PHYICAL REVIEW E VOLUME 59, NUMBER 5 MAY 999 Variational principle for limit cycles of the Rayleigh van er Pol equation R. D. Benguria an M. C. Depassier Faculta e Física, Pontificia Universia Católica e Chile, Casilla 306, antiago 22, Chile Receive 2 October 998 We show that the amplitue of the limit cycle of Rayleigh s equation can be obtaine from a variational principle. We use this principle to reobtain the asymptotic values for the perio an amplitue of the Rayleigh an van er Pol equations. Limit cycles of general Liénar systems can also be erive from a variational principle. 063-65X9902705- PAC numbers: 45.05.x, 47.20.Ky, 02.30.Hq, 02.30.W I. INTRODUCTION Limit cycles or self-excite oscillations appear in a wie class of problems in electronics, biology, astrophysics, to name a few. The first an most stuie examples of equations which exhibit limit cycles are the Rayleigh equation ÿ 3 ẏ ẏ 3 y0, which was introuce to show the appearance of sustaine vibrations in acoustics, an the van er Pol equation 2 ẍx 2 ẋx0, erive for a certain electrical circuit. These systems exhibit a single perioic solution of perio an amplitue epening on. It was soon notice that these two equations are closely relate. Inee, taking the erivative of Rayleigh s equation an calling ẏx, we obtain van er Pol s equation for x. Therefore, the limit cycles for both equations have the same perio an the amplitue x max of the limit cycle of the van er Pol equation is the maximum value of ẏ. More general systems of the form ẍ f xẋx0 were stuie by Liénar 3, who gave conitions on f for the existence of a unique limit cycle. The conitions for the existence an uniqueness of limit cycles for systems of the form ẍ f (x)ẋg(x)0, or generalize Liénar systems, have also been establishe 4. For functions f (x) which o not satisfy Liénar s conitions, the existence, number, an location of limit cycles is an open problem. For small epartures of the Hamiltonian case ẍx0 these questions can be answere by Melnikov s theory 5,6. A new, nonperturbative, approach to solving this problem has been propose recently 7,8. In this paper we will be concerne with the Rayleigh van er Pol equation. There is a large amount of work on this equation, both rigorous an numerical. The perio an amplitue have been etermine mainly by perturbation theory for the small an large limits. Accurate methos 9 have been evelope to obtain regular series for intermeiate values of. The leaing orer results in the limit 0 are the perio T2 an the amplitue of the limit cycle A2. In the limit of the perio is, in leaing orer, T3 2 3 2 ln(2) an the amplitue for the van er Pol equation is again A2. Higher orer corrections for both the small an large 0 limits have been obtaine. Detaile results an aitional references on these an other points can be foun in 0 3, among other books. The purpose of this work is to present an alternative approach by noticing that a variational principle can be formulate for Rayleigh s equation an also for general Liénar equations 3. We erive this variational principle an show how to recover from it the perio an amplitue in the limiting regions of the parameter. By using this variational principle together with appropriate trial functions one can obtain the amplitue an the perio of the limit cycle for arbitrary values of. II. VARIATIONAL PRINCIPLE We shall work on Rayleigh s equation an erive the amplitue an perio for Rayleigh s equation. Given the connection between the two equations mentione in the Introuction, one can also obtain the amplitue of the limit cycle for the van er Pol equation. We shall use a metho employe previously to obtain variational principles for other nonlinear eigenvalue problems 5 7 for a review see 4. Due to the symmetry of Rayleigh s equation, the limit cycle extens between a minimum x min A an a maximum x max A. Moreover, in phase space, if the point (ẏ,y) belongs to the limit cycle, then the point (ẏ,y) also belongs to it. Therefore we may consier the positive upper half ẏ0 of the phase plane, where half a perio will elapse when going from the points (ẏ0,x min )to(ẏ0,x max ). Then the equation for the limit cycle in phase space can be written as the nonlinear eigenvalue problem, p p y p3 3 p y0 with pa0 an p0, where we have calle p(y)ẏ(y). The nonlinear eigenvalue is the amplitue A which appears in the bounary conitions. It is convenient to efine a new variable uy/a in terms of which the equation for p is given by with p()0 an p0. p p u p3 3 p Ru0, 4 063-65X/99/595/48895/$5.00 PRE 59 4889 999 The American Physical ociety
