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

P. Gaspard, X. J. Wang

Research output: Contribution to journalArticlepeer-review


Nonlinear autonomous dynamical systems with a homoclinic 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 a homoclinic 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.

Original languageEnglish (US)
Pages (from-to)151-199
Number of pages49
JournalJournal of Statistical Physics
Issue number1-2
StatePublished - Jul 1987


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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