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

Abstract

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. 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 ≤ 2N following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus:The GALI k is a generalization of the Smaller Alignment Index (SALI) (GALI 2 ∝ 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.

Adjusting chaotic indicators to curved spacetimes

“…In the case of a two dimensional map, SALI always goes to zero, but for chaotic orbits this happens exponentially fast, while for regular orbits SALI ∝ t −q , where q ≈ 2. This kind of power laws, in fact, provide the means for GALI to find the dimension of a torus in multidimensional systems [11]. The advantage of GALI over the other indicators is exactly this ability, but in order to use it, we have to evolve more than two deviation vectors.…”

“…From the category of these indicators, the most renowned is the maximal Lyapunov Characteristic Exponent (mLCE) (see, [4] for a survey). Other similar indicators are: the Fast Lyapunov Indicator (FLI) [6,7], the Mean Exponential Growth of Nearby Orbits (MEGNO) [8,9], the Generalized Alignment Index (GALI) [10,11], the Average Power Law Exponent (APLE) [12,13]. In classical mechanics the above mentioned indicators have been compared and studied for their efficiency several times (see, e.g., [5,12]).…”

The long-term dynamics of the Jovian irregular satellites

“…There are a variety of methods that are used to determine the level of stochasticity of a given orbit and which are either based on the Lyapunov Exponents theory (see a review in Skokos 2010) such as the FLI (Froeschlé et al 1997), the OFLI (Fouchard et al 2002), the MEGNO (Cincotta et al 2003), or the GALI (Skokos et al 2007), or they are based on spectral methods, such as the frequency analysis (Laskar 2005), the spectral number (Ferraz-Mello et al 2005), or the 0-1 test (Gottwald & Melbourne 2009). Using the frequency analysis, a two-dimensional view in the frequency space of the web of resonances of a mapping has been realized for the first time by Laskar (1993).…”

Boxy Orbital Structures in Rotating Bar Models

“…For these reasons the use of GALI 2 is an essential tool for the needs of our investigation. We also note here that the GALI 2 index is closely related to the SALI method (see for example Appendix B of Skokos et al 2007). …”