4890 R. D. BENGURIA AND M. C. DEPAIER PRE 59 Two parameters appear naturally in this problem, RA/ an A. In terms of the function p(u), the perio of the limit cycle is given by u T2A p. Having written the problem in phase space, we may aapt a metho to obtain variational principles evelope for other problems. The metho is simple enough that the full etail can be given here. Let g(u) be a positive continuous, with continuous erivative but otherwise arbitrary function. Multiplying Eq. 4 by g(u) an integrating in u between 0 an, we obtain, after integrating by parts an making use of the bounary conitions on p, R uguu gu,pu u, 5 ĝuexp u vtt exp u p p t, which is a nonsingular positive function. Notice that the minimizing g is uniquely efine moulo a multiplicative constant. Our main result is then Rmin g 2 8 guf vu u, 9 uguu where the minimization is over all positive functions g(u) which are continuous an have continuous erivative in 0,. Alternatively, one may express g an v in terms of pˆ, an consier the more intuitive function pˆ as the trial function. In terms of pˆ the above results rea where gu,pu gp 2 p2 g 3 gp3 Rmin pˆ 6 u e [pˆ /pˆ ]t pˆ 33pˆu u ue [pˆ/pˆ ]t u. gp 2 p2 v 3 p3. Here we have efine v(u)g(u)/ g(u). At each value of u we may regar as a function of p; since g is positive, has a single maximum at a positive value pˆ given by pˆ 2 v v 2 4. 6 To avoi confusion between the true solution of the equation p(u) an the trial functions pˆ (u), we will continue with the use of g(u) an also v), the original trial function. Notice also that we shall be intereste in the maximum value of v(u), which gives the maximum of pˆ, which in turn is the amplitue of the limit cycle of the van er Pol equation. For each trial function g we obtain a value of pˆ an the corresponing approximate perio is given by Therefore (g,p)(g,pˆ ). Replacing this in the ientity Eq. 5, we obtain u T4A vv 2 4. 0 R 2 guf vu u, 7 uguu where F(v)2v(4v 2 )v4v 2. Therefore, given a trial function g(u) an a value of, through Eq. 7 we obtain a boun on R; the original parameters A an are given in terms of an R by AR /2 an R /2. The equality in Eq. 7 will hol whenever the trial function g(u) is such that pˆ is exactly p, the true solution for the limit cycle. From Eq. 6 we see that this will occur for gĝ satisfying from which it follows that g g vp p, If we wish to obtain approximate values for the amplitue an perio, we may choose ifferent trial functions an optimize the boun numerically. Instea, we shall stuy analytically the large an small limits to reobtain the known results in these limits. III. AYMPTOTIC REULT In this section we reobtain the known results in the limits of small an large. Notice that the variational principle 9 is of the form Rmin Ig,g/Jg,g with I g(u)f v(u) u/2 an J ug(u)u. In orer to minimize the above quotient, it is better to introuce a Lagrange multiplier, an extremize instea the quantity Ig,gJg,g Lw,w,uu, where is a Lagrange multiplier an where, rather than using variables g an g, we have introuce the variables u wu vss an wuvu.
PRE 59 VARIATIONAL PRINCIPLE FOR LIMIT CYCLE OF... 489 The Euler-Lagrange equation tw L L w 0, for this problem, is FvvvFvFvu0 or, more explicitly, 3 u v2 vv 2 42v 3 2v 2 2v 2 4u0. 2 This equation for v(u) written in terms of p is, as expecte, the original Eq. 4 in phase space with R/2. A. mall limit Let us consier first the limit of small. In orer to o this, we first obtain the approximate solution v(u) of Eq. 2 for small, which will epen on. For small, Eq. 2 for v becomes 3 u v2 vv 2 4u0 which can be integrate to obtain v 2 vv 2 4 6 u2 C, where C is a constant. Noticing that the above equation can be written as 2pˆ 22u 2 /6C an that the bounary conitions on pˆ are pˆ ()0, evaluating at u or ) we obtain the value for the constant C2/6. Defining /2 we have that for small an finally v 2 vv 2 42u 2 20, vuu 2 u 2. Thus, for small in leaing orer we have I 2 exp u vtt F vu u 2 F vu u an J exp u vtt uu u u vtt. Using the expression 3 for v(u) we obtain 3 I 3 4 4, so that from Eq. 7 FIG.. Graph of the function u(v). R 3 J 4 34, 4 34. 4 The right sie of Eq. 4 is minimize at 4, leaing to the approximation R4/ for small values of. Therefore, AR /2 2, which is the correct result in the small limit. The perio, evaluate from Eq. 0, with v(u) given by Eq. 3, ist 2. B. Large limit Let us examine now the limit of large. For large, we obtain from Eq. 2 u2v 3 22v 2 v 2 4. 