II) can only have a finite number of closed trajectories; hence (I) can only have a finite number of singular points and a finite number of closed trajectories. LEMMA 8. 2. System (I) can only have elementary singular points, and the characteristic roots of its corresponding linear approximate system do not have zero real parts. PROOF. Take any singular point, which we may assume to be (0, 0). If it is not an elementary singular point, then a(P,Q)

I -o

l a(x, y) (0,0) -

,

178


which shows that the two curves P = 0 and Q = 0 have a point of intersection at (0, 0) which is not simple.(2) Thus we can make a slight perturbation of one of them such that its singular point changes from one to more than one. This contradicts the hypothesis that (I) is structurally stable. We now prove that the characteristic roots of its linear approximate system at (0, 0) do not have zero real parts. Expand (I) into dx dt =ax+ by+ P1(x,y),

dy dt =ex+ dy

+ Q 1(x,y),

Jx

2 + y 2 ). Suppose its characteristic root has zero where P 1 and Q1 are o( real part, i.e., a+d = 0; thus system (I) takes the origin as its center or focus. Now we add a perturbed term o:x, 0 < lo:l « 1, on the right side of dxfdt; then the perturbed sytem still takes the origin as its singular point and the real part of its characteristic root is -o:. If (0, 0) is a center of (I) but is a· focus with respect to the perturbed system, it is clear that the center and the neighborhood of the focus cannot be €-homeomorphic, which contradicts the hypothesis that (I) is structurally stable. If (0, 0) is a stable (unstable) focus of (I), then we use Theorem 3.7 in §3 and take o: < 0 (> 0), which can make the origin become an unstable (stable) focus of the perturbed system and can generate new limit cycles in its vicinity; hence these two systems are not €-homeomorphic. In short, the real part of the characteristic root of the linear approximate system of (0, 0) cannot be zero. This lemma is completely proved.

LEMMA 8. 3.

There is no trajectory connecting two saddle points of sys-

tem (I). PROOF. To get a contradiction, assume there is a trajectory of (I) which

connects two saddle points A and B (which may coincide). Construct a family of rotated vector fields dxfdt = P- o:Q,

dyfdt = Q

+ o:P;

(IV)

when o: < 0, the vector field rotates in the clockwise direction and the arrows in Figure 8.1 indicate the directions of the trajectories of (IV). It is easy to see that (IV) still has A and B as its saddle points, but at the same time the separatrix entering B does not come from A, and the separatrix starting from A does not enter B; that is, (I) and (IV) are not €-homeomorphic, which contradicts the hypothesis that (I) is structurally stable. This completes the proof. (2)This implies that these two curves touch tangentially at (0,0), or at least one curve has (0, 0) as a multiple point.


FIGURE

179

8.1

LEMMA 8. 4. Closed traJectories of system (I) are single limit cycles. PROOF.

Let "ft, ... ,

"'N be all the limit cycles of (I), and set

h("tt) =

i( 'll

aP ax

+ aQ) ay

dt = 0.

Now we shall construct a function cp 1(x, y) E C 1 such that 'Pt(x, y) = 0 on i= 0 outside an q-neighborhood N.,., of "ft. where 17 > 0 is small enough that N.,., does not meet with any "fi (i :j: 1). For this, we only have to take "fl, acptfax ~ 0 on "ft. and 'Pl

• 2 1rn(x, y) + ( '1f"2 sm n x, y ) exp [217

'Pt(x,y) =

t

y)] an 2 1rn(x, 211

when (x, y) EN.,.,, 1f"

when (x, y) ¢.

2

N.,.,,

"'l·

where n(x,y) represents the distance of a point (x,y) of N.,., from From 1 n(x, y) E C it is easy to prove that 'Pt(x, y) indeed satisfies the above conditions. Similarly, for any "fi we can define 'Pi(x, y). Now we study the system of equations dy ~ acpl (V) - = P(x y) + c'P1'P2 · · · 'PNdt = Q(x,y). dt ' ax' It is easy to see that (V) and (I) both take "ft. ... , as closed trajectories, and when c > 0 is sufficiently small, they are permissible perturbed systems. Using the fact that h("tt) = 0 and the properties of 'Pl (x, y) we can compute that, for system (V),

"'N

hy("tt) =

i

ccp2 · · · 'PN (

"'ll

a;:l)

2

dt.

