Journal of Statistical Physics

, Volume 48, Issue 1–2, pp 151–199 | Cite as

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

  • P. Gaspard
  • X. J. Wang


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.

Key words

Homoclinic tangency bifurcation theory periodic attractors chaos hyperbolic repellor symbolic dynamics chemical thermokinetics cool flame-ignition oscillations 


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Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • P. Gaspard
    • 1
  • X. J. Wang
    • 1
  1. 1.Faculté des SciencesUniversité Libre de BruxellesBrusselsBelgium

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