SEMINAIRE MERCREDI 3 OCTOBRE 2007
14 heures
Salle Séminaire 5
Centre de Physique Théorique
Marseille-Luminy

Charalampos (Haris) Skokos
Astronomie et Systèmes Dynamiques, Institut de Mécanique
Céleste et de Calcul des Ephémerides (IMCCE),
Observatoire de Paris

Title: Studying the Dynamics of Conservative Dynamical Systems by the
Generalized Alignment Index (GALI) Method

Abstract: We investigate the dynamics of conservative dynamical
systems by studying the evolution of volume elements formed by
deviation vectors about their orbits. The behavior of these volumes is
strongly influenced by the regular or chaotic nature of the motion.
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. More specifically we define the
Generalized Alignment Index of order k (GALI_k), with k>1, as the
volume of a generalized parallelepiped, whose edges are k initially
linearly independent unit deviation vectors from the studied orbit. We
show analytically and verify numerically on particular examples of
Hamiltonian systems and symplectic mappings that, for chaotic orbits,
GALI_k tends exponentially to zero with exponents that involve the
values of several Lyapunov exponents, while in the case of regular
orbits GALI_k fluctuates around non-zero values or goes to zero
following power laws that depend on the dimension of the torus and the
number of deviation vectors initially tangent to it. The GALI_k is a
generalization of the Smaller Alignment Index (SALI), since GALI_2 is
practically equivalent to SALI. However, GALI_k provides significantly
more information on the local dynamics of the system, allows for a
faster and clearer distinction between order and chaos than SALI and
works even in cases where the SALI method faced difficulties. Finally,
exploiting the advantages of GALIs we demonstrate how one can use the
indices for understanding rapidly the global dynamics of various
dynamical systems, as well as identify regular motion on
low-dimensional tori.