Hamilton系统中局域动力学的几何性质：广义排列指数（GALI）方法
Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method
摘要：
我们研究多维哈密顿系统的详细动力学，通过研究由单位偏差向量形成的体积元关于它们的轨道的演化。这些体积的行为受到运动的规则或混沌性质、偏差向量的数目、它们的线性（不）依赖性和李雅普诺夫指数的谱的强烈影响。这些体积的不同时间演化可以用来快速有效地识别动力学的性质，从而引入明确区分$N$维环面上的混沌行为和准周期运动的量。更具体地说，我们引入了广义排列指数的顺序$k$（GALI $_k$）作为广义平行六面体的体积，其边缘是$k$最初线性独立的单位偏离向量从研究的轨道，其幅度归一化为统一的每一个时间步。GALI $_k$是较小比对指数（SALI）（GALI $_2$ $\propto$ SALI）的推广。然而，GALI $_k$提供了显着更详细的信息，当地的动态，允许更快，更清楚地区分秩序和混乱比SALI和工程，即使在SALI方法是不确定的情况下。
英文摘要：
We investigate the detailed dynamics of multidimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on $N$-dimensional tori. More specifically we introduce the Generalized Alignment Index of order $k$ (GALI$_k$) as the volume of a generalized parallelepiped, whose edges are $k$ initially linearly independent unit deviation vectors from the studied orbit whose magnitude is normalized to unity at every time step. The GALI$_k$ is a generalization of the Smaller Alignment Index (SALI) (GALI$_2$ $\propto$ SALI). However, GALI$_k$ provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.
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