On the numerical computation of orbits of dynamical systems: The higher dimensional case

Abstract

Chaotic dynamical systems exhibit sensitive dependence to initial conditions. So, because of round-off error, a computed orbit diverges at an exponential rate from the true orbit with the same initial condition. Nevertheless, we are able to exploit the hyperbolicity of the dynamical system to prove a “finite time” shadowing lemma, from which we deduce that a true orbit shadows the computed orbit for a large number of iterates. An algorithm for the computation of the shadowing error is given and, furthermore, the effect of round-off error on these computations is analyzed in detail. The algorithm is applied to the Hénon map. This paper is a continuation of an earlier paper on one-dimensional maps.