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

P. Gaspard, Xiao-Jing Wang

Research output: Contribution to journalArticle

Abstract

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
Volume48
Issue number1-2
DOIs
StatePublished - Jul 1987

Fingerprint

Mixed Mode
Homoclinic Orbit
Homoclinic Tangency
Ignition
Oscillation
Flame
orbits
dynamical systems
Periodic Orbits
oscillations
ignition
flames
Bifurcation
Dynamical system
trajectories
Trajectory
Contractivity
Regular Sequence
Symbolic Dynamics
Cantor set

Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Physics and Astronomy(all)
  • Mathematical Physics

Cite this

Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems. / Gaspard, P.; Wang, Xiao-Jing.

In: Journal of Statistical Physics, Vol. 48, No. 1-2, 07.1987, p. 151-199.

Research output: Contribution to journalArticle

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