Mathematics and Its Applications
Volume 357
by
G. A. Leonov Department 0/ Mathematics and Mechanics, St Petersburg University, St Petersburg, Russia
I. M. Burkin Tula Technical University, Tula, Russia
and
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
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This is a completely revised and updated translation of the original Russian work Frequency Methods in Oscillation Theory, © Leonov, Burkin, Shepeljavyi. St Petersburg University Press, 1992.
All Rights Reserved © 1996 Kluwer Academic Publishers Softcover reprint of the hardcover 1 st edition 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
Contents
multidimensional analogues ........................................ 1
§1.2. The equation of oscillations of a pendulum ................. 6
§1.3. Oscillations in two-dimensional systems with hysteresis ..... 22
§1.4. Lower estimates of the number of cycles of a two-dimensional
system.................................................... 27
solutions of special matrix inequalities ............................. 34
§2.1. Frequency criteria for stability and dichotomy.............. 34
§2.2. Theorems on solvability and properties of special
matrix inequalities ......................................... 46
Chapter 3. Multidimensional analogues of the van der Pol equation 52
§3.1. Dissipative systems. Frequency criteria
for dissipativity ............................................ 52
§3.3. Third-order systems. The torus principle ................... 80
§3.4. The main ideas of applying frequency methods
for multidimensional systems ............................... 89
§3.5. The criterion for the existence of a periodic solution
in a system with tachometric feedback ..................... 94
§3.6. The method of transition into the "space of derivatives" .... 97
§3.7. A positively invariant torus and the function" quadratic form
plus integral of nonlinearity" ............................... 111
§3.8. The generalized Poincare-Bendixson principle .............. 119
§3.9. A frequency realization of the generalized
Poincare-Bendixson principle........................... .... 123
§3.10. Frequency estimates of the period of a cycle. . . . . . . . . . . . . . . 126
VI Contents
§4.1. Frequency criteria for oscillation of systems
with one differentiable nonlinearity ....................... 130
§4.2. Examples of oscillatory systems .......................... 142
Chapter 5. Cycles in systems with cylindrical phase space........ 148
§5.1. The simplest case of application of the nonlocal reduction
method for the equation of a synchronous machine 149
§5.2. Circular motions and cycles of the second kind
in systems with one nonlinearity ......................... 152
§5.3. The method of systems of comparison.................... 169
§5.4. Examples ............................................... 171
§5.5. Frequency criteria for the existence of cycles of the second
kind in systems with several nonlinearities ................ 180
§5.6. Estimation of the period of cycles of the second kind ..... 196
Chapter 6. The Barbashin-Ezeilo problem........................ 202
§6.1. The existence of cycles of the second kind ................ 204
§6.2. Bakaev stability. The method of invariant conical grids .. , 218
§6.3. The existence of cycles of the first kind in phase systems. . 231
§6.4. A criterion for the existence of nontrivial periodic solutions
of a third-order nonlinear system ......................... 239
Chapter 7. Oscillations in systems satisfying generalized
Routh-Hurwitz conditions. Aizerman conjecture............ ...... 249
§7.1. The existence of periodic solutions of systems with
nonlinearity from a Hurwitzian sector .................... 251
§7.2. Necessary conditions for global stability in the critical
case of two zero roots .................................... 271
§7.3. Lemmas on estimates of solutions in the critical case of one
zero root ................................................ 277
nonautonomous systems ................................. 280
of systems with hysteretic nonlinearities .................. 289
Contents VII
of at tractors and orbital stability of cycles ......................... 304
§S.l. Upper estimates of the Hausdorff measure of compact sets
under differentiable mappings ............................. 304
§S.2. Estimate of the Hausdorff dimension of at tractors of systems
of differential equations ................................... 310
§S.4. Zhukovsky stability of trajectories ......................... 322
§S.5. A frequency criterion for Poincare stability of cycles of
the second kind ........................................... 345
Bibliography ...................................................... 377
Preface
The linear theory of oscillations traditionally operates with frequency representa­ tions based on the concepts of a transfer function and a frequency response. The universality of the critria of Nyquist and Mikhailov and the simplicity and obvi­ ousness of the application of frequency and amplitude - frequency characteristics in analysing forced linear oscillations greatly encouraged the development of practi­ cally important nonlinear theories based on various forms of the harmonic balance hypothesis [303]. Therefore mathematically rigorous frequency methods of investi­ gating nonlinear systems, which appeared in the 60s, also began to influence many areas of nonlinear theory of oscillations.
First in this sphere of influence was a wide range of problems connected with multidimensional analogues of the famous van der Pol equation describing auto­ oscillations of generators of various radiotechnical devices. Such analogues have as a rule a unique unstable stationary point in the phase space and are Levinson dis­ sipative. One of the pioneering works in this field, which started the investigation of a three-dimensional analogue of the van der Pol equation, was K.O.Friedrichs's paper [123]. The author suggested a scheme for constructing a positively invariant set homeomorphic to a torus, by means of which the existence of non-trivial periodic solutions was established. That scheme was then developed and improved for dif­ ferent classes of multidimensional dynamical systems [131, 132, 297, 317, 334, 357, 358]. The method of Poincare mapping [12, 13, 17] in piecewise linear systems was another intensively developed direction.
The application of the Yakubovich - Kalman frequency theorem [130, 154, 178, 267, 323, 372, 376, 382] to the analysis of quadratic forms generating a positively invariant torus led to new problems, the solution of which allowed the formulation of a number of frequency criteria for the existence of cycles in multidimensional analogues of the van der Pol equation [94, 180,278,280,281,338, 339, 341].
The ideas of E.D.Garber and V.A.Yakubovich [127, 381, 383] enable one to obtain frequency estimates of the period and "amplitude" of these oscillations. It should be noted that since frequency criteria for the existence of cycles are based on the Yakubovich - Kalman theorem, then for an estimate of the period the method of a priori integral estimates of V.M.Popov appears to be the most developed at present.
Other nonlinear effects qualitatively different from auto-oscillations in the van der Pol equation are observed in dynamical systems with angular coordinates. One can mention first of all circular motions and cycles of the second kind in the equa-
x Preface
tion of a pendulum. Synchronous electrical machines and electronic systems of phase synchronization are described by the same equations [247, 330, 387J. The founda­ tions of the nonlocal theory of two-dimensional systems with angular coordinates were laid in the works of F.Tricomi and his numerous followers [6, 39, 46, 61, 145, 328, 350J. However, a less rough idealization for synchronous machines and the complication of phase synchronization devices required the investigation of systems of higher dimension.
The synthesis of the Lyapunov direct method and the elements of bifurcation theory, as well as the construction of various comparison systems, turned out to be the most effective. The Lyapunov functions constructed in this case contain cycles and separatrices of the corresponding two-dimensional comparison systems. In this way it became possible to obtain frequency criteria for the existence of circular motions and various types of cycle, which extend the widely known theorems of Tricomi, Amerio and other authors to multidimensional systems [99, 130, 183, 184, 186, 195J.
E.A.Barbashin and J.Ezeilo posed the problem of the existence of a cycle of a third-order differential equation with a cylindrical phase space desqibing various synchronization systems. From the control theory point of view the difficulty of investigating this equation is due to a certain degeneration of its transfer function. It is similar to critical cases in classical stability theory. The frequency criteria for the existence of cycles of the first and second kind are obtained in the works [89, 188, 192, 203], which in particular answer the questions put by Barbashin and Ezeilo.
The third current direction in the applied theory of oscillations is the investi­ gation of cycles in dissipative systems with one locally asymptotically stable equi­ librium. In 1949 M.A. Aizerman [4, 5J put forward the conjecture of stability in the large of multidimensional dynamical systems with one nonlinearity satisfying the generalized Routh-Hurwitz conditions. N.N. Krasovskii [169J was the first to refute this hypothesis, pointing out a two-dimensional system of this class which has solutions going to infinity. V.A.Pliss [296] proved the existence of cycles for a three-dimensional system and he was the first to obtain non-trivial upper estimates for a sector of absolute stability. Futher development of Pliss's method led to fre­ quency criteria for the existence of cycles in multidimensional systems that satisfy the generalized Routh-Hurwitz conditions [179, 280].
Close to the results indicated come the frequency criteria for oscillation in sys­ tems with nonstationary and hysteretic nonlinearities [189, 226], extending the widely-known theorems of A.A. Andronov and N.N. Bautin [14], N.A. Zheleztsov (see [16]), A.A. Feldbaum [120], A.Yu. Levin [241], R.W. Brockett [74], E.S. Pyat­ nitskii [313J to the multidimensional case.
After E.N.Lorenz's [254J discovery of strange attractors a great many experimen­ tal and theoretical works appeared [18, Ill, 135, 137, 268, 269, 275, 289, 315, 316, 335, 344, 345, 347], which made it clear that stochastic oscillations are widespread in finite-dimensional dynamical systems. In this case cycles do not have any sig­ nificance in the system under consideration because of their instability and hence their physical unrealizability, even if they do exist in such attractors. Global char­ acteristics such as various dimensions of at tractors were advanced [275, 352). Note
Preface Xl
that the dimension of a strange attractor in which chaotic oscillations occur is as important a quantitative characteristic of oscillations as its frequency in the case of ordinary periodic oscillation. The work of A.Douady and J.Oesterle [112] was an important step in obtaining frequency estimates of the Hausdorff dimension [68, 69]. So there arose a close relationship between the procedure used in the articles mentioned and the works of G.E.Borg, F.Hartman, C.Olech, G.A.Leonov [73, 143, 208~210, 212, 284], in which the orbital stability of trajectories is investigated. It turned out that the problems of estimating the Hausdorff dimension and investigat­ ing orbital stability reduced to the local study of compressing properties of a shift operator along the trajectories of the systems under consideration.
By now it had become clear that, on the one hand, analytical methods developed for upper estimates of the Hausdorff dimension of at tractors are a part of the modern theory of stability of motion. And on the other hand, the interpretation of the Hausdorff measure of compact sets mapped by a shift operator along trajectories as an analogue of the Lyapunov function allows one to obtain new results in the classical theory of stability. Such understanding especially stimulated the introduction of the notion of weakly contracting systems [148~ 150, 152] and the investigations of A. Douady and J. Oesterle [112], R. Smith [340], R. Temam [351, 352], A.V. Babin and M.l. Vishik [22, 23]. Applying the frequency theorem of Yakubovich and Kalman [382], it is possible to give estimates of the Hausdorff dimension a frequency form [68,69,84, 189].
