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Detection of the onset of numerical chaotic instabilities by lyapunov exponents

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  • Alicia Serfaty De Markus

Abstract

It is commonly found in the fixed-step numerical integration of nonlinear differential equations that the size of the integration step is opposite related to the numerical stability of the scheme and to the speed of computation. We present a procedure that establishes a criterion to select the largest possible step size before the onset of chaotic numerical instabilities, based upon the observation that computational chaos does not occur in a smooth, continuous way, but rather abruptly, as detected by examining the largest Lyapunov exponent as a function of the step size. For completeness, examination of the bifurcation diagrams with the step reveals the complexity imposed by the algorithmic discretization, showing the robustness of a scheme to numerical instabilities, illustrated here for explicit and implicit Euler schemes. An example of numerical suppression of chaos is also provided.

Suggested Citation

  • Alicia Serfaty De Markus, 2001. "Detection of the onset of numerical chaotic instabilities by lyapunov exponents," Discrete Dynamics in Nature and Society, Hindawi, vol. 6, pages 1-8, January.
    • Handle: RePEc:hin:jnddns:185919
    DOI: 10.1155/S1026022601000127
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    Cited by:

    1. Peixoto, Márcia L.C. & Nepomuceno, Erivelton G. & Martins, Samir A.M. & Lacerda, Márcio J., 2018. "Computation of the largest positive Lyapunov exponent using rounding mode and recursive least square algorithm," Chaos, Solitons & Fractals, Elsevier, vol. 112(C), pages 36-43.

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