Analysis of Round Off Errors with Reversibility Test as a Dynamical Indicator

TL;DR

This paper investigates how round off errors affect the analysis of dynamical systems, proposing the reversibility error as a simple and effective indicator to distinguish between regular and chaotic behaviors.

Contribution

It introduces the reversibility error as a new, straightforward dynamical indicator for analyzing round off errors in orbit computations, comparable to established variational methods.

Findings

Reversibility error effectively distinguishes regular and chaotic orbits.

Round off error correlations differ significantly in regular versus chaotic systems.

The method accurately maps resonance webs and weakly chaotic regions in high-dimensional maps.

Abstract

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analysed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator. The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same. Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described.