I,,,. J Non-Lmew Mwhonru. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Vol. 20. NO 4. pp. 325-338. 19X5 0020-7462/E 1603.00 + HI Pnmed in Great Bntam Pergamon Press Ltd zyxwvutsrqp DYNAMICS OF A SYSTEM EXHIBITING THE GLOBAL BIFURCATION OF A LIMIT CYCLE AT INFINITY W. L. KEITH and R. H. RAND Department of Theoretical and Applied Mechanics, Thurston Hall, Cornell University, Ithaca, NY 14853, U.S.A. (Received 26 July 1984; received .for publication 7 March 1985) Abstract-This paper concerns the dynamics of a class of non-linear oscillators of the form: x” + x - .sx’(l - ax’ - bx’2) = 0. The non-linear term contains two parameters a and b which may be varied to give the Rayleigh and Van der Pal differential equations as special cases. The existence and approximation of limit cycles in this system are investigated using the Poincare-Bendixson theorem and the Lindstedt perturbation method. Analysis of the system at infinity is used to study the global bifurcation through which the limit cycle is created from four saddle-saddle connections between equilibrium points at infinity. Center manifold theory is used to determine the stability of the equilibrium points at infinity. Numerical integration is used to verify the analytical results. It is shown that an arbitrarily small perturbation to the damping term of the Rayleigh equation results in points close to the stable limit cycle escaping to infinity. INTRODUCTION We consider the non-linear differential equation: X” + X - Ex’(1 - zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ax2 - bx”) = 0 or i zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON x'=y y’ = -x -t &Y(l- ,x2 - by2) (1) where E > 0 (but not necessarily small), and where the natural frequency of the E = 0 equation has been taken as unity. When a = 0 and b = l/3 we have Rayleigh’s equation ([l xU + x - &X’(l - x’2/3) = 0 and when a = 1 and b = 0 we have Van der Pol’s equation x” -t x - &X’(l - x2) = 0. I, p. 81): zyxwvutsrqponmlkjihgfedcb PI: (2) (3) Note that (3) may be obtained from (2) by differentiating (2) with respect to time t and taking the variable x in (3) to be x’ in (2). Both (2) and (3) are known to exhibit stable limit cycles for all values of E > 0 ( [3], p. 3). Equatibn (1) possesses exactly one equilibrium point in the finite phase plane located at the origin for all a, b and E.In view of the assumption that E > 0, it is an unstable node (E > 2) or unstable focus (0 < E < 2). Note that since equation (1) is invariant under replacement of (x, y) by (- x, -y), the flow (1) is point-symmetric about the origin (i.e. it is invariant under a rotation of n radians). It turns out that equation (1) exhibits a limit cycle in the x, y phase plane for some values of the parameters. We shall be interested in determining regions in the a,b parameter plane for which a limit cycle exists. We will investigate this question by applying a variety of techniques, including examining the system at infinity (i.e. for large x, y). Information gained from the behavior of the system at infinity will be shown to explain the global bifurcations which lead to the creation of the limit cycle. The symbolic manipulation system MACSYMA [4] has been used extensively in this work. 3’5