And finally the introduction of Lyapunov functions into estimates of the Haus­ dorff dimension of at tractors by generalizing the estimates of Douady and Oesterle [214, 215] made it possible to suggest a combination of classical theorems of the second Lyapunov method [101, 109, 130, 171, 259, 323] and theorems of Hartman, Olech and Smith [142, 143, 340].
Since "nonlinear frequency reasoning" is a rather difficult branch of the applied theory of differential equations, the authors have tried to present it as simply as possible for a majority of readers. With this aim there are two introductory chapters. In the first chapter two-dimensional oscillation systems and their multidimensional analogues are considered and discussed. In the second chapter a short summary of the main results on frequency criteria for absolute stability and quadratic matrix inequalities is given.
The third chapter is devoted to the investigation of multidimensional analogues of the van der Pol equation. The fourth chapter gives frequency estimates of the period and amplitude. In the fifth and the sixth chapters a frequency approach to the study of dynamical systems with cylindrical phase space is presented.
The seventh chapter considers problems connected with the conjecture of Aizer­ man.
In the eighth chapter attention is concentrated on estimates of the Hausdorff di­ mension of attractors and methodologically close questions of Poincare and Zhukovsky stability of trajectories.
The beginning of the third and the fifth chapters may seem unnecessarily long for the specialist. But we intend this book for the reader who has just begin to study the frequency analysis of nonlinear systems.
xu Preface
It should be noted that the authors have focused only on oscillations in au­ tonomous systems. This is due to the fact that mathematically rigorous methods of frequency analysis of forced nonlinear oscillations do not exceed for the time being the bounds of the classical theory of absolute stability, and are well discussed in the literature [244, 267, 384].
The two-digit system of numbering formulae, theorems, definitions, examples and figures is used in the book. When they are mentioned in other chapters a figure denoting the number of the respective chapter is added.
Authors are greatly indebted to Dinara Kh. Ibragimova and Elmira A. Gurmu­ zova for their help in the preparation of the manuscript in English. Special thanks go to Iury K. Zotov and Inga I. Ryzhakova who helped to make a camera-ready copy.
The work was carried out with the financial support of the Russian Fund for Fundamental Research (93-011-135).
CHAPTER 1
Classical Two-Dimensional Oscillating Systems and their Multidimensional Analogues
As mentioned in the Preface, the starting point for the development of fre­ quency methods of investigating nonlinear oscillations is the qualitative theory of two-dimensional dynamical systems and such elements of absolute stability theory as the Popov method of a priori integral estimate~and frequency theorems on solv­ ability and properties of solutions of quadratic matrix inequalities.
The first two chapters are devoted to presenting results obtained in those direc­ tions of investigation that we need in future. At present these directions are widely discussed both in review articles 1 [244, 312, 368, 384] and in other publications [5, 16,20,42,43,44, 49, 101, 128, 130, 140, 177, 178, 255, 257, 258, 267, 270, 274, 290, 302, 309, 319, 323, 346, 348, 359, 365, 393]. In this connection we will cite only the formulations of necessary results with reference to the works where their proofs can be found.
§1.1. The van der Pol Equation
The van der Pol equation describes oscillating processes in electronic auto­ oscillators. Using Kirchhoff's law, under a number of assumptions one can obtain for one of the simplest schemes of such an auto-oscillator the following mathematical model of processes [57, 316] taking place in it:
LCx + [rC - MS(x)]x + x = 0, (1.1 )
where L, C, r are inductance, capacitance and series resistance respectively in an oscillation circuit; M is mutual induction between an anode circuit and an oscillating one containing an electronic lamp net; x is the voltage across its net (or across the capacitor plate); S( x) is steepness of the lamp characteristic. In many cases the function S( x) is approximated by a polynomial chosen according to the lamp type and experimental data. The simplest dependence S( x) = So - S2X2 is often used. In this case (1.1) takes the form
(1.2)
where Wo = L~' a = (MSo - rC)w6, (3 = M3::~~C' Equation (1.2) is called the
lWe shall note the recent review: Wassim M. Haddad, Jonathan P. How, Steven R. Hall and Dennis S. Berstein (1994) 'Extensions of mixed-I-' bounds to monotonic and odd nonlinearities using absolute stability theory', Int. J. Control, vol. 60, No.5, 905-951.
2 Chapter 1.
van der Pol equation [19, 350, 351, 353]. It is often written in a simpler "reduced" form:
X+6(x2-1)x+x=0, (1.3)
in which 6 = (MSa - rC)wa. The change of variables t' = wat, x' = J13x would suffice to pass from (1.2) to (1.3).
Under respective idealization (1.3) (or(1.2)) can describe the processes in other important systems as well. For instance, van der Pol was first interested in this equation in connection with the theory of relaxation oscillations of a symmetric multivibrator into whose circuit self-inductions are inserted [16].
Van der Pol was the first to discover such auto-oscillations [361, 362]. By ap­ proximate graphic integration, giving the parameter 6 specific numerical values and applying the method of isoclines, van der Pol obtained the now widely-known "phase portrait gallery" [16, Fig. 255] in a phase plane of the system
dx dy 2 dt =y, dt =-X-6(X -l)y, (1.4 )
equivalent to (1.3). All figures contain one unstable equilibrium and one closed trajectory (a stable cycle) to which the remaining trajectories tend. In this case for small 6 (c = 0.1) the cycle almost takes the shape of the circumference, i.e. the corresponding auto-oscillations closely resemble harmonic ones, but for large 6 (6 = 10) it has a significantly stretched form which corresponds to the so-called relaxation oscillations. In fact, when observing the change of an auto-oscillation form with increasing 6 with the help of numerical integration of (1.3), we find that it changes from a quasi-sinusoidal one to a relaxation one (see, for example, [316, Fig. 14,4]).
It is clear that all the trajectories of the system (1.4) as t --t +00 "are immersed" into a certain bounded domain D containing the cycle and do not leave it again. The system (1.4) with the property in question is usually called dissipative (see §3.1). The domain D is correspondingly called a dissipative domain.
Thus in the 20s the methods of graphical and numerical integration made it possible not only to obtain the phase portrait of the system (1.4) experimentally and discover a stable cycle, but also to analyze the character of the change of a cycle dependent on the parameter 6.
These results fostered the development of new methods for analytical investi­ gation of the van der Pol equation and its various two-dimensional generalizations [242, 243, 245]. Thus we have obtained not only existence theorems of stable peri­ odic solutions for the van der Pol equation, but a number of estimates of the period of oscillations having asymptotic character as a rule [177] (when the value of param­ eter 6 is sufficiently large or small) and also estimates of the" amplitude" of a cycle [16].
We can compare the "boom" of the 30-40s in the development of the theory of oscillations of nonlinear two-dimensional systems after the first experimental results of van der Pol with the nowadays intensive study of strange at tractors of multidi­ mensional dynamical systems after their experimental discovery by E.Lorenz [254, 347].
Classical Two-Dimensional Oscillating Systems 3
Let us formulate a theorem on the existence of a stable periodic solution of the van der Pol equation [177]. First we recall some universally accepted definitions.
Consider the autonomous system
dx Tt=f(x), xElRn,fEC 1. (1.5)
Definition 1.1. A vector c is called an equilibrium (a stationary solution, a stationary point, a singular point) of the system (1.5) if x (t) == c is a solution of this system.
Definition 1.2. Let x o(t) be a solution of the system (1.5) defined in the infinite interval [0, +(0). The solution xo(t) is said to be Lyapunov stable iffor any number c > 0 we can find a number 5 > 0 such that for any solution x (t) of the system (1.5) Ix (0) - xo(O)1 < 5 implies that Ix (t) - xo(t)1 < c for all t > O.
Otherwise the solution xo(t) is said to be Lyapunov unstable.
Definition 1.3. Let x (t) be a solution of the system (1.5) defined in some interval T (finite or infinite). The totality of points r = {x (t) : t E T}, r c IR n, is said to be a trajectory of this solution.
Definition 1.4. Let r c IR n be a certain set. The magnitude p(xo, r) = inf Ix ° - x I is called the distance from the point x ° E IR n to the set r.
XEf
(1.6)
equivalent to the van der Pol equation (1.3), which is derived from (1.3) by intro­ ducing a new variable:
Theorem 1.1 [177]. The system (1.6) has a closed trajectory r in the phase plane (x, y) to which all its solutions (x( t), y( t)) different from the unstable equilib­ rium (x(t) == 0, y(t) == 0) tend as t -t +00, i.e. lim p((x(t), y(t)), r) = o.
t .... +oo The most interesting two-dimensional generalizations of the van der Pol equation
in the autonomous case are the Lienard equation [101, 177, 319]
for which there exist analogues of Theorem 1.1, and the equation
cX + (x 2 - 1)i: + x = a
in which we observe the effect of uneven increase of the" amplitude" of a cycle under the change of parameter a in the neighbourhood of the value ao ~ 1 - c/8 - 3c2/32
4 Cbapter 1.
if the value of parameter e is sufficiently small (the so-called" French duck" effect) [157,397].
Among the two-dimensional generalizations of the van der Pol equation in the nonautonomous case we should mention the equation
x + k( x 2 - 1):i: + x = bAk cos At
(k is a large parameter, b and A are constants), for which the result of Cartwright and Littlewood [250-252] on the existence of infinitely many unstable periodic solutions is known.
Let us turn to the discussion of a multidimensional analogue of the van der Pol equation. Consider the autonomous system (1.5). Suppose that all the solutions x (t, xo) (x (0, x 0) = x 0) of this system are defined for t E [0, +00).
Definition 1.5. We call a closed trajectory r c IR n, different from an equilib­ rium, a cycle of the system (1.5).
Definition 1.6 [175]. We say that a set Bo C IR n attracts a set Be lR. n ifVe > 0 it is possible to indicate h(e, B) such that for any x 0 E B we have x (t, xo) E De(Bo) Vt 2: t1(e, B), where De(Bo) is the totality of all balls of radius e with centres at points of Bo.
