A Fast Shadowing Algorithm for High Dimensional ODE Systems

Wayne Hayes and Kenneth R. Jackson


A true trajectory of a chaotic dynamical system lying near an approximate trajectory is called a shadow. Finding shadows of numerical trajectories of ODE systems is very compute intensive, and until recently it has been infeasible to study shadows of higher-dimensional systems. We study the shadowing algorithm introduced by Grebogi, Hammel, Yorke and Sauer in 1990 and extended to arbitrary Hamiltonian systems by Quinlan and Tremaine in 1992, and introduce several major optimizations resulting in speedups of over 100. This algorithm is used to shadow gravitational N-body systems with 150 phase space dimensions.

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