"'ll

Since acptfax ~ 0 and is continuous on and 'Pi(x, y) > 0 on the right side of the above formula is not equal to 0, and has the same sign as c. Now assume that 11 is an externally stable cycle of system (I). From the theory of rotated vector fields in §3, we know that in the outer neighborhood of there exists a curve r such that all the trajectories of (I) intersecting

"'t


180

r

move in the exterior-to-interior direction. Take E > 0 small enough so that r is still a curve without contact with respect to (V), and (V) does not have singular points between "Y1 and r. On the other hand, since E > 0, "11 has become an unstable cycle of (V); hence in the outer neighborhood of "11 there appear closed trajectories of (V) different from 11 ,, .. , "fN, which is impossible. The lemma is completely proved. Combining Lemmas 8.1-8.4, we know immediately that conditions (III) are necessary for system (I) to be structurally stable. REMARK. Lemma 8.4 also tells us that systems possessing a multiple cycle can, after a slight perturbation, make the multiple cycle split into at least two cycles. This important fact has not been mentioned in §§2 and 4. Using similar methods, Ding Tong-ren (148] proved that if a multiple cycle has even (odd) multiplicity, then the system can, after any arbitrarily small perturbation, split the multiple cycle into an even (odd) number of single cycles (not less than 2). THEOREM 8. 2. Conditions (III) are sufficient for system (I) to be structurally stable.

For this we first introduce the concept of singular trajectory and regular trajectory. DEFINITION 8.2. If the positive direction and the negative direction of a trajectory are orbitally stable, or the positive direction of this trajectory is orbitally stable and the negative direction runs out of the region B, then the trajectory is called a regular traj'ectory. If a trajectory has at least one direction in B which is orbitally unstable, then it is called a singular traj'ectory. It is easy to see that a center or central focus is a regular trajectory, but any other singular point is a singular trajectory since it always has one direction which is orbitally unstable. Periodic cycles and compound limit cycles are regular trajectories, but other closed trajectories are singular trajectories. A separatrix entering or leaving a saddle point must be a singular trajectory. If both ends of a trajectory run to a saddle point or a focus or approach a limit cycle, or if its positive direction does so but its negative direction runs out of the region B, then this trajectory must be regular, since all the trajectories in its vicinity possess the same properties. Thus, as far as a structurally stable system is concerned, all its singular points, limit cycles and separatrices are singular trajectories, and there are no other singular trajectories. We now turn to the proof of Theorem 8.2. Since (I) satisfies conditions (III), there exist only a finite number of singular trajectories (including singular points) in the region B, which divide B into a finite number of subregions, called normal regions; and the interior of each normal region is completely


181

filled with regular trajectories. There are two kinds of normal regions: one kind takes the boundary of B (or part of it) as part of its boundary, and another kind does not contain any boundary point of B on its boundary. DEFINITION 8.3. Stable nodal points, foci, and limit cycles are called sinks, and unstable nodal points and foci and the boundary of B are all called sources. It is easy to see the boundary of every normal region has at least one source and one sink; for otherwise the regular trajectory in this normal region would have nowhere to go as t-+ ±oo.( 3 ) LEMMA 8. 5. The boundary of every normal region has one and only one source and one sink.

