Detecting order and chaos in Hamiltonian systems by the SALI method

2004/04/30 by Ch Skokos, Ch Antonopoulos, T C Bountis +1 · 14 citations
Physics and Astronomy · #Chaos control and synchronization #Quantum chaos and dynamical systems #Quantum many-body systems #astro-ph #nlin.CD

paper · pdf · doi:10.1088/0305-4470/37/24/006

published as J.Phys.A37:6269-6284,2004 · 18 pages, 10 figures, accepted for publication in J. Phys. A

arxiv created 2004/04/30 · openalex publication_date 2004/06/03 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30

Abstract

We use the smaller alignment index (SALI) to distinguish rapidly and with certainty between ordered and chaotic motion in Hamiltonian flows. This distinction is based on the different behaviour of the SALI for the two cases: the index fluctuates around non-zero values for ordered orbits, while it tends rapidly to zero for chaotic orbits. We present a detailed study of SALI's behaviour for chaotic orbits and show that in this case the SALI exponentially converges to zero, following a time rate depending on the difference of the two largest Lyapunov exponents σ 1 , σ 2 i.e. . Exploiting the advantages of the SALI method, we demonstrate how one can rapidly identify even tiny regions of order or chaos in the phase space of Hamiltonian systems of two and three degrees of freedom.

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