In particular, if Bo is the equilibrium c of the system(1.5), then the set B is called the domain of attraction of the equilibrium mentioned.
Definition 1. 7 [175]. The smallest non-empty closed set M attracting any bounded set B C lR. n is called a minimal global B-attractor of the system (1.5).
It is easily seen from Theorem 1.1, for example, that the part of a phase plane bounded by a cycle M is a minimal global B-attractor of the system (1.4) (i.e. De(M) is a domain of dissipativity).
Definition 1.8 [175]. The smallest non-empty closed set if attracting any point x E lR. n is called a minimal global attractor of the system (1.5).
It is clear that if C M. In particular, for the system (1.4) if consists of a cycle and an unstable state of equilibrium.
We now define a multidimensional analogue of the van der Pol equation as the system (1.5) with a minimal global attractor containing a cycle and a unique Lya­ punov unstable equilibrium.
In investigating specific systems the linear and nonlinear parts are naturally dis­ tinguished. Many such systems are then described by the system of vector equations
~: = A x + be, (J = C *x , (1. 7)
(1.8)
where x E lR. n, (J E lR. I, e E lR. m; A, b ,c are constant matrices of order n x n, n x m, n X I respectively; cp( (J) is a nonlinear vector-function of the vector argument (J.
Note that formally any system (1.5) can be written in the form (1.7), (1.8) for A = 0, b = I, c = I, cp = f, m = n, and in particular the system (1.6) equivalent
Classical Two-Dimensional Oscillating Systems 5
to the van der Pol equation, where
In control theory (1.7) is interpreted as a mathematical description of some linear block at the input of which a signal ~ = ~(t) is fed and at the output of which the signal a = a( t) is registered.
An important concept characterizing the properties of the linear block (1.7) is the matrix of transfer functions (or for the case m = I = 1 a scalar transfer function).
Definition 1.9. A complex-valued matrix function
(1.9)
where p is a complex variable, is called a transfer matrix (for m = I = 1 a transfer function) of the linear part of the system (1. 7), (1. 8) .
Definition 1.10. A function ( matrix function)
(1.10)
where w E (-00, +(0) is a real variable and i = p, is called a frequency response of the linear part of system (1.7) .
It is natural that in (1.9) and (1.10) the values of the variables p and iw must not coincide with the eigenvalues of A. In the remaining part of the plane of the complex variable p the function X(p) is analytic and can be recovered from the values of the frequency response x( iw).
The meaning of a frequency response [130] in terms of input and output is well known. Suppose that m = I = 1 and a harmonic signal of linear part ~ = ~o exp( iwt) enters the input of the linear part of the system (1. 7). Then, under the appropriate choice of the initial state Xo, the output a there will also be a periodic signal a = = -x( iw )~o exp( iwt). Hence it follows that Ix( iw)1 determines the ratio of the signal amplitudes of the input and output of the system (1.7), and argx(iw) defines the phase difference of these signals. Thus the frequency response of the system (1. 7) can be experimentally defined by changing 2 the values of the frequency of the input signal from -00 to +00 and by measuring the ratio of the amplitude at the output and input, as well as their phase difference.
Under certain conditions the frequency response X( iw) completely defines the properties of the linear block (1. 7). This fact and the possibility of experimentaly obtaining the frequency response for a particular system with unknown parameters in the mathematical model (1.7) determined the important role of a frequency re­ sponse in control theory. We recall also that the transfer function X(p) is invariant under a nonsingular linear transformation of the phase space.
Let us formulate some more concepts characterizing the linear part of (1.7): controllability, observability, stabilizability, non degeneracy [130, 309, 367].
2This procedure is correct because lim x( iw) = O. w-+oo
6 Chapter 1.
Definition 1.11. A paIr (A, b) IS called controllable if rank lib, A b, ... , A n-1b II = n.
Definition 1.12. A paIr (A,c) IS called observable if k II A * A *(n-I) 11-ran c , c , ... , c - n.
Definition 1.13. The matrix A is called stable or Hurwitzian if any of its eigenvalues has a negative real part.
Definition 1.14. A pair (A, b ) is called stabilizable if there exists an (n x m)­ matrix s such that the matrix A + b s * is Hurwitzian.
Definition 1.15. Let l = m = 1, i.e. let X(p) be a scalar function. The transfer function X(P) is nondegenerate if it cannot be represented in the form of a ratio of polynomials with the degree of the denominator less than n.
Controllability and observability of systems were given detailed consideration in the books [130, 309, 367]. Here we state without proof only the statements we need in future.
Theorem 1.2 (on criteria for controllability). The following conditions are equivalent, and each of them is equivalent to controllability of a pair (A, b):
1) The relations z*Akb = 0, k = 0,1, ... ,n-1, for a vector z E ((:1 can be satisfied only for z = o.
2) The relations A *z = Aoz, b *z = 0, satisfied for some complex number Ao and a vector z, are possible only for z = 0.
3) For any complex number A, rank IIA - AI, b II = n.
Corollary 1.1. If a pair (A, b ) is controllable, then for any (n X m )-matrix s the pair (A + b s *, b ) is also controllable.
It is clear from Definition 1.12 that a pair (A, c) is observable if and only if the pair (A *, c) is controllable. Thus any condition for a pair (A, b) to be controllable becomes the condition for a pair (A, c) to be observable after replacing A and b by A * and c.
Theorem 1.3. Let l = m = 1. For a pair (A, b) to be controllable, and a pair (A, c ) to be observable, it is necessary and sufficient that the transfer function X(p) be nondegenerate.
In conclusion we note that with respect to the nonlinearity of cp( 0") in (1.8) we usually suppose that it satisfies all conditions ensuring the existence and uniqueness of solutions x (t) of the system (1. 7), (1.8) on [0, +(0) and their continuous depen­ dence on the initial data. The cases of hysteretic and discontinuous nonlinearities will be specified each time.
§1.2. The Equation of Oscillations of a Pendulum
Consider a mathematical pendulum in the form of a mass point M of mass m suspended on an inextensible weightless thread of length l to a fixed point O. It is well known (see, for example, [39] ) that the equation of motion of the mathematical
Classical Two-Dimensional Oscillating Systems 7
pendulum can be represented in the form
mlB + kB + mgsinB = N, (2.1 )
where B is the angle between the line OM and the vertical axis passing through the point of suspension 0; N is a constant force directed along the tangent to the trajectory of motion of the point M; k is a coefficient of proportionality which defines the viscosity of the medium; 9 is free fall acceleration. Assuming in Eq.(2.1) that a = k / (ml), b = 9 / I, L = N / (ml), we write the equation of oscillations of the pendulum in the form
B + aB + b sin B = L. (2.2)
This equation may describe the dynamics of a synchronous machine in the crudest idealization (in the so-called" zero approximation") [387], the work of the simplest phase locked loop (with RC-circuit as a low pass fillter and with a sinusoidal phase detector characteristic) [247, 330], and the dynamics of a search system of phase locked loop as well [330]. For the synchronous machine B(t) is the phase difference of a rotating magnetic field and a rotor, and for the systems of phase locked loop B(t) is the phase difference of standard and controlled oscillators. In addition, (2.2) can be the equation of Josephson's junctions [246].
Thus (2.2) describes a sufficiently large class of objects different in their physical nature. Moreover in analysing (2.2) many common properties belonging to all such objects become apparent.
Equation (2.2) and the system equivalent to it
B = 1], i] = -a1] - bsinB + L (2.3)
have been well studied [39]. Some results have been often used in what follows will be given later.
We draw attention to one important peculiarity of the system (2.3), namely, to the correctness of the following simple statement.
Proposition 2.1. If (B(t), 1](t)) is a solution of the system (2.3), then for any integer j the function (B( t) + 2j7r, 1]( t)) is also a solution of the system (2.3).
The natural requirement on the phase space of a mathematical model of a real system is that to every physical state of the system there should correspond one and only one point of this space. In this case the plane (B, 1]) cannot be used for such a phase space of the system (2.3). Indeed, the pendulum state is defined by the angle B of its deviation from the low vertical position and its velocity 1] = B. But in changing B to 27r the physical state of a pendulum becomes such that it does not differ from the initial one. Hence, in a plane (B, 1]) there are infinitely many points corresponding to the same physical state of a pendulum. These are the points which are at a distance of 27rj (j an integer) from each other along the B axis.
The requirement for uniqueness will be observed if we introduce the residue classes modulo 27r( B mod 27r, 1]) forming the ring of residue classes modulo 27r {( B mod 27r, 1])}. This set possesses the natural structure of a smooth manifold which is diffeomorphic to the cylindrical surface <C X IR. 1 situated in IR. 3. Therefore
8 Chapter 1.
the space {( () mod 27f, TJ)} is usually called cylindrical. Sometimes, implying a dif­ feomorphism between {( () mod 27f, TJ)} and C x IR 1, we talk about the surface of the cylinder {( () mod 27f, TJ n [20]. The space IR 2 = {( (), 1] n is called a covering space for {( () mod 27f, 1] n·
It follows from Proposition 2.1 that {( () mod 27f, 1] n is a phase space for the system (2.3). In many cases it is convenient to represent the trajectories of the system (2.3) on the phase cylinder C x IR 1, piling up a phase portrait of trajectories in the covering space IR 2 = {( (), TJ n on it.
Consider now the possible motions of the pendulum and the corresponding tra­ jectories on the phase cylinder" at the level of common sense" so as to clarify the definitions introduced later.
Suppose that a = 0 (there is no resistance of the medium) in the system (2.3). Then two types of pendulum motion are possible: (1) periodic undamped oscillations around an equilibrium (the moment of the rotating force is less than the maximal moment of gravity); (2) circular rotations of the pendulum with increasing instanta­ neous angular velocity around the point of suspension (the moment of the rotating force is larger than the maximal moment of gravity). If a =J 0, then it is obvious that a motion of type (1) is impossible, but in return along with (2) the following kinds of motions are possible: (3) damped oscillations; (4) circular motions around the point of suspension with periodically repeated instantaneous angular velocity.
