How does the Smaller Alignment Index (SALI) distinguish order from chaos?

Abstract

The ability of the Smaller Alignment Index (SALI) to distinguish chaotic from ordered motion, has been demonstrated recently in several publications.\cite{Sk01,GRACM} Basically it is observed that in chaotic regions the SALI goes to zero very rapidly, while it fluctuates around a nonzero value in ordered regions. In this paper, we make a first step forward explaining these results by studying in detail the evolution of small deviations from regular orbits lying on the invariant tori of an {\bf integrable} 2D Hamiltonian system. We show that, in general, any two initial deviation vectors will eventually fall on the ``tangent space'' of the torus, pointing in different directions due to the different dynamics of the 2 integrals of motion, which means that the SALI (or the smaller angle between these vectors) will oscillate away from zero for all time.

To appear in Progress of Theoretical Physics Supplement

Cited by in corpus (26)

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• Application of the MEGNO technique to the dynamics of Jovian irregular satellites
• Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi--Pasta--Ulam lattices by the Generalized Alignment Index method
• Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate
• Regular and chaotic orbits in barred galaxies - I. Applying the SALI/GALI method to explore their distribution in several models
• Self-consistent models of cuspy triaxial galaxies with dark matter haloes
• Chaotic Dynamics of N-degree of Freedom Hamiltonian Systems
• Stability of Simple Periodic Orbits and Chaos in a Fermi -- Pasta -- Ulam Lattice
• Application of the SALI chaos detection method to accelerator mappings
• Efficient integration of the variational equations of multi-dimensional Hamiltonian systems: Application to the Fermi-Pasta-Ulam lattice
• Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method
• Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems
• Interplay Between Chaotic and Regular Motion in a Time-Dependent Barred Galaxy Model
• Weak Chaos and the "Melting Transition" in a Confined Microplasma System
• Boxy Orbital Structures in Rotating Bar Models
• Efficient control of accelerator maps
• Effects of chaos on the detectability of stellar streams
• Self-consistent triaxial models
• Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method
• Nonlinear dynamics and onset of chaos in a physical model of a damper pressure relief valve
• Linear And Nonlinear Arabesques: A Study Of Closed Chains Of Negative 2-Element Circuits
• Further results on the power-law decay of the fraction of the mixed eigenstates in kicked-top model with mixed-type classical phase space
• Energy-recurrence Breakdown and Chaos in Disordered Fermi-Pasta-Ulam-Tsingou Lattices