Symbolic dynamics and relaxation oscillations

Physica ID (1980) 227-235 © North-Holland Publishing Company

SYMBOLIC DYNAMICS AND RELAXATION OSCILLATIONS* John G U C K E N H E I M E R Division of Natural Sciences, University of California, Santa Cruz, CA USA Received 18 September 1979 Revised 4 December 1979

This paper describes the application of qualitative methods of dynamical systems theory to a specific problem. It examines the forced Van der Pol equation as an example of a relaxation oscillation with aperiodic solutions. The technique of symbolic dynamics, particularly for one-dimensional mappings, is used to give a complete topological characterization of the set of these aperiodic solutions for parameter values for which the equation appears structurally stable. These results are a mathematical interpretation of numerical computations and are not the result of rigorous analysis.

asymptotic behavior exhibited by all solutions. Adequate tools to do so were not then available. Since this work was completed, there has been an enormous expansion of the subject of dynamical systems. Smale's creation of his "horseshoe" [4], was motivated by Levinson's work. It is our intention here to use further techniques from dynamical systems to give a more complete analysis of the phenomena which appear in the forced Van der Pol equation. We do not try to prove here that the description we present accurately portrays the properties of either the forced Van der Pol equation or the equation studied by Levinson. The thesis of Levi [5] does provide the required analysis to substantiate this picture for another piecewise linear analogue of (1) and its perturbations. Our description here is consistent with everything currently known about these equations, all of whose solutions seem to have the same qualitative structure. It represents the asymptotic behavior of these solutions in the simplest possible way. This work complements Levi's by giving a more complete representation of the dynamics of the limit sets. In Levi's approach, it is more difficult to determine such information as the periods of the veriodic orbits.

1. Introduction

The work of Cartwright and Littlewood [1], and Levinson [2] on forced relaxation oscillations provided an impetus to the development of a modern theory of dynamical systems [3]. This work began with the observations by Cartwright and Littlewood during World War II that solutions of the forced Van der Pol equation jl - k(1 - y2) )~ + y = blzk cos/zt

(1)

have remarkable properties for certain ranges of the coefficients b,/~ and k when k is large. In particular, for these parameter ranges, they noticed that (1) has two stable limit cycles whose periods differ and that there are an infinite number of other periodic solutions. Levinson studied a simpler, piecewise linear analog of the forced Van der Pol equation and gave more concise proofs of these results for the equation he studied. Neither Levinson nor Cartwright and Littlewood carried through a complete analysis of the *This research was partially supported by the National Science Foundation. 227

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J. Guckenheimer/Symbolic dynamics and relaxation oscillations

Traditional techniques in the theory of nonlinear oscillations [6] rely upon asymptotic or perturbation analysis of equations when they are close to being linear. Typically, the results are valid for "small" values of a parameter with the meaning of "small" unspecified. Calculations give approximate values for specific solutions of the equation being studied. On the other hand singular perturbation techniques piece together particular solutions of highly nonlinear equations. Our results differ both in character and technique. The results are asymptotic in that they rely upon the analysis of a singular, limiting situation for a class of equations. The limit in our case cannot be immediately derived from simple computations with the equations being studied. It is expressed geometrically by assuming that there is a field of directions in which the solutions of the equations contract infinitely sharply. The limit has the character of the "constrained differential equations" studied by Takens [7]. This limit cannot be realized by the solutions of any system of ordinary differential equations, so that the analysis is necessarily approximate. The techniques we use are geometric and rely heavily upon symbolic dynamics [8]. We use no power series, trigonometric series, or averaging. Our analysis illustrates a new kind of application to be expected from the modern theory of dynamical systems for studying specific differential equations.

2. The geometry of relaxation oscillations In this section, we review the geometry associated with the Van der Pol and forced Van der Pol equations. We discuss the shape of the solutions of these equations and the hypotheses that are required for our analysis of their asymptotic properties. The Van der Pol equation is the second-order differential equation

(2)

J / - e ( 1 - x 2) 2 + x =0.

Here ¢ E [0, ~] is a parameter which is of special interest both when it is small and when it is large. When e = 0, one obtains the equations of simple harmonic motion, while one obtains a "relaxation oscillation" when e is very large. The relaxation oscillation is characterized by abrupt changes in the velocity of solutions of the equation. We prefer to write the Van der Pol equation as a pair of first-order equations, rather than working with (2). Consider )'1 = e[Y2 - (y]13 - YO], )'2 = ( - yJe).