A closed curve not surrounding the cylinder (but surrounding a singular point of the system (2.3)) (Fig.l.1,a) corresponds to the a motion of type (1) on the phase cylinder. A closed curve on the phase cylinder which surrounds it (Fig.I.1, b) corre­ sponds to a motion of type (4). To differentiate these two types of periodic motions we call them periodic ones (cycles) of the first and the second kind respectively. Mo­ tions of type (2) are called circular. Trajectories which also surround the cylinder, but are not necessarily closed on it (Fig. I.1,c), correspond to them.
a 8 c
Fig. I. I.
Under the development of a phase cylinder on the surface ((), 1]) a cycle of the first kind remains a closed curve (Fig. 1.2,a), and a cycle of the second kind is converted into the graph of a 27f-periodic function F(()) with respect to () (Fig.I.2,b). The
Classical Two-Dimensional Oscillating Systems 9
circular motion is converted into a curve F1(B) not necessarily periodic with respect to B (Fig.1.2,b).
Finding out such relations between the parameters a, L, b of the system (2.3) is of interest. If these are satisfied, one or other of the mentioned types of pendulum motion may arise. Suppose that in (2.3) 0 ::::; L ::::; b. Then there exists Bo such that o ::::; Bo ::::; 7r /2 and L = b sin Bo. In this case system (2.3) can be transformed into a system of type
B = 'fJ, r, = -a'fJ - b(sinB - sinBo).
9
Fig.1.2.
Having performed the change of variables B - Bo = x, B = y, we get the system
:i; = y, if = -ay - f(x),
where f( x) = b[sin( Bo + x) - sin Bol. Having performed the change T = -/bt, we write (2.1) in the form
jj + alB + sin () -, = 0,
(2.4)
(2.5)
where al = a/-/b, , = L/b. This equation is also equivalent to the system (2.4) in which f( x) = sin x - , and instead of a we write al.
Representation of the system (2.3) in the form (2.4) often occurs in the literature. As we have mentioned, its properties have been thoroughly investigated. Later on we shall have to refer many times to the system (2.4), and therefore we now turn our attention to the properties of its solutions.
Consider the system (2.4), where f( x) is a continuously differentiable 27r-periodic function. The singular points (stationary points) of system (2.4) are the points defined by
y = 0, f(x) = o. (2.6)
Thus all the stationary points of this system (if there are any) are situated on the x-axis in
y
Fig.1.3
10 Chapter 1.
the plane (x,y) and coincide with the zeros of f(x). For definiteness, every where later on we assume that f( x) has no more than two zeros in the period [0, 21f) (Fig.1.3).
Let
(J(x)? + (J'(X))2 -1= 0, Vx E JR I, (2.7)
l.e. in particular, the mean value of f( x) during a period is non-positive. This requirement does not lose generality, because if To > 0 it is sufficient to make the change of variables Xl = -x and we arrive at the case considered.
Let us also assume that a > O. This supposition immediately results in the lack of cycles of the first kind, i.e. solutions of the system (2.4) periodic in time. Indeed, examining the function
Fig.1.4.
V(x,y) = y2 + 21x f(x)dx
and estimating its derivative with respect to system (2.4) we obtain V = -2ay2 < 0 for y -1=
-1= o. Let (x(t), y(t)) be a peri­ odic solution of the system (2.4). Since this solution is not en­ tirely disposed in the set y = 0, then for this solution the func­ tion V[x(t),y(t)] must have
a negative increment during the period on the one hand, but on the other hand it is not changed. This contradiction proves the lack of cycles of the first kind.
In fulfilling condition (2.7) and the supposition a > 0, different versions of the behaviour of trajectories of the system (2.4) as a whole are possible. Below they are given in the form of separate theorems [39J.
Theorem 2.1. Let f(x) < 0 V x E JRI. Then all solutions of the system (2.4) are not bounded with respect to the coordinate x, and for any solution there exist numbers T and c > 0 such that for all t 2': T
dx y(t) = dt 2': c. (2.8)
In this case there exists a unique solution (x(t),y(t)) for which
x(O) = x(T) + 2h, x'(O) = x'(T) (2.9)
with some integer k and T> 0 (Fig.1.4).
Classical Two-Dimensional Oscillating Systems 11
a
8
Fig.1.5.
12 Chapter 1.
For a mathematical pendulum such a situation holds if L > b> 0 (see Eq.(2.1)). In this connection, any motion of a pendulum with increasing time changes to a rotary one around the point of suspension (into a circular motion). Mathematically a circular motion is given by condition (2.8). The solution defined by condition (2.9) is a cycle of the second kind.
Theorem 2.2. Let f(xo) = f(Xl) = 0, where XO,Xl E [0,211'). Then there exists acr(To) such that
a) for a > acr(To) any solution of the system (2.4) as t --+ +00 tends to some equilibrium (Fig.1.5,a);
b) for 0 < a < aer(To) the system (2.4) has equilibria stable locally and solutions satisfying conditions (2.8), (2.9) (Fig.1.5,b);
c) for a = aer(To) the system (2.4) has solutions tending to saddle equilibria as t --+ +00 and as t --+ -00. In a cylindrical phase space these solutions correspond to the loop of the separatrix of saddle equilibrium (Fig. 1.5,b ). All other solutions as t --+ +00 tend either to the equilibrium or to the loop of the separatrix.
The quantity acr(To) is called a critical or bifurcational value of the parameter. It is clear that aer(To) decreases with decreasing To, and if To --+ 0, then aer(To) --+ O.
Later we often have to consider the system (2.4) with f(x) = sinx -" where I is a certain non-negative number. Using our common analysis of the behaviour of trajectories of the system (2.4), we denote their possible types in this particular case in accordance with the value of parameter I, assuming a > 0 to be fixed. The following theorem is true.
Theorem 2.3. There exists ler (a) such that 1) if I > 1 for any trajectory of the system (2.4), then condition (2.8) IS
fulfilled; 2) If 1 > I > ler( a), the system (2.4) has equilibria locally stable and solutions
satisfying conditions (2.8) and (2.9); 3) if 0 < I < ler (a), any solution of the system tends to some equilibrium as
t --+ +00 .
10
In the theory of phase locked loop [330],er (a) is called a capture band of the system. The graph of ler(a) is shown in Fig. 1.6 [330]. Nu­ merous investigations beginning with the work of F.Tricomi [354] are de­ voted to estimates of ler(a). More­ over, besides analytic estimates there are estimates using various numeri­ cal and approximation methods [330, 349].
Later on we shall also need some 102. oct facts concerning the behaviour of
Classical Two-Dimensional Oscillating Systems 13
For definiteness, suppose that f(xo) = 0, f'(xo) < 0, where Xo E [0,211"}- Then x = Xo, y = 0 evidently defines an equilibrium of the system (2.4). Hence
it follows from the condition f'(xo) < 0 that the roots of the eqution p2 + ap+ + f'(xo) = 0 are real numbers of opposite signs, i.e. x = Xo, Y = 0 is an unstable saddle singular point.
Consider the case a > aCT' Then, as mentione~ abov~, all solutions of the system (2.4) are bounded. Denote by (xo(t),yo(t)) and (xo(t),yo(t)) the trajectories of the system (2.4) tending to a singular point x = Xo, Y = 0 as t -+ +00 (Fig.1.5,a). These trajectories are called separatrices of the saddle system (2.4).
Assume (for defini~eness) that in some neighbourhood of the point t = +00 the
conditions Yo(t) > 0, yo(t) < 0 are satisfied.
Theorem 2.4[39,130] Conclusions
lim Yo(t) = +00, Yo(t) > 0 for t E (-00, +00), t-t-oo
lim Yo(t) = -00, Yo(t) < 0 for t E (-00, +00). t-+-oo
are correct.
(2.10)
(2.11 )
Dividing term by term the second equation of the system (2.4) by the first one, we get the following first-order equation:
dy f(x) - - -a---dx - y , or
dy Ydx +ay+f(x) =0. (2.12)
Suppose that all the solutions of (2.12) are boundedly by (a > aCT)' Then the following proposition is true.
Proposition 2.2. There exists an integral curve y(x) of (2.12) defined for x E (-00, +00) and passing through a point x = Xo, Y = 0 such that
lim [y(xW = +00. Ixl-++oo
(2.13)
Indeed, as the curve y(x) ,!!e m~ take an integral curve of (2.12) "sewn" from
trajectories (xo( t), Yo( t)) and (xo( t), Yo(t)) ofthe system (2.4). Since forthese trajec­ tories the relations (2.10) and (2.11) are satisfied, then to prove (2.13) it is sufficient to show that the curve y( x) has no vertical asymptotes.
Assuming the contrary, there exists a number (3 such that
lim ly(x)1 = +00. x-+(3
(2.14)
But then lim ly'(x)1 = +00 . Moreover, passing to the limit as x -+ (3 in the equality x-+(3
dd Y = -a - f((x)) and taking into consideration (2.14), we get limy'(x) = -a. The x y x x-+(3
last contradiction proves the relation (2.13).
14 Chapter 1.
Now consider the case 0 < a < aCT'
Proposition 2.3. There exists a solution y( x) of (2.12) satisfying the initial condition y(xo) = 0 and such that y(x) > 0 for x > Xo.
Indeed, this solution corresponds to the trajectory (x(t), y(t)) of the system (2.4) which tends as t -+ -00 to an unstable singular point x = Xo, Y = 0 and is such that y(t) > 0 in some neighbourhood of t = -00 (Fig.1.5,b).
Finally, for a = aCT (see Theorem 2.2) (2.12) has a pseudo-periodic solution generated by trajectories of the system (2.4) tending singular points as t -+ -00
and t -+ +00 (Fig.1.5,b). In a cylindrical phase space it corresponds to the loop of the separatrix adjoining
an unstable equilibrium as t = -00 and t = +00 (Fig.l.l,b). Thus the loop of the separatrix is the unique nontrivial 3 bifurcation of the system
(2.4). Let us now turn to the second-order system
:i; = y - bf(x), iJ = -ay - f(x), (2.15)
which is a generalization of the system (2.4). The equations describing widely spread phase locked loop systems with a pro­
portionally integrating filter can be written in the form (2.15) [247, 330, 366]. It turns out that adding the term -bf(x) to the first equation of the system (2.4)
substantially extends the range of its possible bifurcations. Considerable efforts were undertaken to discover this fact [48, 136]. The paper [47] contains detailed and complete investigation of the system to which system (2.15) can be reduced for b ;::: O. Some results of this work used in the chapters to follow are given below.
