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The 0-1 Test for Chaos: A Review

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Chaos Detection and Predictability

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

Abstract

We review here theoretical as well as practical aspects of the 0-1 test for chaos for deterministic dynamical systems. The test is designed to distinguish between regular, i.e. periodic or quasi-periodic, dynamics and chaotic dynamics. It works directly with the time series and does not require any phase space reconstruction. This makes the test suitable for the analysis of discrete maps, ordinary differential equations, delay differential equations, partial differential equations and real world time series. To illustrate the range of applicability we apply the test to examples of discrete dynamics such as the logistic map, Pomeau–Manneville intermittency maps with both summable and nonsummable autocorrelation functions, and the Hamiltonian standard map exhibiting weak chaos. We also consider examples of continuous time dynamics such as the Lorenz-96 system and a driven and damped nonlinear Schrödinger equation. Finally, we show the applicability of the 0-1 test for time series contaminated with noise as found in real world applications.

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Notes

  1. 1.

    One may either use off the shelf routines provided for example in Numerical Recipes [61] or build-in routines in MATLAB [49].

  2. 2.

    One may use e 2π i j ω∕n rather than e ij ω for a rescaled domain.

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Acknowledgements

GAG acknowledges support from the Australian Research Council. The research of IM was supported in part by the European Advanced Grant StochExtHomog (ERC AdG 320977).

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Authors and Affiliations

  1. School of Mathematics & Statistics, The University of Sydney, Sydney, 2006, NSW, Australia

    Georg A. Gottwald

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Ian Melbourne

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  1. Georg A. Gottwald
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Correspondence to Georg A. Gottwald .

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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

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Gottwald, G.A., Melbourne, I. (2016). The 0-1 Test for Chaos: A Review. 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_7

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