(3)

Eqs. (3) are equivalent to (2), as is seen by setting y~ = x and Y2= (~xs - x) + ~/~. When the parameter • is large, the solutions of (3) are pictured in fig. 1. We shall assume throughout the remainder of this paper that e is large without further comment. Away from the curve 3' defined by y2 = ( y ] / 3 - Yl), the vector field is almost horizontal. Above y solutions move to the right, below they move to the left. Thus, any nonzero solution moves quickly to the vicinity of % In the right half plane solutions follow 3' downard; in the left plane they follow y upward. When a solution approaches a point where y has a horizontal tangent, it jumps quickly to the other point

"YI3/3 "YI

Fig. 1. The flow of the unforced Van der Pol equation.

J. GuckenheimedSymbolic dynamics and relaxation oscillations

of 3' along this horizontal line. All nonzero solutions tend asymptotically to a limit cycle which lies close to a curve pieced together from two segments of 3" and two horizontal segments tangent to 3'. There is a large difference in the velocity along the horizontal segments of the limit cycle and those which follow 3". The forced Van der Pol equation can be written J/-e(1

- - X 2 ) .~ J t - X :

be

cos At,

or

))1 = e(y:

))2 =

( --

- (y~13 - Y0),

(4)

1/e)(ym)+ a cos At.

This is a nonautonomous equation. To study its integral curves, one must introduce time as a third coordinate. Since the dependence of (4) on t is periodic, the time variable can be treated as periodic, yielding a system of three equations defined on R z x S~: Yl = e(y2 - (y~]3 -

Yl)),

Y2= ( - l/e)(yl) + a cos AO,

(5)

0---1. We regard 0 as the 2¢r periodic angular coordinate on S'. To study the asymptotic properties of (5), we note that the plane defined by 0 = 0 is carried into itself by following the flow of (5) for 2¢r units of time. This time 2¢r map is a diffeomorphism F:Re--~R~. The map F is the cross-section or Poincar6 map of (5). The asymptotic properties of solutions to (4) or (5) can be studied by examining F as the generator of a discrete dynamical system. This means that we study the iterates of F. The dynamical system defined by F has asymptotic properties which are more complicated than those of the Van der Pol equation (3). A

229

complete description of the trajectories of F analogous to that given above for the Van der Pol equation is sensitive to the values of the various parameters in the equation. Much of the available information about solutions is based upon observation more than proof, though asymptotic descriptions of some trajectories have been given recently [9]. Nevertheless, there is a high degree of confidence that the following description is accurate. Solutions of (4) still have the general character that they alternately follow segments of 3' and horizontal segments tangent to 3". The effect of the additional sinusoidal term in (4) is that solutions move back and forth along 3'. A sensitive part of the motion occurs when a trajectory reaches a point where it is about to jump horizontally at approximately the time when it begins to move across 3'. The trajectory either completes the motion across 3', jumps horizontally to the other segment of y, or does something intermediate. During this sensitive part of the motion trajectories close to the limit cycle can be spread apart widely from one another. This phenomenon leads to the complicated dynamics of the forced Van der Pol equation. It is easy to watch analogue or numerical solutions of the forced Van tier Pol equation on an oscilloscope screen. One sees stable limit cycles of F with odd periods. Jumps of solutions of (4) to the left and to the right tend to occur at values of t which differ by odd multiples of or. The variation of Y2 along a typical solution is oscillatory until it reaches a critical value -+~ of y l / 3 - yl. The amplitude of these oscillations is of the order of a while the change in extreme values from one cycle to the next is of order ~-I. Thus, the jumps tend to occur near extreme values of the oscillation and successive jumps are spaced by a time interval which is approximately an odd multiple of or. Therefore the periods of the solutions one observes are odd multiples of 2~r because the time between jumps is half the period of the limit cycle. For some parameter ranges, one sees stable periodic