Assume that the parameters a, b and the 21l'-periodic function f( x) in system (2.15) satisfy the conditions
a 20, b 2 0, f(x) = fl(X) -" ,20. (2.16)
Moreover, the function h(x) is continuously differentiable, fl (-x) = - h(x), N(x) has exactly two zeros on the interval (0, 21l'), and the relation
12K h(x)dx = 0 (2.17)
is satisfied. For definiteness we suppose fHO) > o. If ,0 = max fl(X), then for xE[0,211']
0:::; , < ,0 the system (2.15) has exactly two equilibria (xo(T), 0) and (XI(T), 0), 0 < < xo(T) < XI(T), in the period [0,21l'). In addition, a characteristic polynomial of the system (2.15) linearized at a point (Xi(T),O) (i = 0,1) is of the form
p2 + [a + bf~(Xi(T))]p + (ab + l)fnxi(T)] = O.
Since f{[xo(T)] > 0 for, E [0,')'0], the equilibrium (xo(T), O) is always a stable focus or a stable node. It follows from the inequality f{ [Xl (T)] < 0 that the point (Xl (T), 0) is always a saddle singular point. For, > ')'0 the system (2.15) has no equilibrium.
3We recall that another bifurcation is connected with vanishing equilibrium.
Classical Two-Dimensional Oscillating Systems 15
In the system (2.15) we perform the change of variables
Y1 = [y - bf(x)](ab+ It1/Z, r = (ab+ l)l/Zt. (2.18)
Then it takes the form
x = Y1, 1/1 =,- JI(x) - [a + (3J'(X)]Yb (2.19)
where 00= a(ab+ 1)-l/Z, (3 = b(ab+ It1/z. According to (2.16) and (2.17)
r" 00:2:0, (3:2: 0, /:2: 0, 10 JI(x)dx = o. (2.20)
Under the assumptions (2.20) the system (2.19) was thoroughly investigated in [47]. Since the change of variables (2.18) retains the topological structure of trajectories ofthe system (2.15) on the phase cylinder, for investigating this structure under different relations between the parameters of the system we can use the results of [47]. We give without proof the results from [47] necessary for us later on in the form of separate theorems.
Theorem 2.5. For (3 > 0 the system (2.19) has no cycles of the first kind, and none of the second kind situated in the half-plane Y1 < o.
Theorem 2.6. Let a = O. The following statements are true. a) For / = 0 every solution of the system (2.19), with the exception of separa­
trices tending to saddle singular points as t ~ +00, tends to one a solution of the stable equilibrium as t ~ +00 (i.e. the system is globally asymptotically stable) (Fig.I.7,a).
b) For 0 <, < ,((3) the system (2.19) has a unique unstable cycle of the second kind. In this case the separatrices of saddle points are situated as shown in Fig.l. 7,b.
c) For ,((3) < , < 1 the system (2.19) has no cycles of the second kind, and separatrices of saddle points are situated as shown in Fig.I.7,c.
d) For, > 1 the system (2.19) has neither singular points nor cycles, and for each trajectory of it the condition x(t) :2: c > 0 for t :2: to is satisfied (Fig.1.7,d).
Theorem 2.7. Let a > O. Then for every fixed value (3 > 0 the plane of parameters (00,(3) can be divided into four domains (Fig.1.8).
a) For parameters (a,,) from the domain d1 the system (2.19) is globally asymp­ totically stable (Fig.I. 7,a).
b) For parameters (a,,) from the domain dz the system (2.19) has a unique cycle of the second kind, locally stable (Fig.1.9,a).
c) For parameters (a,,) from the domain d3 the system (2.19) has at least two cycles of the second kind, one of which is locally stable in the small and the other is locally unstable (Fig.1.9,b).
d) For parameters (a,,) from the domain d4 (i.e., for, > /0) the system (2.19) has no equilibria, all its solutions satisfy the condition x(t) :2: c > 0 for t :2: to, and moreover the system has a unique cycle of the second kind (Fig.1.9,c).
16 Chapter 1.
Fig.1.7.
It follows from Theorems 2.5 - 2.7 that the system (2.19) (and hence also the system (2.15)) under the fulfilment of conditions (2.16) and (2.17) is either globally asymptotically stable or has at least one solution for which
i(t) ~ c; > 0 for t ~ to. (2.21 )
Classical Two-Dimensional Oscillating Systems 17
In the special case when it (x) sm x, for the search of relations between the parameters of the system (2.19) under which this system has a circular solution, the re- af sults of the numerical analy- sis established in [48J may be used. Fig.l.l0 shows the de- pendence of magnitude "'( (the d capture band) on 0'2 for the fixed ~
value of parameter n = 0'(3. In this connection the curves cor- responding to the boundary of 10 , the domain d3 are denoted by
Fig.l.8.
dotted lines, and those corresponding to the boundary of the domain d2 by conti-
c :r -----
18 Chapter 1.
nuous ones. The digits on the curves denote the value of parameter n for which the curve is drawn.
We emphasize once more that Theorems 2.5 - 2.7 do not give a complete de­ scription of the possible behaviour of trajectories of the system (2.15) under different relations between its parameters. Thus, for example, the system (2.15) for b < 0 may have cycles of the first kind [349], but this is impossible, as we have verified above, for b 2: O. However, the information contained in these theorems will be quite enough for us in what follows.
The properties of trajectories of dynamical systems of the second order (2.4) and (2.15) that we have considered turn out to be typical also for multidimensional dynamical systems describing different pendulums [39], electronic [247, 330] and electromechanical systems of synchronization, Josephson junctions [246], vibrators [58] systems of angular stabilization [156].
All these objects are described by differential equations
dcr dz m I dt = <1>(cr,z), dt = W(cr,z), z E ~ ,cr E ~ , (2.22)
and by their various generalizations, including infinite - dimensional ones as well [247,330,343]. Here <1>(., .), W(-,·) are vector functions of vector arguments cr and z, and their components CPi( cr, z) and 1jJj( cr, z) are periodic with respect to components cri(i = 1, ... , I; j = 1, ... , m) of the vector cr. Without loss of generality we may suppose that the period with respect to all angular coordinates cri is the same and equal to 271'. The following statement is true.
Proposition 2.4. If z (t), cr(t) is a solution of the system (2.22), then z (t), cr(t)+ +2L71' is also a solution of the system (2.22), where L is an arbitrary vector with integer-valued components.
For the system (2.22) as well as for the system (2.3) along with the Euclidean phase space ~ m+l we can introduce a cylindrical phase space by considering residue classes modulo 271' (cr}mod 271' , ... , crlmod 271' ,Z}, . . . ,zm) forming the ring: {( cr}mod 271', ... ,crlmod 271' Z}, ... , zm)} . It follows from Proposition (2.3) that {( cr}mod 271' , ... , crzmod 271' ,Z}, ... , zm)} is a phase space for the system (2.22).
The coordinates cri are called angular [39] or cyclic [301], and the system (2.22) is often called a system with cylindrical phase space [39]. In the theory of syn­ chronization, systems of the form of (2.22) are called phase systems [25, 28], and in the theory of power systems they are called systems of pendulum like [356]. In mechanics such systems are called systems with angular coordinates. Later on we shall use all these terms.
In solving various problems of global analysis of such systems it often turns out to be convenient to work in some space: ~ m+Z [180] or {( cr}mod 21l', ... ,crzmod 271', Z}, ... , zm)} [81, 232]. Therefore the choice of phase space is more due to the method of investigation used than to the requirement for uniqueness of the position of an object in the space of states (phase space). Thus, the term "a system with cylindrical phase space" reflects only the case when this system has cylindrical phase space.
Classical Two-Dimensional Oscillating Systems 19
~~------r---------+---------+--------+--------~~ ~--____ ~ ________ L-______ -L ______ ~ ______ ~~
20 Chapter 1.
Sometimes it is more appropriate to consider a system of pendulum type written in the form
dx dt = f(x), (2.23)
than the system (2.22). An important example of such a case is shown larer in discussing the general notation for multidimensional phase systems with one scalar nonlinearity.
Definition 2.1. The system (2.23) is called a phase system if there is a nonzero vector d E lR. n such that f(x + d) = f(x). Magnitude d*x is called an angular or phase coordinate of the system (2.23).
Let us give definitions of a circular solution and a cycle for the system (2.23) that generalize to multidimensional phase systems the similar notions we have considered for a pendulum.
Definition 2.2. The solution x( t, xo) of the system (2.23) is called circular with respect to the phase coordinate d*x if on some interval (T, +00)
![d*x(t,xo)] ~ E,
where E is some positive number.
Definition 2.3. The solution x(t,xo) is called a cycle of the second kind with respect to the phase coordinate d*x if there exists a number T > 0 and an integer k 1:- 0 such that
X(T,XO) - Xo = kd.
In particular, Ch. 5 is devoted to the establishment of criteria for the existence of circular solutions and cycles of the second kind in multidimensional phase systems. In Ch. 6 conditions for the existence of cycles of the first kind in multidimensional systems with an angular coordinate are obtained.
Consider the general notation for phase systems with one scalar nonlinearity by which, for example, the dynamics of widely spread phase locked loop systems are described. Such systems can be written in the form [130]
x=Px+q~, a=r*x, ~=cp(a), (2.24 )
where P is a constant (n x n )-matrix; q and r are constant n-dimensional vectors. It is clear that the system (2.24) is a special case of the system (2.23).
We denote by X (p) = r*(P - pItlq the transfer function of the linear part of the system (2.24) from the input ~ to the output (-a). Assuming X (p) to be nondegenerate, we show that without loss of generality the matrix P in (2.24) can be regarded as singular and the function cp( a) as periodic.
Phasability of the system (2.24) with respect to Definition 2.1 means the exis­ tence of an n-vector d 1:- 0 such that for all x E lR. n
Pd + qcp(r*x + r*d) = qcp(r*x),
Classical Two-Dimensional Oscillating Systems 21
equivalent to which is Pd + q<p(a + r*d) = q<p(a), (2.25)
where a E IR 1.
