Geometrical properties of local dynamics in Hamiltonian systems: The Generalized Alignment Index (GALI) method

We investigate the detailed dynamics of multi-dimensional 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 with respect to the orbit studied whose magnitude is normalized to unity at every time step. We show analytically and verify numerically on particular examples of N-degree-of-freedom Hamiltonian systems that, for chaotic orbits, GALI_k tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, GALI_k fluctuates around non-zero values for 2 ≤ k ≤ N and goes to zero for N < k ≤ 2 N following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus: ∝ t^{- 2 (k - N) + m} if 0 ≤ m < k - N, and ∝ t^{- (k - N)} if m ≥ k - N. The GALI_k is a generalization of the Smaller Alignment Index (SALI) (GALI₂ ∝ 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.