A procedure for finding numerical trajectories on chaotic saddles

Abstract

Examples are common in dynamical systems in which there are regions containing chaotic sets that are not attractors. If almost every trajectory eventually leaves some region, but the region contains a chaotic set, then typical trajectories will behave chaotically for a while and then will leave the region, and so we will observe chaotic transients. The main objective that will be addressed is the “Dynamic Restraint Problem”: Given a region that contains a chaotic set but does not contain an attractor, find a chaotic trajectory numerically that remains in the regionfor an arbituarily long period of time. Systems with horseshoes have such regions as do systems with fractal basin boundaries, as does the Hénon map for suitably chosen parameters. We present a numerical technique for finding trajectories which will stay in such chaotic sets for arbitrarily long periods of time, and it leads to a “saddle straddle trajectory”. The method is called the “PIM triple procedure” since it is based on so called PIM triples. A PIM (Proper Interior Maximum) triple is three points ( a, c, b) in a straight line segment such that the interior point c (i.e. c is between a and b) has the maximum escape time, that is, its escape time from the region is greater than the escape time of both a and b. “Proper” means the segment from a to b is smaller than a previously obtained segment. We show rigorously that the PIM triple procedure works in ideal situations. We find it works well even in less than ideal cases. This procedure can also be used for the computation of Lyapunov exponents. Furthermore, the “accessible PIM triple procedure” (a refined PIM triple procedure for finding accessible trajectories on the chaotic saddle) will also be discussed.