Invariant Curves for Numerical Methods

The problem of finding periodic orbits of dynamical systems numerically is considered. It is shown that if a convergent, strongly stable, multi-step method is employed then under some suitable conditions, there exist invariant curves. The result also shows that the rates of convergence toward the invariant curves are roughly the same for different methods and different step sizes. Introduction. The problem of locating the periodic orbits of dynamical systems numerically has been considered by many authors (see, e.g., [3,4,8]). In many instances, especially when the periodic orbits of an one-parameter family of dynamical systems are to be followed, the existence of these orbits is more important than their exact locations. This paper arises from the author's dissertation [2], Its purpose is to show that when a strongly stable method is employed, then under suitable conditions, there exist invariant curves. This result is stronger than the one obtained in [1] in that the method could be multi-step and the periodic orbit need not be stable. The result also shows that reducing the step size or employing more sophisticated methods do not necessarily improve the rate of convergence. The problem. Let woo, (1) x e R", f(x) is as smooth as needed, be a dynamical system which possesses a periodic orbit F with characteristic multipliers ju,,, jun satisfying fil = 1, |juy| < 1 for j = 2,...,/ and |juy| >1 for j = I + 1 Suppose that (1) is approximated by a convergent, k-step method of the form k k *„, + ! = L ajXm+l_j+ h £ bjf(xm+l_j) + h2F(h,xm,...,xm+1_k) (2) 7=1 7-1 * Received September 21, 1984. ©1985 Brown University