Detecting order and chaos in Hamiltonian systems by the SALI method

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

Research output: Contribution to journalArticlepeer-review

102 Citations (Scopus)


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 σ1, σ2 i.e. SALI ∝ e-(σ1-σ2)t. 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.

Original languageEnglish
Pages (from-to)6269-6284
Number of pages16
JournalJournal of Physics A: Mathematical and General
Issue number24
Publication statusPublished - Jun 18 2004

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)

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