We recall that non degeneracy of the transfer function X (p) implies controllability of the pair (P , q) and observability of the pair (P, r). Therefore q # ° and rOd # 0. Indeed, assuming that rOd = 0, we immediately deduce from (2.25) that
r*pkd = 0, k = 1,2, ... , n. (2.26)
According to Theorem 1.2 it follows from (2.26), the condition rOd = 0, and the observability of the pair (P, r), that d = 0. However, in Definition 2.1 dolO. This contradiction proves that rOd # 0.
Let us assume that 1= q*Pd(r*dlqI2t1 and write (2.24) in the form
x = (P - lqr*)x + q~l' * a = r x, 6 = <p(a) + lao (2.27)
We show that the function 1j;( a) = <p( a) + la is ~-periodic with period ~ = rOd. For this purpose we take use the following equality resulting from (2.25):
* q*Pd <p(a + r d) - <p(a) = -Tclr.
We have
q*Pd * - Tclr + lr d = 0.
We now show that the matrix P -lqr* is singular. Indeed, from (2.25) and the .6.-periodicity of 1jJ(a) = <p(a) + la it follows that
(P -lqr*)d = q[1j;(a) -1jJ(a + .6.)] = 0.
Since dolO, the last equality means that det (P - lqr*) = 0. Since any phase system (2.24) can be written in the form (2.27), for the phase
system with nondegenerate transfer function X (p) it can be assumed without loss of generality that det P = ° and <p( a) is a ~-periodic function.
It is easily seen that the converse is also true. If in (2.24) det P = ° and <p(a) is a ~-periodic function, then (2.24) is a phase system. Indeed, denoting by S the eigenvector of P corresponding to the zero eigenvalue, we get (2.25) with d = ~S(r*Stl.
On the assumption that det P = ° the system (2.24) can be reduced by a nonsingular linear transformation to the system
22 Chapter 1.
where A is an (n - 1) X (n - 1 )-matrix; gl, al are (n - 1 )-dimensional vectors; g2 and a2 are numbers. Since the pair (P, r) is observable, the pair (A, a l ) is also observable, and hence A *al i: O. From (2.28) we have
Assuming that Xl = Z, gl = b, A *al = C, a~gl + a2g2 = p, the system (2.28) can be written as
(2.29)
So a phase system with one scalar nonlinearity and non degenerate transfer func­ tion X (p) can always be represented in the form (2.29), where A is an (n - 1) X
x(n - I)-matrix; band care (n - I)-vectors; p is a number; cp(o-) is a ~-periodic function. We emphasize that having distinguished an equation for an angular coor­ dinate 0- we have thereby reduced the system to the form (2.22).
Later in obtaining frequency criteria for the existence of circular solutions and cycles of the second kind we shall mainly use the notation for a phase system with one nonlinearity in the form (2.29).
§1.3. Oscillations in Two-Dimensional Systems with Hysteresis
In different fields of engineering the devices described by equations with hysteretic nonlinearities are widely spread. The graphs of hysteretic functions represent as a rule several (sometimes infinitely many) branches, for which certain rules of transi­ tion from one branch to another are given. Let us restrict ourselves to consideration of so-called typical hysteretic nonlinearities. They are: a play [14l (Fig.l.l1, a), a relay with hysteresis, and a relay with hysteresis and insensitivity [16, 120l (Fig. 1.11,b, c). The rules of transition from one branch to another mentioned above are given in the figures by arrows.
We denote the value of a hysteretic function by cp[o-(t),CPolt, assuming that cp[., 'It is a functional given on a direct product {o-( t)} X {CPo} and depending on a parameter t 2: to. Here {o-(tn is some set of continuous functions, and {CPo} is a number set depending on 0-( to) . {o-( tn is often called an input set, and {cpo} is a set of initial states.
In particular, for a play in Fig.1.11a a set of initial states {CPo} = E(o-(to)) for some o-(to) is shown. Thus, cp[o-(t), cpolto = CPo, and if cpo belongs to the interior of E(o-(to)) the equality cp[o-(t), cpolt = CPo is also fulfilled for t E [to, TJ, where T is the first instant of time when CPo belongs to the boundary of E(o-(T)). Let CPo = max 1],
where 1] E E(o-{T)) and also o-(t) is differentiable and a-(T) < O. Then according to the arrow on the graph (Fig.l.ll, a) we move along the upper straight line (i.e. for t such that cp[o-(t),cpolt = 8 + o-(t) ) until a-(t) changes sign. If at a point T
magnitude a-(t) changes sign (more precisely a-(t) > 0 for t > T and close to T ),
then preserving continuity we begin moving along the horizontal straight line (i.e. cp[o-(t),cpolt = cp[o-(t),cpolr) until we again go out to the boundary of E(o-{t)). If we again go out to a maximal point of E( 0-( t)), then we repeat the algorithm for defining
Classical Two-Dimensional Oscillating Systems 23
cp[a(t)'cpok The arrows described above in the case of going out to a minimal point of E(a(t)) enable one to move along the inclined straight line only for er(t) > o.
E(6(t.g
Fig. 1.1 1.
Thus we have described a "play" functional given on differentiable functions a(t) for which er(t) changes sign at the zeros of er(t).
In a similar way, the functionals of "relay with hysteresis" and "relay with hys­ teresis and zone of insensitivity" are described.
The necessity of creating a sufficiently developed theory of hysteretic functions becomes apparent already on typical nonlinearities: in the case with a "play" it is necessary to extend a set of "inputs" {a(t)} to continuous ones or, at least, giving up differentiability, to avoid requiring er(t) to change sign at zeros of er(t). In the case with a relay there also arises the question of interpreting values of cp at the points a = ±o. Later on it will often appear natural to interpret cp( 0) and cp( -0) as segments [0, M] and [-M, 0] in passing to multifunctions with convex ranges.
It is quite natural that hysteretic functions describing a play and a relay, which are elements of more complex technical systems, occur in differential equations defin­ ing the dynamics of these systems.
24 Chapter 1.
In the works of A.A.Andronov and N.N.Bautin [14], N.A. Zheleztsov (see [16]) and A.A.Feldbaum [120], which are now classical, the system of equations
(3.1 )
has been considered. Here Andronov and Bautin investigated the system (3.1) with a "play", Zheleztsov the system (3.1) with a "hysteretic relay", and A.A.Feldbaum the system with a hysteretic relay and insensitivity. In these works by the method of Poincare mapping it has been shown that the system (3.1) unlike the case when ~ = cp(O') (see the arguments in §1.2) may have a stable cycle which "surrounds" a stationary set (compare with the van der Pol equation in §1.1).
A.A.Andronov and N.N.Bautin [14] gave the critical value of parameter 0:, equal to (3.04to.5 , such that for 0: > (3.04to. 5 the system (3.1) with nonlinearity of play type is stable in the large and for 0 < 0: < (3.04to. 5 it has a unique nontrivial periodic solution (cycle), the closed trajectory of which is symmetric with respect to the origin of the phase plane. All points of a stationary set of the system (3.1) belong to a domain bounded by this trajectory, therefore for brevity we can say that a trajectory of a periodic solution" surrounds" a stationary set.
N.A.Zheleztsov showed [16] that the system (3.1) with nonlinearity of hysteretic relay type always has a nontrivial periodic solution whose trajectory also "sur­ rounds" a stationary set.
Investigating the system (3.1) with nonlinearity of a hysteretic relay with insen­ sitiveness type (Fig.1.11, b) A.A.Feldbaum found [120] a necessary and sufficient condition for stability in the large:
If we violate this condition, the system (3.1) has a unique nontrivial periodic solution whose trajectory "surrounds" a stationary set.
Note also the works [241, 313], where the stability of Eq. (3.1) with a nonsta­ tionary function cp( t, 0') is investigated.
We now turn to a multidimensional analogue of the system (3.1) and consider the system
dx dt = Ax + b~, 0' = c*x, ~ = cp[O'(t), CPo]t, (3.2)
where x E ~n; A is an (n x n)-matrix; b,c are n-vectors; cp[O'(t),CPo]t is a value (branch) of a hysteretic function. Let us clarify the notion of a hysteretic function in the general case and the notion of a solution of the system (3.2) with a hysteretic function on the right-hand side. However, we do not pose the problem of describing the theory of systems with hysteretic functions completely and strictly, referring those who wish to get acquainted with such theory to the fundamental monograph of M.A.Krasnoselsky and A.V.Pokrovsky [168]. When we give later the definition of a hysteretic function adopted in our book, we note that for simplicity we may have in mind the most frequently occurring hysteretic functions of a relay with hysteresis type (Fig.1.11, b, 1.11, c) and a play type (Fig.1.11, a).
Classical Two-Dimensional Oscillating Systems 25
First of all, we recall the notion of a semi continuous multifunction [130]. We restrict ourselves to the case of a scalar function cp( a) (a E lR 1, cp E lR 1).
The scalar function cp( a) is single - valued at a point ao if the set cp( ao) con­ sists of one point. Otherwise, the function cp( a) will be called multivalued at ao. The function whose graph is represented in Fig.1.11,b is single-valued for lal > 8 and multivalued for 2. lal < 8. Moreover its values at the points a = ±8 are seg- ments: cp(-8) = cp(8) = E[±8] = [-M,M]. For a E (-8,8) 1 we can write cp(a) = E[a] = {-M,M}. The function whose graph is shown in Fig.1.11, a is multivalued for all a E lR 1.
In this case cp(a) = E[a] = [a - 8, a + 8]. The function whose (f graph is represented in Fig.1.12 is also multivalued at the -1 point a = O. Here by definition E[O] = [-1,1].
Let E be some set in the plane. We call the set of points -2 y satisfying the inequality inf p( x, y) < 6, where p( x, y) is the
xEE Fig. 1.12.
Euclidean distance between the points x and y, the 6-neighbourhood of E.
Definition 3.1 [130]. The function cp(a) is called semi continuous at a point ao if from any 6> 0 we can find 8(6, ao) such that cp(a1) belongs to the 6 - neighbourhood of cp( ao) if a1 is located in the 8 - neighbourhood of ao.