230

J. Guckenheimed Symbolic dynamics and relaxation oscillations

orbits of period ( 2 n - 1) and (2n + 1) occuring simultaneously. It was this phenomenon which originally attracted the attention of Cartwright and Littlewood. These stable periodic orbits have been described asymptotically as ~-*0 by Grasman, Veling and Willems [9]. 3. Maps of the annulus and circle If one considers an annulus D of moderate size which contains the limit cycle of the Van der Pol equation (3), then the Poincar6 map F of (5) maps this annulus into itself in such a way that the image is very thin. Fig. 2 gives a schematic sketch of D and F(D). We would like to replace F by a map which has rank 1 and is more easily studied. If there were an invariant one-dimensional foliation across D whose leaves are contracted, then this could be done rigorously. Such a foliation would be represented by a set of "polar coordinates" in D with the property that radial curves would be mapped to radial curves. The folding of the annulus along its circumference precludes the global existence of such a foliation since, along the short branches of the fold, the orientation of radial and angular directions is reversed. Still there might be an invariant foliation around enough of the annulus to include all of the nonwandering set of F in D apart from the attracting periodic orbits. This is proved by Levi in the example he studies. In any case, we shall assume that the image F(D) of D is thin enough that F can be approximated by a map with a one-dimensional image. The actual thickness of the image annulus F(D) is of the order of exp(-2w~) around most of its circumference.

Define G:D->D to be a smooth rank 1 map which is close to F (in the C ~ sense). The map G is assumed to have a number of additional properties. Since this derivative of G has constant rank l, G-I(p) is a nonsingular curve for each p ~ D. We assume that each of these curves is connected and joins the two boundary components of D. Assume further that there is a curve S, homeomorphic to S z, which intersects each of the sets G-~(p) transversely in a single point. Define the map or:D-> S which maps each of the sets G-t(p) to its intersection with S. Then g -- ¢rGIs is a map of S into itself whose dynamics determine those of G in a straight-forward way. It is the map g which we study in some detail. After analyzing these maps of the circle, we discuss their relationship with the map F and the Van der Pol equation. For concreteness, we begin by analyzing a specific map g which seems to have qualitative properties corresponding to values of the parameters for which the forced Van der Pol equation has stable limit cycles of periods 6¢r and 10~r. The graph of the map g is shown in Fig 3. We assume that it has the following properties. There are four critical points of g, all of which are assumed to lie in stable periodic

l Fig. 2. The annulus D and its image under the Poincar6 map of the forced Van der Pol equation.

b,

Fig. 3. The graph of the one-dimensional approximation f to the Poincar6 map of the forced Van d©r Pol equation with its stable periodic orbits.

J. Guckenheimer[Symbolic dynamics and relaxation oscillations orbits. (That critical points are periodic is a nongeneric property, but it simplifies our description without changing the nonwandering set of g.) There are two stable periodic orbits of periods 5 and 3, whose points we label a~, a2, a3, a4, as and b~, b2, b3 respectively. We assume that the ordering of these points on the circle is albla2b2a3b3a4b4asal with g(ai) = ai+j for 1 -< i -< 4, g(as) = al, g(b0 = b2, g(b2) = 133and g(b3) = b~. The critical points are b2, a3, b3 and as. These assumptions determine the nonwandering set of g through the use of symbolic dynamics [8]. Symbolic dynamics keeps track of which images of intervals intersect which other intervals as we iterate g. Partition the circle into eight intervals: I~ = [a~, bd, I2 = [b~, a2], I3 = [a2, b2], I4 = [b2, a3], I5 = [a3, a4], I6 = [a4, b3], I7 = [b3, as] and Is = [a~, ad. Our assumptions about g imply that the images of I~. . . . . Is are given by g(I~) = I3, g(I2) = I4, g(I3) = I~ U I6, g(I4) = I6, g(Is) = I6 O I7, g(I6) = Is L/I~, g(I7) = I~, and g(Is)=I~ L/ I2. The images of I4 and I7 have reversed orientation. This information about images can be embodied in the transition matrix: 0

A=

0

1

0

0

0

0

00 O 0 0 0 0 0 1

O 0 0 0 0 0 0

10 0 0 0 0 0 0

1 0 0 0 0 0

O 1 I 1 0 0 0

O 0 0 0 0 1 0 0 0

a# =

231

0 if Ij~ g(Ii) if I t C g(Ii)"

The transition matrix A allows us to calculate the image gk(I~) of I~ for the kth iterate of g. Indeed, the transition matrix of gk is A k. This is seen easily from the formula for Ak:

(At)ii-~

E i! . . . . .