5 To fix the value of we coul procee as above. However, the same results can be obtaine in a more intuitive manner which we evelop here. The function u(v) is plotte in Fig.. Its maximum occurs at v0 an its value is 8. ince the maximum value of u is recall that with our choice of variables u ranges from towe must have 8. With this choice for, the maximum value for v which occurs at u is 3/2. We have then our first conclusion, namely that the maximum of v at large is 3/2 an the corresponing 2 maximum for p is (v max v max 4)/22. This maximum for p of the Rayleigh equation correspons to the limiting amplitue of the van er Pol equation. The amplitue for the Rayleigh equation can be rea from the variational principle. For large values of the function ĝ(u) given by Eq. 8 is highly peake at the maximum of u v(t)t, i.e., at u since in this case v0). Therefore, for large, from Eq. 9 we obtain R 2 Fv 2 F0 2 3,
4892 R. D. BENGURIA AND M. C. DEPAIER PRE 59 FIG. 2. Numerical results for the amplitue of the Rayleigh equation as a function of. The asymptotic behavior A 2/3 can be notice here. an the asymptotic behavior for the amplitue is A 2 3. 6 In Fig. 2 we show the results of the numerical integration of Rayleigh s equation where we verify this asymptotic behavior. The perio can be calculate from Eq. 0. In fact, by changing the inepenent variable from u to v, an by using Eq. 5, we obtain 0 T4A 3/2 vv 2 4 u v v4a 9 8 3 4 arcsinh3/4. Using Eq. 6 in Eq. 7 we finally obtain T32 arcsinh3/432 ln2 7 which is the correct leaing orer for the perio for large. The two functions v(u) which we have obtaine as limiting values may also be use as convenient trial functions to obtain numerical bouns on R. In Fig. 3 we show the boun obtaine using as a trial function the small approximation for v(u). With this, the simplest trial function, we obtain at 4 an error of less than.7%. Improvements of the maximum value of p, or equivalently on the amplitue of the van er Pol equation, can be obtaine only by improving the trial function. IV. CONCLUION We have shown that the equation satisfie by the limit cycle of Rayleigh s equation erives from the variational FIG. 3. Bouns on the amplitue of the limit cycle of the Rayleigh equation obtaine with a simple trial function. principle Eq. 9 which enables one to obtain the amplitue, in principle, with any esire accuracy. Once sufficient accuracy is obtaine, the perio an amplitue for the van er Pol equation follow from it. We have reobtaine the limiting values for small an large eviations from the linear problem analytically. Our purpose in this work has been to show the existence of the variational principle, an the metho of erivation. It is clear that a variational principle can also be formulate for Liénar systems of the form given in Eq. 3. The conition for the existence of a unique limit cycle are that f is even, F(x) 0 x f (s)s satisfies F0 for 0x x 0, F0 for xx 0, F is monotone increasing for xx 0, an F as x. These properties guarantee that the limit cycle of the associate equation ÿf(ẏ)y0 obtaine calling xẏ an integrating can be erive from a variational principle as well. everal questions remain to be answere. First, for other types of functions f the equation may possess more than one limit cycle. It is irect to show that the phase space equation for the limit cycles of ÿf(ẏ)y0, for any polynomial F, can be erive from a variational principle. This means that all the limit cycles correspon to an extremum of a certain functional. It is an open question whether it is possible to count or estimate the positions of such extrema. Finally, the possibility of extening these results to generalize Liénar systems of the form ẍ f (x)ẋg(x)0 is another problem that remains open. ACKNOWLEDGMENT This work was supporte by Fonecyt Project Nos. 960450 an 990429 Chile. The work of R.B. has been supporte by a Cátera Presiencial en Ciencias Chile an by the John imon Guggenheim Memorial Founation. Lor Rayleigh, Philos. Mag. 5, 229 883. 2 B. van er Pol, Philos. Mag. er. 7 2, 978 926. 3 A. Liénar, Rev. Gen. Electr., 23, 90 928. 4 N. Levinson an O. K. mith, Duke Math. J. 9, 384 942. 5 L. Perko, Differential Equations an Dynamical ystems pringer-verlag, New York, 99. 6 M. A. F. anjuán, Phys. Rev. E 57, 340 998. 7 H. Giacomini an. Neukirch, Phys. Rev. E 56, 3809 997. 8 H. Giacomini an. Neukirch, Phys. Rev. E 57, 6573 998. 9 C. M. Anersen an J. F. Geer, IAM oc. In. Appl. Math.
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