### 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 language | English (US) |
---|---|

Pages (from-to) | 151-199 |

Number of pages | 49 |

Journal | Journal of Statistical Physics |

Volume | 48 |

Issue number | 1-2 |

DOIs | |

State | Published - Jul 1 1987 |

### Fingerprint

### Keywords

- 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

### Cite this

*Journal of Statistical Physics*,

*48*(1-2), 151-199. https://doi.org/10.1007/BF01010405