### 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 σ_{1}, σ_{2} 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 language | English |
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Pages (from-to) | 6269-6284 |

Number of pages | 16 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 37 |

Issue number | 24 |

DOIs | |

Publication status | Published - Jun 18 2004 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Physics A: Mathematical and General*,

*37*(24), 6269-6284. https://doi.org/10.1088/0305-4470/37/24/006

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

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 37, no. 24, pp. 6269-6284. https://doi.org/10.1088/0305-4470/37/24/006

}

TY - JOUR

T1 - Detecting order and chaos in Hamiltonian systems by the SALI method

AU - Skokos, Ch

AU - Antonopoulos, Ch

AU - Bountis, T. C.

AU - Vrahatis, M. N.

PY - 2004/6/18

Y1 - 2004/6/18

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=3042620995&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042620995&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/37/24/006

DO - 10.1088/0305-4470/37/24/006

M3 - Article

VL - 37

SP - 6269

EP - 6284

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 24

ER -