标题： 李雅普诺夫和可逆性误差不变量指标 显示英文标题

标题： Lyapunov and Reversibility error invariant indicators

摘要： 在这篇综述中，我们展示了关于李雅普诺夫误差和可逆性误差（\cite{Faranda2012}）的调查，并提出了一种使其不受初始条件选择影响的一般化方法。 我们首先定义一个过程为映射或流随时间的演化，然后引入给定过程的协方差矩阵，并利用其迹来计算LE和RE。 协方差矩阵的行列式用于计算所有高阶不变指标。 通过这种方式，在这里介绍的框架内提出了两组不变指标，一组用于可逆性，另一组用于李雅普诺夫误差，分别。 在各自的集合中，LE和RE已经被用作\textit{一阶}不变指标。 新的不变指标集具有相同的基本含义，可逆性误差集用于表征轨道上连续小扰动的动力学演化，而LE用于研究两个相邻轨道之间初始位移的演化。 我们还重新获得了可逆性误差方法（REM），它是RE的一个特例，其中加性噪声被舍入误差所取代。 已经证明REM是一种实用、准确且快速的动力学指标，其结果与由给定幅度随机噪声引起的RE结果相当。 RE不变量的行为取决于正Lyapunov指数的总和。 我们在两个被广泛研究的哈密顿系统上展示了这些指标的准确性和可靠性。 在这个新的框架内，可以全面地表征哈密顿系统。 例如，提供了吉布斯熵、其渐近行为以及计算受扰轨线保真度的方法。 显示英文摘要

摘要： 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.