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

Ch. Skokos·Ch. Antonopoulos·T. C. Bountis·M. N. Vrahatis
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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.

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Ch. Skokos, Ch. Antonopoulos, T. C. Bountis, M. N. Vrahatis. 2003-01-27. How does the Smaller Alignment Index (SALI) distinguish order from chaos?. https://doi.org/10.1143/ptps.150.439

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