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 journalArticle

91 Citations (Scopus)

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 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
Volume37
Issue number24
DOIs
Publication statusPublished - Jun 18 2004
Externally publishedYes

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Hamiltonians
Chaos theory
Hamiltonian Systems
chaos
Chaos
Alignment
alignment
Orbits
Orbit
orbits
Largest Lyapunov Exponent
Chaotic Motion
Zero
Phase Space
degrees of freedom
Degree of freedom
exponents
Tend
Converge
Demonstrate

ASJC Scopus subject areas

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

Cite this

Detecting order and chaos in Hamiltonian systems by the SALI method. / Skokos, Ch; Antonopoulos, Ch; Bountis, T. C.; Vrahatis, M. N.

In: Journal of Physics A: Mathematical and General, Vol. 37, No. 24, 18.06.2004, p. 6269-6284.

Research output: Contribution to journalArticle

Skokos, Ch ; Antonopoulos, Ch ; Bountis, T. C. ; Vrahatis, M. N. / Detecting order and chaos in Hamiltonian systems by the SALI method. In: Journal of Physics A: Mathematical and General. 2004 ; Vol. 37, No. 24. pp. 6269-6284.
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