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Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems

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Abstract

Nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated. We study the bifurcation sequences of the mixed-mode oscillations generated by the homoclinicity, which are shown to belong to two different types, depending on the nature of the Liapunov numbers of the basic periodic orbit. A detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame-ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation. Chaotic cool flame oscillations are also observed in the same model. When the control parameter crosses a critical value, this chaotic set of trajectories becomes globally unstable and forms a Cantor-like hyperbolic repellor, and the ignition mechanism generates ahomoclinic tangency to the Cantor set of trajectories. The complex bifurcation diagram may be globally reconstructed from a one-dimensional dynamical system, thanks to the strong contractivity of thermokinetics. It is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system.

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Authors and Affiliations

  1. Faculté des Sciences, Université Libre de Bruxelles, B-1050, Brussels, Belgium

    P. Gaspard & X. J. Wang

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  1. P. Gaspard
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  2. X. J. Wang
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Gaspard, P., Wang, X.J. Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems. J Stat Phys 48, 151–199 (1987). https://doi.org/10.1007/BF01010405

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  • DOI: https://doi.org/10.1007/BF01010405

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