aioil aiti2 ai2i3... Clit_lit Ik_ I = I

with i0 = i and ik = j. Each term is zero unless all factors are 1, and then Iit+~C g(I 0 for 0 -< ! < k. This implies Ii C g~(I;). T h e r e f o r e , (Ak)ij is the number of times that gk(Ii) covers the interval I t. The number of unstable fixed points of g~ is given by the total number of allowable k-step transitions, each of which begins where it started. This is E(Ak)u, the trace of A k. Since gt is contracting at a fixed end point, each time gk(ID covers I~, there is at least one fixed point in the interior of Ii. Our assumption that g has only two stable periodic orbits implies that there will be exactly one fixed point in the interior of Ii. Table 1 gives the number of unstable fixed points of gk for

k_<15. The (one-sided) subshift of finite type ~ with transition matrix A is a topological space (again denoted E) together with a map o-[3,8]. The space consists of all sequences i={ii[I~ii--8}~=0 with the property that a~it+~= I for all I. A distance function can be defined by

0

:¢

d(i, j) = ~ 812-Itf

with which is characterized by Table I Number of unstable fixed points of gk for k_< 15, and the number of unstable periodic orbits in the subshift with transition matrix A 2

k Number of unstable fixed points gt

1 0

2 0

3 3

4 16

5 5

6 3

7 28

8 64

9 39

10 45

!1 176

12 307

13 260

14 392

15 1028

Number of unstable periodic orbit% period k

0

0

1

4

1

0

4

6

4

4

16

24

20

26

68

232

J. Guckenheimer/ Symbolic dynamics and relaxation oscillations 0

81 =

if it = jt if it # j~"

This metric is complete on X. The map tr:X--*X is defined by t r ( i ) = j with Jt = i~+~. It shifts the indices on a sequence, omitting the first term. The map tr is topologically equivalent to the map g restricted to the unstable part of its nonwandering set. This means that there is a homeomorphism h from X to the unstable part of the nonwandering set of g such that ho- = gh. Put somewhat differently, there is a unique nonattractive orbit in the nonwandering set of g for each sequence i with Iit+, C g(Ii), l -> 0. Thus, the subshift with transition matrix A allows us to characterize the nonwandering set of the map g. Note that A 9 has only positive entries. This implies that tr has a dense orbit in X. H e n c e the unstable part of the nonwandering set of g cannot be d e c o m p o s e d into closed invariant pieces. It is a b a s i c set [3] for g. As a hyperbolic basic set for a dynamical system, it is endowed with stochastic-like features such as sensitive dependence on initial conditions. It is topologically mixing and the set of periodic orbits is dense. The results described above for g are also valid for G. Its nonwandering set consists of two stable periodic orbits together with a set on which G is topologically equivalent to the one-sided subshift with transition matrix A. Furthermore, if F is sufficiently close to G, then its nonwandering set will consist of two stable periodic orbits and a basic set on which it is topologically equivalent to the two-sided shift with transition matrix A. The two-sided shift with transition matrix A is defined just as before, except that sequences are indexed by the set of all integers instead of the nonnegative integers. In particular, the numbers of periodic orbits are the same. This gives us a good description of the nonwandering set of F, which we assume looks qualitatively like a Poincar6 map for the forced Van der Pol equation. It is easy to modify the preceding description so that it applies when the parameters of the

forced Van der Pol equation have values such that there are stable periodic orbits of periods (2n + 1)2rr and (2n - 1)27r. For the corresponding map g:SI--->S~, the points of the stable periodic orbits can be labelled al . . . . . a2,+l and bl . . . . . b2,-i so that they have the following ordering on the circle: at < bt < a2 < bE < ... < a. < bn < an+l < an+2 < bn+t < an+3 < bn+2 < • • • < azn
There are parameter values of eq. (5) so that the nonwandering set of its Poincar6 map conists of two stable periodic orbits of periods ( 2 n - 1) and (2n + l), one unstable fixed point, and a saddle-like basic set topologically equivalent to the (two-sided) subshift of finite type with transition matrix A,. The matrix A, is a 4n x 4n matrix of O's and l's with a o = 1 if and only if j = i + 2 ( m o d n ) or ( i , j ) = ( 2 n - 1 , 2n + 2), (2n + 1, 2n + 2), (4n - 2, I) or (4n, 1). This conjecture is consistent with the conjecture that the forced Van der Pol equation is structurally stable [3] for these parameter values.

