判定轨道混沌的几个指标

评述判定天体轨道混沌性质的几个指标,包括Poincare截面方法、Lyapunov指数、局部Lyapunov指数及其谱分布、快速Lyapunov指标、较小排列指标、0-1指标和频谱分析法等,讨论它们的优缺点和适用范围;强调相对论系统中建立坐标不变指标的重要性,例如,利用投影算符实现“1+3”时空分解而建立的独立于坐标规范的Lyapunov指数来处理弯曲时空是方便的。

Several Diagnostic Indexes for Orbital Chaos

In this paper we review in detail some methods for distinguishing between a regular orbit and a chaotic one in a Newtonian dynamical system, which contain Poincare section, Lyapunov exponents, local Lyapunov exponents and their spectral distributions, fast Lyapunov exponent index, smaller alignment index, 0-1 test, frequency map analysis, and so on. In particular, merits, demerits and application of these diagnostic indexes are discussed. In principle, these indexes from the Newtonian frame can also be applied to relativistic gravitational systems in general. However, there may still exist some problems because they are not coordinate invariant. As a result, it is vital to understand the behavior of a relativistic gravitational system by a eovariant way. For example, it is convenient to employ our way for the calculation of Lyapunov exponents with gauge invariance by use of the "1+3" splitting of a curved spacetime and the projected norm.