The functions whose graphs are represented in Fig. 1. 11 are semicontinuous. The function in Fig.1.12 is not semicontinuous.
Further, C(to, ao) denotes the family of all functions a(t) continuous on [to, ()Q) such that a(to) = ao.
Definition 3.2 (of a hysteretic function). Let the following conditions be ful­ filled.
1. With any number ao E lR 1 there is associated a set E[aol C IR 1;
2. To each triple of numbers (to, ao, CPo) : to E lR 1, CPo E E[ ao], there corresponds an operator W(to, ao, CPo) which associates with any function a(t) E C(to, ao) a certain, generally speaking, multivalued function cp[a(t), CPo]t given on [to, to+6) = ~ (6 depends on to,ao,cpo,a(t)):
(3.3)
3. There hold the relations
cpo E cp[a(t), CPo]to; cp[a(t), CPo]t C E[a(t)] for t E ~;
4. If a1(t) == a2(t) for t E [to,t1] C~, then cp[a1(t),CPO]t == cp[a2(t),CPO]t for these values of t;
5. If t1 E ~, CP1 E cp[a(t), CPo]tJl then cp[a(t), CPo]t = cp[a(t), CP1lt for t 2: t1, t E ~. Then we say that a hysteretic function W is specified.
Thus, a hysteretic function represents a family of operators each corresponding to a certain initial state (to,ao, CPo ). The operator W(to, ao, CPo) (see (3.3)) is called
26 Chapter 1.
henceforth a branch of the hysteretic function W. The set of points in the plane
(O",'P) fw = U {(O",'P): 0" = 0"0, 'P E E(O"o)}
(ToElle
is called the graph of the function W. Note that any semicontinuous function 'P( 0") whose values are segments of JR. 1 is a special case of the hysteretic function.
We now turn to the system (3.2) and agree on how to interpret the solution of this system.
Definition 3.3. A solution of the system (3.2) fo on the interval [to, T] with initial data to,xo,eo is a pair of functions x(t) E JR.n, e(t) E JR.l possessing the following properties.
1) x(t) is absolutely continuous on [to, TJ, x(to) = xo, e(t) is Lebesgue summable on [to, TJ, e(to) = eo E E[c*xo];
2) For almost all t from [to, T] the relations
x(t) = Ax(t) + be(t), e(t) E 'P[o-(t),eo]t
are fulfilled. 3) O"(t) = c*x(t) for all t E [to, T].
The theory of generalized differential equations is given a detailed investigation in the monograph [130]. For the case of systems (3.2) with hysteretic nonlinearities this theory is concretely defined by M.Yu.Filina [121]. In particular, in [121] the question of the existence and continuability of solutions of the system (3.2) with hysteretic nonlinearities of a sufficiently wide class is solved. Leaving aside the most general case, we mention without proof one result of Filina concerning systems with hysteretic functions of a relay with hysteresis type and a play type.
Theorem 3.1 [120]. If in system (3.2) the hysteretic function
coincides with any of the functions whose graphs are represented in Fig.l.11, then the solution of such a system with any initial data to, Xo, eo E E[c*xo] exists and is continuable on [to, +00).
In Chapter 5 in clarifying the conditions for the existence of a periodic solution of (3.2) with nonlinearity of a play type we need one more property of this system: the right-side uniqueness and continuous dependence of its solutions on initial data. In the monograph [168] it is shown (Theorem 2.2) that in the case of arbitrary continuous functions O"l(t) and O"z(t) for the operator W(to,O"o,'Po) (a play), the estimate (we give the estimate for the case of a play represented in Fig. 1. 11 ,a)
IW(to,O"l,'Pl)O"l(t) - W(to, 0"2, 'P2)0"2(t) I ~ max {1'P1 - 'P21, 11001(t) - 0"2(t)llto,r}
holds for t E [to, T]. Here 1100(t)lltD,T = max 100(t)l. tEltD,T]
Using this estimate it is possible to prove the correctness of the following state­ ment.
Classical Two-Dimensional Oscillating Systems 27
Theorem 3.2 [168]. In the system (3.2) with hysteretic nonlinearity whose graph is represented in Fig.1.ll, a the right-side uniqueness and continuous dependence of solutions on initial values holds on any finite time interval.
To conclude this section we concentrate on the question of stationary solutions of the system (3.2). If a matrix in this system is nonsingular, then all its stationary solutions are of the form x = -A -1 h'P, where 'P is the ordinate of any intersection point (CT, 'P) of the hysteretic nonlinearity graph rw with a "characteristic straight line" CT + c* A -lh'P = 0 (see Fig.1.11). Depending on the location of a character­ istic straight line, the system (3.2) with nonlinearity whose character is shown in Fig.1.ll,b may have two or four equilibria; in Fig.l.ll,a there are infinitely many such positions (stationary segment).
Later, in Chapter 7, we shall be interested in the case when the matrix A is singular (has a zero eigenvalue of multiplicity 1). In this case, as we can easily see, a characteristic straight line coincides with the axis CT (has equation 'P = 0). Then the system (3.2) with nonlinearity represented in Fig.1.11,b has exactly two equilibria, and with nonlinearity as in Fig.1.11, a or Fig.1.11,c infinitely many such positions (stationary segment). The coordinates of points belonging to a stationary segment satisfy the relations
Ax = 0, Ic*xl S; 8. (:3.4 )
In the case of an observable pair (A,c) the set defined by (3.4) is bounded.
§1.4. Lower Estimates of the Number of Cycles of a Two-Dimensional System
The present section demonstrates two important principles used in the theory of two-dimensional systems: they are the principle of a ring [319, 325] and the principle of Chaplygin and Kamke [102, 218].
Multidimensional analogues of these principles are used in this book. And here, on the basis of these principles, the recurrent procedure of estimation from below of the number of two- dimensional cycles of a system is developed [218].
Let the nonlinear system (1.5) with n = 2 be given, for which conditions for uniqueness and continuous dependence of solutions on the initial data are supposed to be fulfilled.
The annulus principle. Let D be an annular domain in ~ 2, bounded by closed Jordan curves embedded in each other. If this domain is positively(negatively) invariant for the trajectories of the system under consideration and does not contain singular points of it, then the domain D contains at least one periodic solution (cycle) of the system (1.5).
The assertion formulated is a corollary of the Poincare-Bendixson theorem [319]. Consider two first-order equations
dy/dx = f(x, y), dz/dx = g(x, z), (4.1 )
28 Chapter 1.
where f(x,y) and g(x,z) are continuous functions on some domain D in the plane {x, y}.
The Chaplygin-Kamke principle of comparison. Let the relation f(x, u) > > g(x,u) be fulfilled for x E [a,iJ), (x,u) ED. If y(x) and z(x) are the solutions of equations (4.1), defined on [a,iJ) and satisfying the conditions y(a) ~ z(a), (y(x),x) E D, (z(x),x) E D), then y(x) > z(x) for any x E (a,iJ). The proof of this principle is almost evident and is contained, for example, in [102].
Consider a piecewise-linear system
:i; = y, y = py - Ksigny - x, 0 < p < 2, K > 0, (4.2)
whose solution is understood as in §1.3. It is well known [16, 158] that the system (4.2) has a periodic solution (cycle) whose initial condition is y(O) = 0, x(0)=-K-2K{exp[p1l'(J4-p2 t 1]_q-1. The cycle surrounds a segment of equilibria of this system. In addition, for T = 211'(4 - p2t1/2 we have y( T) = 0, X(T) = K+2K{exp[p1l'(J4=p2)-1]_q-1 (Fig. 1.13).
Fig. 1.13.
In the half-space {y > O} a cycle (x(t),y(t)) is a solution of the system :i; = y, y = py - K - x. Since g(t) = 2py2(t) ~ 0 for g(t) = (x(t) + K)2+ +y2(t), then g(t) increases monotonically on [O,T] and gmax = g(T) = = (x( T) + K)2 = (2K)2 exp [2p1l' ( J4=P2t1] x {exp[p1l'( J 4 - p2t1] _ I} -2.
It is evident that for any trajectory (x(t),y(t)) ofthe system (4.2) in the half­ space {y > o} the relation iJ( t) = 0 is fulfilled in the half-line AB : y = p-1 (x + K) (Fig.1.13). Therefore for a cycle max ly(t)1 = IABI sin LBAC = (1 + p2t1/21ABI <
t -
~ y'gmax(1 + p2t1/2 = 2K exp[p1l'( J4=P2t1] {exp[p1l'( J4 - p2t1] - q -1 X
X (1 + p2t1/2. Assuming that
( 4.3)
t ( 4.4)
It is well known [16, 158] that a segment of equilibria of the system (4.2) is stable, and a cycle of this system is unstable. Moreover, the domain in a plane bounded by a cycle of the system (4.2) is a minimal global B-attractor of this system as t - -00 (see Fig.1.14 and Definition 1.7).
Fig. 1.14. Fig. 1.15
Consider now another piecewise linear system
x = y, if = ->.y + IIsign y - x, 0 < >. < 2, II> o. ( 4.5)
This system is reduced to the form x = YI, if = >'YI - IIsign YI - x by the substitution t = -tl, Y = -YI. Therefore it also has a periodic solution (cycle) (x(t), y(t)), for which the estimate
max ly(t)1 ~ J(>')II t
(4.6)
holds. But the domain bounded by the cycle of this system is its minimal global B-attractor as t - +00 (Fig. 1.15).
The following lemmas will play an important role in further constructions.
Lemma 4.1. Let the estimate max ly(t)1 ~ "f hold for a cycle of the system t
(4.5). Suppose that py - K < ->.y + II, 0 ~ Y ~ "f. ( 4.7)
Then a cycle of the system (4.5) is situated inside a cycle of the system (4.2).
Lemma 4.2. Suppose that for a cycle of the system (4.2) the estimate max Iy( t) I ~ "f holds and also the relation (4.7). Then a cycle of the system (4.2) is
t
situated inside a cycle of the system (4.5).
Proof of Lemma 4.1. By virtue of (4.7) and a comparison principle the vectors of the field of the system (4.2) at all points of a cycle of the system (4.5) ( except
30 Chapter 1.
the two points of intersection of the cycle with the set {y = o} ) are directed strictly inside the cycle.
