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Long-time behavior of solutions and chaos in reaction-diffusion equations

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  • Soltanov, Kamal N.
  • Prykarpatski, Anatolij K.
  • Blackmore, Denis

Abstract

It is shown that members of a class (of current interest with many applications) of non-dissipative reaction-diffusion partial differential equations with local nonlinearity can have an infinite number of different unstable solutions traveling along an axis of the space variable with varying speeds, traveling impulses and also an infinite number of different states of spatio-temporal (diffusion) chaos. These solutions are generated by cascades of bifurcations governed by the corresponding steady states. The behavior of these solutions is analyzed in detail and, as an example, it is explained how space-time chaos can arise. Results of the same type are also obtained in the case of a nonlocal nonlinearity.

Suggested Citation

  • Soltanov, Kamal N. & Prykarpatski, Anatolij K. & Blackmore, Denis, 2017. "Long-time behavior of solutions and chaos in reaction-diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 91-100.
    • Handle: RePEc:eee:chsofr:v:99:y:2017:i:c:p:91-100
    DOI: 10.1016/j.chaos.2017.03.057
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    References listed on IDEAS

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    1. Xavier Barrachina & J. Alberto Conejero, 2012. "Devaney Chaos and Distributional Chaos in the Solution of Certain Partial Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-11, December.
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    Cited by:

    1. Maximenko, Vladimir A. & Hramov, Alexander E. & Koronovskii, Alexey A. & Makarov, Vladimir V. & Postnov, Dmitry E. & Balanov, Alexander G., 2017. "Lyapunov analysis of the spatially discrete-continuous system dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 228-237.

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