Lyapunov and Reversibility error invariant indicators

TL;DR

This paper surveys Lyapunov and Reversibility Error indicators, introduces their invariant generalizations, and demonstrates their effectiveness in characterizing Hamiltonian system dynamics with improved robustness and new analytical tools.

Contribution

The paper proposes a generalization of Lyapunov and Reversibility Error indicators to make them invariant to initial conditions, and introduces higher-order invariant indicators for dynamical analysis.

Findings

Invariant indicators effectively characterize Hamiltonian dynamics.

Reversibility Error Method (REM) is practical and accurate.

Indicators depend on positive Lyapunov exponents.

Abstract

In this review, we present a survey of the Lyapunov Error and Reversibility Error (\cite{Faranda2012}), and we propose a generalization to make them invariant to the choice of initial conditions. We first define a process as the evolution in time of a a map or a flow, we then introduce the covariance matrix of a given process, and use their trace to compute LE and RE. The determinant of the covariance matrices is used to compute all invariant indicators of higher order. In this way, two set of invariant indicators are proposed within the framework introduced here, one for the Reversibility and one for the Lyapunov Errors, respectively. LE and RE which have been used in the literature, are the \textit{first-order} invariant indicators in their respective sets. The new sets of invariant indicators have the same fundamental meaning, the set for RE is used to characterize the dynamical evolution of a continuous small perturbation over an orbit, while LE is used to study the evolution of an initial displacement between two nearby orbits. We obtain again also the Reversibility Error Method (REM), which is a particular case of RE, where the additive noise is replaced by the round-off errors. REM has been proved to be a practical, accurate, and fast dynamical indicator and the results are comparable with the RE induced by a random noise of given amplitude. the behaviour of RE invariants depends on the sum of the positive Lyapunov exponents. We showcase the accuracy and reliability of those indicators on two well studied Hamiltonian. Within this new framework, a full dynamical characterization of a Hamiltonian system can be obtained. For instance, the Gibbs entropy, its asymptotic behaviour and a method to compute the fidelity over the perturbed orbit, are provided.