A FAST SHADOWING ALGORITHM FOR HIGH-DIMENSIONAL ODE SYSTEMS.

Numerical solutions to chaotic dynamical systems are suspect because the exponential divergence of nearby trajectories causes numerical errors to be exponentially magnified with time. To bolster confidence in the reliability of such numerical solutions, we turn to the study of shadowing: a shadow is an exact trajectory of a chaotic dynamical system that stays close to a numerical trajectory for a long time. Finding shadows of numerical trajectories is very computationally intensive, and until recently it has been infeasible to study shadows of higher-dimensional systems. This paper introduces several optimizations to previous algorithms, which collectively achieve an average speedup of almost two orders of magnitude with no measurable loss in effectiveness. We test the algorithm on systems with up to 180 dimensions. Its application to large gravitational N-body integrations has already led to a deeper understanding of the reliability of galaxy and cosmological simulations. The source code (available from the first author) provides to the experienced user a generic interface capable of searching for shadows of any set of ordinary differential equations. [ABSTRACT FROM AUTHOR]