### Abstract

The ability of the Smaller Alignment Index (SALI) to distinguish chaotic from ordered motion, has been demonstrated recently in several publications. [Ch. Skokos, J. of Phys. A 34 (2001), 10029. Ch. Skokos, Ch. Antonopoulos, T. C. Bountis and M. N. Vrahatis, in Proceedings of the 4th GRACM Congress on Computational Mechanics, ed. D. T. Tsahalis (Univ. Patras, Patras, 2002), Vol. IV, p. 1496; in Libration Point Orbits and Applications, ed. G. Gómez, M. W. Lo and J. J. Masdemont (World Scientific, 2003), in press, nlin.CD/0210053.] 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 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.

Original language | English |
---|---|

Pages (from-to) | 439-443 |

Number of pages | 5 |

Journal | Progress of Theoretical Physics Supplement |

Volume | 150 |

Publication status | Published - 2003 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Progress of Theoretical Physics Supplement*,

*150*, 439-443.

**How Does the Smaller Alignment Index (SALI) Distinguish Order from Chaos?** / Skokos, Charalampos; Antonopoulos, Chris; Bountis, Tassos C.; Vrahatis, Michael N.

Research output: Contribution to journal › Article

*Progress of Theoretical Physics Supplement*, vol. 150, pp. 439-443.

}

TY - JOUR

T1 - How Does the Smaller Alignment Index (SALI) Distinguish Order from Chaos?

AU - Skokos, Charalampos

AU - Antonopoulos, Chris

AU - Bountis, Tassos C.

AU - Vrahatis, Michael N.

PY - 2003

Y1 - 2003

N2 - The ability of the Smaller Alignment Index (SALI) to distinguish chaotic from ordered motion, has been demonstrated recently in several publications. [Ch. Skokos, J. of Phys. A 34 (2001), 10029. Ch. Skokos, Ch. Antonopoulos, T. C. Bountis and M. N. Vrahatis, in Proceedings of the 4th GRACM Congress on Computational Mechanics, ed. D. T. Tsahalis (Univ. Patras, Patras, 2002), Vol. IV, p. 1496; in Libration Point Orbits and Applications, ed. G. Gómez, M. W. Lo and J. J. Masdemont (World Scientific, 2003), in press, nlin.CD/0210053.] 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 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.

AB - The ability of the Smaller Alignment Index (SALI) to distinguish chaotic from ordered motion, has been demonstrated recently in several publications. [Ch. Skokos, J. of Phys. A 34 (2001), 10029. Ch. Skokos, Ch. Antonopoulos, T. C. Bountis and M. N. Vrahatis, in Proceedings of the 4th GRACM Congress on Computational Mechanics, ed. D. T. Tsahalis (Univ. Patras, Patras, 2002), Vol. IV, p. 1496; in Libration Point Orbits and Applications, ed. G. Gómez, M. W. Lo and J. J. Masdemont (World Scientific, 2003), in press, nlin.CD/0210053.] 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 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.

UR - http://www.scopus.com/inward/record.url?scp=0345411904&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0345411904&partnerID=8YFLogxK

M3 - Article

VL - 150

SP - 439

EP - 443

JO - Progress of Theoretical Physics Supplement

JF - Progress of Theoretical Physics Supplement

SN - 0375-9687

ER -