Numerical Shadowing Using Componentwise Bounds and a Sharper Fixed Point Result

Abstract

Shadowing provides a means of obtaining global error bounds for approximate solutions of nonlinear differential equations with interesting dynamics, in particular, for initial value problems with positive Lyapunov exponents. Shadowing breaks down in the presence of zero Lyapunov exponents, although some results such as shadowing with rescaling of time have been obtained. Using a reformulation of the original differential equations and an improved fixed point result we take advantage of componentwise local error bounds to use relatively smaller error tolerances in nonhyperbolic and contractive directions (i.e., in directions corresponding to zero and negative Lyapunov exponents). The result is a decrease in the shadowing global error.