Open AccessHow Does the Smaller Alignment Index (SALI) Distinguish Order from Chaos?
Author(s) -
Ch. Skokos,
Chris G. Antonopoulos,
Tassos Bountis,
Michael N. Vrahatis
Publication year - 2003
Publication title -
progress of theoretical physics supplement
Language(s) - English
Resource type - Journals
ISSN - 0375-9687
DOI - 10.1143/ptps.150.439
Subject(s) - integrable system , chaotic , hamiltonian system , torus , physics , hamiltonian (control theory) , phase space , invariant (physics) , tangent , tangent space , mathematical physics , zero (linguistics) , motion (physics) , order (exchange) , mathematical analysis , mathematics , classical mechanics , geometry , quantum mechanics , computer science , mathematical optimization , linguistics , philosophy , finance , artificial intelligence , economics
The ability of the Smaller Alignment Index (SALI) to distinguish chaotic fromordered motion, has been demonstrated recently in severalpublications.\cite{Sk01,GRACM} Basically it is observed that in chaotic regionsthe SALI goes to zero very rapidly, while it fluctuates around a nonzero valuein ordered regions. In this paper, we make a first step forward explainingthese results by studying in detail the evolution of small deviations fromregular orbits lying on the invariant tori of an {\bf integrable} 2DHamiltonian system. We show that, in general, any two initial deviation vectorswill eventually fall on the ``tangent space'' of the torus, pointing indifferent directions due to the different dynamics of the 2 integrals ofmotion, which means that the SALI (or the smaller angle between these vectors)will oscillate away from zero for all time.Comment: To appear in Progress of Theoretical Physics Supplemen
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