Obstructions to Shadowing When a Lyapunov Exponent Fluctuates about Zero
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TL;DR
A dynamical mechanism for the unshadowability of chaotic dynamical systems that lie close to computer-generated trajectories is explained through a theoretical model and identified in simulations of a typical physical system.
Abstract
We study the existence or nonexistence of true trajectories of chaotic dynamical systems that lie close to computer-generated trajectories. The nonexistence of such shadowing trajectories is caused by finite-time Lyapunov exponents of the system fluctuating about zero. A dynamical mechanism of the unshadowability is explained through a theoretical model and identified in simulations of a typical physical system. The problem of fluctuating Lyapunov exponents is expected to be common in simulations of higher-dimensional systems.