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The Smaller (SALI) and the Generalized (GALI) Alignment Indices: Efficient Methods of Chaos Detection

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Book cover Chaos Detection and Predictability

Part of the book series: Lecture Notes in Physics ((LNP,volume 915))

Abstract

We provide a concise presentation of the Smaller (SALI) and the Generalized Alignment Index (GALI) methods of chaos detection. These are efficient chaos indicators based on the evolution of two or more, initially distinct, deviation vectors from the studied orbit. After explaining the motivation behind the introduction of these indices, we sum up the behaviors they exhibit for regular and chaotic motion, as well as for stable and unstable periodic orbits, focusing mainly on finite-dimensional conservative systems: autonomous Hamiltonian models and symplectic maps. We emphasize the advantages of these methods in studying the global dynamics of a system, as well as their ability to identify regular motion on low dimensional tori. Finally we discuss several applications of these indices to problems originating from different scientific fields like celestial mechanics, galactic dynamics, accelerator physics and condensed matter physics.

The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-662-48410-4_9

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-662-48410-4_9

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Notes

  1. 1.

    We note that the value of \(\lambda _{1}\) is independent of the used norm.

  2. 2.

    We note that throughout this chapter we use the hat symbol (\(\,\hat{}\,\)) to denote a unit vector.

  3. 3.

    For Hamiltonian systems the time is a continuous variable, while for maps it is a discrete one counting the map’s iterations.

  4. 4.

    We note that throughout this chapter the logarithm to base 10 is denoted by \(\log\).

  5. 5.

    We note that here, as well as in several, forthcoming figures in this chapter, the evaluation of the LEs is done only for confirming the theoretical predictions for the time evolution of the SALI (Eq. (5.12) in the current case) and later on of the GALIs, and it is not needed for the computation of the SALI and the GALIs.

  6. 6.

    Note that this parallelogram is not the usual 2d parallelogram on the plane because its sides (the deviation vectors) are not 2d vectors.

  7. 7.

    For a brief introduction to the notion of the wedge product the reader is referred to the Appendix A of [84] and the Appendix of [81].

  8. 8.

    A proof of the second equality of (5.14) can be found in the Appendix B of [84].

  9. 9.

    It is worth mentioning here that other chaos indicators, like the Orthogonal Fast Lyapunov Indicator (OFLI) and its variations [7, 8], are quite successful in performing this task as they were actually designed for this purpose.

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Acknowledgements

Many of the results described in this chapter were obtained in close collaboration with Prof. T. Bountis, Dr. Ch. Antonopoulos and Dr. E. Gerlach. This work was partially supported by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: ‘THALES’. Ch. S. would like to thank the Research Office of the University of Cape Town for the Research Development Grant which funded part of this study, as well as the Max Planck Institute for the Physics of Complex Systems in Dresden for its hospitality during his visit in December 2014–January 2015, when part of this work was carried out. In addition, Ch. S. thanks T. van Heerden for the careful reading of the manuscript and for his valuable comments. We are also grateful to the three anonymous referees whose constructive remarks helped us improve the content and the clarity of the chapter.

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, South Africa

    Charalampos (Haris) Skokos

  2. Center for Applied Mathematics and Theoretical Physics (CAMTP), University of Maribor, Krekova 2, SI-2000, Maribor, Slovenia

    Thanos Manos

  3. School of Applied Sciences, University of Nova Gorica, Vipavska 11c, SI-5270, Ajdovščina, Slovenia

    Thanos Manos

  4. Institute of Neuroscience and Medicine Neuromodulation (INM-7), Research Center Jülich, 52425, Jülich, Germany

    Thanos Manos

Authors
  1. Charalampos (Haris) Skokos
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  2. Thanos Manos
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Correspondence to Charalampos (Haris) Skokos .

Editor information

Editors and Affiliations

  1. University of Cape Town, Department of Mathematics and Applied Ma, Rondebosch, South Africa

    Charalampos (Haris) Skokos

  2. School of Mathematics and Statistics, University of Sydney, Sydney, Australia

    Georg A. Gottwald

  3. Observatoire de Paris, IMCCE, Paris, France

    Jacques Laskar

Appendix: Pseudo-Codes for the Computation of the SALI and the GALI k

Appendix: Pseudo-Codes for the Computation of the SALI and the GALI k

We present here pseudo–codes for the numerical computation of the SALI (Table 5.1) and the GALI k (Table 5.2) methods, according to the algorithms presented in Sects. 5.2 and 5.3.1 respectively.

Table 5.1 Numerical computation of the SALI
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Table 5.2 Numerical computation of the GALI k
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Skokos, C.(., Manos, T. (2016). The Smaller (SALI) and the Generalized (GALI) Alignment Indices: Efficient Methods of Chaos Detection. In: Skokos, C., Gottwald, G., Laskar, J. (eds) Chaos Detection and Predictability. Lecture Notes in Physics, vol 915. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48410-4_5

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