Suppose that on the boundary of the normal region G there exist two sinks M1 and M2. We shall obtain a contradiction. Use a smooth arc C lying entirely in G to connect Mt and M2. Since Mt is stable, the regular trajectories passing through points on C close to M1 all enter M1 (as t -+ +oo); the totality of all the points on C entering M 1 form a corresponding open set 81. Similarly, the totality of all the points on Centering M2 form a corresponding open set 82. The sum of 82 and S 1 cannot be equal to C, since there must exist a point N on C such that the trajectory passing through N as t -+ +oo enters a third sink M3. We may as well assume N is a boundary point of the open set 8 1 ; but the totality of all the points on C entering M3 also form an open set, and so the trajectory passing through the points in the vicinity of N in 81 will enter M3. This contradicts the definition of N. The lemma is completely proved. From this lemma it is easy to see that normal regions of structurally stable systems can only have the following three types: 1. There are one source and one sink on the boundary (Figure 8.2(a)-(d)). 2. There are one source, one sink, two saddle points, and four separatrices (Figure 8.3). 3. There are one source, one sink, and three separatrices (Figure 8.4(a), (b)).(4) (In Figures 8.3 and 8.4, S=source and H=sink.) PROOF.

LEMMA 8. 6. If system (I) satisfies conditions (III), then for sufficiently

small 8 a system (II) which satisfies p(I, II) < 8 also satisfies conditions (III) and has the same number and same type of singular trajectories as (I). ( 3 )Since we have assumed that conditions (III) hold, the regular trajectory in the normal region cannot be a closed trajectory. ( 4 )The source in Figures 8.3 and 8.4 can be a limit cycle or boundary of B; the sink lnay also be a limit cycle. For simplicity, here they are each represented by one point.

182


(a)

(b)

(c)

(d) FIGURE 8.2 H

s FIGURE 8.3 PROOF. For a singular point Oi we can construct a very small simple closed curve ci which contains oi but does not contain any other singular points; the index of Ci is the index of Oi. When 6 > 0 is sufficiently small, the vector fields determined by (II) and (I) on Ci do not take opposite directions. Hence the indices of Ci are the same in these two systems. Moreover, since we


(a)

183

(b) FIGURE 8.4

o

only have to take the radius of Ci and very small, for any singular point of (II) in the interior of Ci, the real part of the characteristic roots of its linear approximate system is not zero, and has the same sign as the characteristic root of (I) at Oi; hence (II) has one and only one elementary singular point in the interior of Ci, whose index is the same as that of Oi. However, we should note that when Oi is a (critical) nodal point of (I), there may appear in the interior of Ci a focus of (II). After we remove the interiors of Ci from the region B, IP(x, y)l + IQ(x, y)l has a positive lower bound in the remaining region; hence, when is sufficiently small, the quantity

o

IP(x, y) + p(x, Y)l + IQ(x, y) + q(x, y)l also has a positive lower bound in that region. Thus we have proved that (II) and (I) have the same number of elementary singular points possessing the same index in B. For every limit cycle ri of (I), we can construct a small neighborhood ~ of r i such that its inner and outer boundary curves are curves without contact. When o > 0 is sufficiently small, the boundary curves of R; with respect to (II) are still curves without contact possessing the same properties. Moreover, in R; there are no singular points of (II); hence (II) has at least one limit cycler; in every R;. Consider that H(s) ds does not change sign (for fJ > 0 sufficiently small), and the adjacent limit cycles possess different stability. We know (II) has only one limit cycle r; in every R;. Moreover, when fJ > 0 is sufficiently small, in the vicinity of an elementary singular point of (I) there are no limit cycles of (II), and in the other parts of B also no limit cycles of (II) appear; hence (I) and (II) have the same number and the same type of limit cycles. Finally, since a solution is continuous with respect to its initial conditions and its parameter, we can easily see that if (I) has a separatrix "Yl from a

J


184

B

D*0

A FIGURE 8.5

s

FIGURE 8.6

source A 1 to a saddle point B and separatrices "Y2 and "Y3 from the saddle point B to the sinks A2 and Aa respectively, and the singular points of (II) in the neighborhood of Ai and B are Ai and B* respectively, then (II) also has a separatrix "Yi from Ai to the saddle point B*, and separatrices "Y2 and "Y3 from the saddle point B* to A2 and Aj respectively. Except for these separatrices, which correspond one-to-one to the separatrices of (I), (II) cannot have any other separatrices. The lemma is completely proved. According to Lemma 8.6, there is a one-to-one correspondence between the finite number of normal regions of (I) and (II), and corresponding normal regions have the same type. In the following we construct a homeomorphic transformation which maps a normal region G1 of (I) to the corresponding normal region G 2 of (II) in such a way that their trajectories correspond to each other, and then combine the point transformations of different normal regions to form a homeomorphic transformation of the region B into itself. Now we use the above-mentioned normal region of type 2 to illustrate the construction of this transformation; the construction for regions of types 1 and 3 is about the same.