4. Bifurcations Bifurcations are changes in the qualitative features of the solutions of a system of

J. Guckenheimer/Symbolic dynamics and relaxation oscillations differential equations which depends upon parameter(s). Bifurcations occur for the forced Van der Pol equation because the periods of the stable periodic orbits for its Poincar6 map change. The number of periodic orbits also changes. Sometimes there is one stable periodic orbit and sometimes there are two of the sort described above. General theorems about dynamical systems preclude the possibility that the homoclinic behavior [3] described in the previous section could arise from a system without it unless there is an infinite succession of intermediate behaviors. One would like to analyze these changes, but there are some aspects we cannot yet describe. It is possible to obtain some information about the bifurcations of one-dimensional maps of the circle which approximate t h e Poincar6 map of the Van der Pol equation in the way discussed in the previous section. For reasons discussed below, there are features of this analysis which are not likely to agree with the actual behavior of the forced Van der Pol equation. The analysis of maps of the circle relies very heavily upon examining the orbits of their critical points. These bound the intervals on which the map is monotonic, and from them one can infer a lot of information about other orbits [10]. Since the orbits of the critical points of a map of the circle can be varied independently from one another, at least as many parameters as critical points are needed to find a universal family of such maps. The maps which correspond to Poincar6 maps of the forced Van der Pol equation enjoy certain symmetry properties which make it unlikely that they represent a versal family [11]. These symmetry properties are the result of the following: if H: R2-*R 2 is the time lr map for (4) and F: R2--*R2 is the time 2,r map of (4), or equivalently the Poincar6 map of (5), then F ( p ) = - H ( - H ( p ) ) for all p E R E. Levi's analysis use this fact. The best that one can hope for is that the bifurcations of Poincar6 maps are generic for maps which have this particular form. We cannot expect that generi-

233

city within this restricted class of symmetric maps will correspond entirely to genericity within the class of all maps of the circle. It is a common feature of bifurcation problems in physics and engineering that symmetry properties force bifurcation diagrams to occur which would not be expected were the symmetry not present. A simple example is the buckling of a beam under increasing loads. One indication of the restrictions forced by the symmetry of the Van der Pol equation is that the number of critical points of the circle maps we are led to study is divisible by 4. We do expect here that the symmetry will affect primarily the order of bifurcations in a family but not the nature of the bifurcations of individual orbits. Ignoring symmetry, we now describe these. Consider now a smooth map f~, : S l ~ S 1 which depends upon a single parameter /~. There are two ways that a periodic orbit of f~, can change its qualitative properties as a function of /~. These are through bifurcations which are called saddle-nodes and flips [12, 13]. At a saddle-node p of period n, there is a curve Y of periodic orbits of period n in S I × R passing through (P, ~0) and having a nondegenerate tangency with the line ~ = p~0. The conditions characterizing the saddle-node are dfT~0(p)/dx = 1 plus nondegeneracy conditions. If we delete the point p from % then one of its two components represents a stable periodic orbit, and the other is unstable. See fig. 4. A flip p of period n at parameter value /~0 is characterized by f~0(P) d/dx = - 1 together with nondegeneracy conditions. The periodic orbit containing p changes its stability at t~, and a periodic orbit 2n and the

P[ ~ Stable Unstab~ Orbit

Orbit

/~o

Fig. 4. The bifurcation diagram of a saddle-node bifurcation.

J. Guckenheimer/Symbolic dynamics and relaxation oscillations

234

P ~

P0tiod 2n

Fig. 5. The bifurcation diagram of the flip bifurcation.

opposite stability branches from the orbit of p. See fig. 5. If a one-parameter family selected from the forced Van der Pol equation satisfies the appropriate non degeneracy conditions at bifurcations of periodic orbits, then saddle-nodes and flips are the only kinds of bifurcations of periodic orbits. One would like to describe the sequence of periods of bifurcating periodic orbits in a one parameter family within the forced Van der Pol equation. This is not currently possible for two reasons. The first is that there are distinct differences between the bifurcation behavior of maps of the circle and diffeomorphisms of an annulus into itself which approximate them [13, 14]. The diffeomorphisms of the annulus are expected to have many more stable periodic orbits, indeed an infinite number for many parameter values. The maps of the circle seem to have relatively few stable periodic orbits for any particular parameter value. These differences are probably reflected in the differences in the order of bifurcations of various periodic orbits. The second difficulty in describing the bifurcation sequences for the forced Van der Pol arises from the symmetry discussed above. The orbits of the critical points of a map of the circle determine much of the qualitative information about all the orbits, the monotone equivalence classes defined by Milnor and Thurston [10]. Thus, to compute information about bifurcation sequences, one needs to compute the way the relative order of the points in the orbits of the critical points changes. There is no theory which