Therefore the cycles of the systems (4.5) and (4.2) do not intersect. Suppose that
Fig. 1.16.
a cycle of the system (4.5) is situated outside a cycle of the system (4.2) (Fig.1.16).
Then for a cycle of the system (4.2) the estimate max Iy(t) I < I
t holds.
But then the domain bounded by a cycle of the system (4.2) cannot be a minimal global B-attractor of this system as t -+ -00. Lemma 4.1 is proved.
Proof of Lemma 4.2. As in the proof of Lemma 4.1, we verify that cycles of the systems (4.2) and (4.5) do not intersect. Suppose that a cycle of the system (4.2) is situated outside a cycle of the system (4.5). Analysing the field of directions of the system (4.5) in a cycle of the system (4.2) we arrive at a contradiction with the fact that the domain bounded by a cycle of the system (4.5) is a minimal global B-attractor of this system as t -+ +00.
Lemma 4.2 is proved. Let the nonlinear system
x=y, y=-rp(y)-x ( 4.8)
be given, where r.p(y) is an odd function, differentiable for a of- O. In what follows we shall consider two cases:
1) rp( a) has a discontinuity of the first kind for a = 0 and lim rp( a) = rp( +0) > O. <7->+0
The solution of the system (4.8) is understood here as in §1.3. 2) rp(a) is differentiable for a = 0 and rp'(O) < O. We introduce the following positive numbers: 0' < 2, f3 < 2, Ai < 2,
Pi < 2,Ki,vi,lbI2i,,2i+b satisfying the relations: Ij < Ij+I for j = 1,2, ... ,
rp(+O)f(f3) < 11,
KJ(Pi) < 1Mb i = 1,2, .... (4.11)
Here f( z) is the function defined by (4.3). We shall also assume that for the function rp( a) the inequalities
rp( a) 2:: Aia - Vi, Ya E [0,,2i]'
rp( a) :::; -pia + Ki, Ya E [0, IMll
( 4.12)
(4.13)
hold, and in case 1) the relations
-ao-+cp(+O) ~ cp(o-) ~ -~o-+cp(+O), '10- E [0,/1]' (4.14)
Theorem 4.1. [218] Let case 1) holds. If in addition the inequalities (4.9), (4.14) and (4.10), (4.12) hold for i = 1, then the system (4.8) has at least two cycles.
If the inequalities (4.9), (4.14), and also (4.10), (4.12) and (4.11), (4.13) for i = 1,2, ... , n, hold, then the system (4.8) has at least 2n + 1 cycles.
If the inequalities (4.9), (4.14), and also (4.10), (4.12) for i = 1,2, .. ,n and (4.11), (4.13) for i = 1,2, ... , n - 1 hold, then the system (4.8) has at least 2n cycles.
Theorem 4.2 [218]. Let case 2) hold. If the inequalities (4.10), (4.12) and (4.11), (4.13) for i = 1,2, ... ,n hold, then the system (4.8) has at least 2n cycles.
If the inequalities (4.10), (4.12) for i = 1,2, ... , nand (4.11), (4.13) for i = 1,2, ... , n - 1 hold, then the system (4.8) has at least 2n - 1 cycles.
We shall give the proof of Theorem 4.1. Theorem 4.2 is proved similarly.
E x amp I e 4.1. Consider the van der Pol equation (1.3). We carry out a change of variables, assuming that
Xl =:i; +£ lx(x2 -l)dx, Y = -X.
Then (1.3) will be written in the form of the system
which coincides with (4.8). It is obvious that, for cp(y) = y3/3 - y, conditions (4.10) and (4.12) for sufficiently large v and / and some .\ > 0 are satisfied. Therefore, by Theorem 4.2, (1.3) has at least one nontrivial periodic solution.
Proof 0 f The 0 rem 4.1. Consider the systems
:i; = y, if = ay - cp(+O)sign y - X,
:i; = y, if = ~y - cp(+O)sign y - X,
:i; = y, if = -)..iY + visign Y - X,
:i; = y, if = PiY - lI:isign y - x.
(4.15 )
(4.16)
( 4.17)
(4.18)
Each of these systems has a cycle which by virtue of (4.4), (4.6) and suppositions (4.9)-( 4.11) satisfies one of the inequalities
ly(t)1 ~ Ii, 'It E IR \
ly(t)1 ~ /2i, 'It E IR \
ly(t)1 ~ /2i+l, 'It E IR 1.
( 4.19)
(4.20)
(4.21)
32 Cbapter 1.
It follows from (4.14) that a > (3. From here it is easy to deduce that a cycle of the system (4.16) surrounds a cycle of the system (4.15). Moreover, from (4.14) and the comparison principle it follows that the ring formed by these cycles is negatively invariant for trajectories of the system (4.8). Consider now the systems (4.16) and (4.17) (for i = 1). Both have cycles. It follows from (4.12), (4.14) and (4.19) that we are under the hypotheses of Lemma 4.2. Therefore the cycle of the system (4.17) for i = 1 surrounds the cycle of the system (4.16). Hence it follows from (4.12) and the comparison principle that the vectors of the field of the system (4.8) on the cycle of the system (4.17) are directed inside this cycle. Thus we have obtained two rings embedded in each other.
The inner ring is negatively invariant, and the out her one is positively invariant for the trajectory of the system (4.8).
Thus according to the ring principle, at least one cycle of the system (4.8) is contained in each of these rings. That is, the system (4.8) has at least two cycles.
Suppose now in addition that the function cp( 0") satisfies condition (4.13) for i = 1, and assume that the system (4.18) (i = 1) has a cycle. Since by virtue of (4.20) a cycle of the system (4.18) satisfies the condition Iy( t) I ::::; 12, then it follows from (4.12) and (4.13) that we are under the hypotheses of Lemma 4.1. Thus, a cycle of the system (4.18) surrounds a cycle of the system (4.17). Analysing the direction field of the system (4.8) on this cycle and using the comparison principle we verify that another (negatively invariant for solutions of the system (4.8)) ring has appeared, whose boundaries are the cycles of the systems (4.17) and (4.18) for i = 1. This means that the system (4.8) has at least three cycles. Developing similar arguments for i = 2,3, ... , n, we verify the correctness of the conclusion being proved.
Theorem 4.1 is proved.
E x amp 1 e 4.2. In studying a flutter of an aeroplane M.V. Keldysh [158] considered the system (4.8) with cp(O") = -1l0" + (~+ Iw2)sign 0", where 1l,~,K are positive numbers. The obvious geometric constructions show that in this case the inequalities (4.14) and (4.9) will be fulfilled with arbitrary a > 0 and (3 < Il satisfying
~ f((3) < Il - (3. K
( 4.22)
The inequalities (4.10) and (4.12) for i = 1 hold for sufficiently large //1, 12 and some Al > o. According to Theorem 4.1, the system considered in [5] has at least two cycles if (4.22) holds. In the case when Il (and consequently (3 also) is small, (4.22) may be given the form 4~K < (3(Il- (3)7r. Assuming now that (3 = 0.51l, we obtain Il > 4~7r-l/2. The last estimate is close to one of Keldysh, Il > 8J2K~( 7rV3)-1/2, obtained in [5] by the harmonic balance principle [158].
E x amp 1 e 4.3. Choosing Ai = Pi = 1, it is possible to construct the function cp( 0") so that the system (4.8) has at least the given number of m cycles. Indeed, as cp(O") it is sufficient to take a polynomial cp(O") = -0" + a30"3 - a50"5 + a70"7 - ... -
-a2m+l0"2m+l with positive coefficients ai such that ai+2 ~ ai (i = 3, ... , 2m - 1). Here there are found numbers //i, Ki"i, satisfying the inequalities (4.12), (4.13) and the relations 1.69//i < 12i, 1.69Ki < IMI. (Here f(l) < 1.69). Thus the system (4.8)
Classical Two-Dimensional Oscillating Systems 33
with nonlinearity in the form of the indicated polynomial of degree 2m + 1 will have at least m cycles, embedded in each other.
The similar result for 'P(O") = c;P(O"), where P(O") is a certain polynomial of degree 2m + 1 and c; is a small parameter, has been obtained in work [248] (look also review [253]).
We stress that the result obtained is closely connected with a known problem of Hilbert on the number of cycles of two-dimensional systems with polynomial right-hand sides [220].
Mention here the lower estimates of the number of limit cycles, obtained by other methods in works [41, 147, 151,287,388].
CHAPTER 2
Frequency Criteria for Stability and Properties of Solutions of Special Matrix Inequalities
§2.1. Frequency Criteria for Stability and Dichotomy
The properties of stability and dichotomy of the nonlinear systems under consider­ ation are of interest for us because they eliminate the existence of bounded solutions not tending to equilibrium in such systems, in particular, cycles.
The mathematical theory of stability nowadays has available a whole arsenal of methods of investigation and numerous results of their application, represented in the monographs [36,37,101,104,109,171,176,256,259,263,266] and in works devoted to one of its contemporary directions, the theory of absolute stability [5, 126, 130, 178, 234, 244, 267, 270, 302, 312, 367, 368, 385].
The frequency approach in the theory of absolute stability dates back to the works of Y.M.Popov [304-306]. At present the formulations of criteria for stability, dichotomy and instability in terms of a frequency response of the linear part of the system have become traditional. The most famous among them, the Popov criterion and the circle criterion [71, 75, 307, 308, 321, 374, 380] admit a simple geometric interpretation and are convenient in practice. The so - called "off the axis circle criterion" of Cho and Narendra [105, 270], followed by a whole series of graphic criteria obtained recently [30-33, 249], have the same advantages. We mention also the criteria for absolute stability obtained with the help of the nonlocal reduction method [193, 194, 237] and criteria for absolute stability of forced oscillations [129, 160, 197,385].
Frequency criteria for stability and dichotomy of multidimensional phase systems [130] are closely connected with the fourth and fifth chapters of our book. These criteria are analogues of a series of classical theorems in the theory of absolute stability.
Frequency criteria for stability and dichotomy use the same language of a fre­ quency response of the linear part of the system as the formulations of criteria for the existence of a cycle, and together with them give in the parameter space of the system "bilateral" estimates of