-------

. B on the boundary~ Df the Suppose first that the source A and the smk normal region G 1 of type 2 are not limit cycles, and let the saddle points in G 1 be Do and D1 (Figure 8.5). Let b~} (0:::;; .X:::;; 17) represent the totality of trajectories in G 1 , where "Yo and "Y., represent the separatrices from A to Do and D 1 respectively, which have extensions "Yb and "Y~ (also separatrices) to B respectively. From a point M 0 on "YO to a point M., on "Y., we construct an arc without contact Co. in this normal region, and let the point of intersection of


185

"t>. and Co. be M>., and denote by A the arc length from Mo toM>. along Ca. We now define a homeomorphic transformation ~ from the normal region G 1 (excluding the points A and B) to a curvilinear rectangle 61 in the (s, A)plane such that the image of a trajectory 1>.• in G1 is a line segment A= A* in 6 1 . Suppose we measure an arc length s• from M>.· on 1>.• to a point M~: (negative towards A, and positive towards B), and set the image of M~: to be (s*, A*). No matter whether A and Bare foci or nodal points, the arc length along 1>. towards A orB always approaches a finite limit; hence the left and right ends of 61 are bounded. Suppose that under this transformation ~ the image of Do is a point D0 on A = 0, and the image of D 1 is a point Di on the line A = TJ. AB before, we can define a homeomorphic transformation "Ill on the normal region G2 (except the two points, source A and sink B) of type 2 of (II) to a curvilinear rectangle 6 2 in the (s, X)-plane. Suppose that under this transformation the images of the saddle points Do and D1 in G2 are the point D~ on the line X= 0 and the point D~ on the line X= Tj respectively. Now it is not difficult to establish a homeomorphic transformation 8 between 61 and 62 such that A = 0, D0, A = TJ, and Di correspond to X= 0, D~, X= Tj, and D~, and the line segment parallel to the s-axis in 61 corresponds to the line segment parallel to the s-axis in 6 2 • Combining the three homeomorphic transformations ~. 8, and w- 1 , and setting the images of A and B to be A and B respectively, we get a homeomorphic mapping between the normal regions G1 and G2. When A is a limit cycle, if we follow the above method of construction, then the left sides of 61 and 62 will become unbounded, and so it is not obvious how to define the transformation between points of A and the points of the corresponding limit cycle A in order for the map between G1 and G2 to be homeomorphic. Now suppose R is a very small open neighborhood of A, whose boundary r (two simple closed curves) is a curve without contact with respect to (I) and (II) (the directions of trajectories crossing it are the same), and the interior of R has a unique limit cycle A of (II). We would like to establish a homeomorphic transformation from R into itself such that the image of A is A, and the trajectories of (I) become the trajectories of (II). On A we measure from a fixed point to a point whose arc length is a, and construct a normal line T/ai the points of intersection of trajectories of (I) with T/a are arranged in order. Let i = ±1, ±2, ... represent the points of intersection of TJa with every trajectory 1b (exterior of A) or 1b' (interior of A) in this order. Here the first point of intersection is on r, the positive integer i represents the point of intersection of 1b and TJa, and the negative


186

"'a·

integer -i represents the point of intersection of 1b' and Thus in R U r all the points except A can be represented by (a,b,i) and (a,b',-i), where i is a positive integer. Suppose we have defined a homeomorphic transformation T 1 of B \ R( 5 ) into itself, which changes trajectories of (I) into trajectories of (II), and T1 on the boundary r of R determines the transformation

(a, b, 1) -+ (

Theory of Limit Cycles

Theory of limit cycles

Limit Cycles of Differential Equations

Limit cycles of differential equations