currently allows one to do this for the specific one-parameter families of circle maps which correspond to the forced Van der Pol equation. Nonetheless, there is some information we do have at hand. Simulations and asymptotic analysis indicate that there are one-parameter families of forced Van der Pol equations in which parameter intervals yielding stable limit cycles periods ( 2 n - 1)lr and (2n + 1)Tr lie between intervals yielding just a stable limit cycle of period (2n - 1)~r and just a stable limit cycle of period (2n + 1)Tr [9]. Between these intervals must be parameter values in which the saddlelike periodic orbits for the case described in section 3 are stable. In passing from an interval with the two stable limit cycles and a nonwandering set homeomorphic to the subshift with transition matrix An to an interval with just one stable limit cy.cle (and one unstable limit cycle), almost all of the unstable periodic orbits of the subshift must disappear somehow. Based upon the analysis of generic bifurcations for maps of the circle, this should happen through flip and saddle-node bifurcations. In particular, any periodic orbit p of period n for a circle map for which dfn(p)/dx < 0 must become a stable periodic orbit over some parameter interval before it disappears. Each periodic orbit either disappears in a flip after becoming stable, or is paired with another periodic orbit of the same period as it disappears at a saddle-node. One of the pair was stable just before the bifurcation. In this sense, over half of the periodic orbits become stable before they do disappear. Thus one can obtain a minimal estimate for the number of periodic orbits of various periods which must each be stable over some parameter interval. Table I gives the number of unstable periodic orbits in the subshift with transition matrix A2. We summarize the results of this section in the following proposition.

Proposition: Assume the conjecture stated at the end of

J. Guckenheimer/ Symbolic dynamics and relaxation oscillations

section 3. Consider a o n e - p a r a m e t e r family of forced Van der Pol equations such that the conjecture applies to the left endpoint of the p a r a m e t e r interval and such that the right endpoint of the p a r a m e t e r interval yields a s y s t e m of equations without homoclinic orbits. Then there are an infinite n u m b e r of distinct periodic orbits of the family, each of which is stable in some p a r a m e t e r interval. We consider two periodic orbits distinct in this proposition if they are not part of a one p a r a m e t e r family of periodic orbits.

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[7] F. Takens, Constrained equations, a study of implicit differential equations and their discontinuous solutions, Mathematisch lnstituut Rijksuniversiteit Groningen. Report ZW-75-03 (1975). [8] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics 470 (Springer-Verlag, Heidelberg, Berlin, G6ttingen, 1975). [9l J. Grasman, E. J. M. Veling and G. M. Willems, Relaxation oscillations governed by a Van der Pol equation with periodic forcing term, Siam J. Appl. Math, 31 (1976) 667-676. [I0] J. Milnor and W. Thurston, On iterated maps of the interval I and II, mimeographed, Princeton (1977). [II] V. I. Arnold, Lectures on Bifurcations in Versal families, Russ. Math. Surveys 27 0972) 54--123. [12] J. Guckenheimer, Bifurcation of quadratic functions, Annals N.Y. Acad. Sci. 316 (1979) 78-85. [13] J. Guckenheimer, On the bifurcation of maps of the interval, Inventiones Mathematicae 39 0977) 165-178. [14] S. E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. I.H.E.S. 50 (1979) I01-152. [15] C. Hayashi, Nonlinear Oscillations in Physical Systems (McGraw-Hill, N e w York, 1964). [16] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra (Academic Press, N e w York, London, 1974). [17] P. J. Holmes and D. A. Rand Bifurcations of the forced Van der Pol oscillator,Q. Appl. Math. (1978) 495-509. [18] J. E. Littlewood, On non-linear differentialequations of the second order: Ill.The equation ~;- k(l - y2)y + y = bl~k cos(/~t +a) for large k, and its generalizations,

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