### 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 1987 |

### Fingerprint

### 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

*Journal of Statistical Physics*,

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

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

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 48, no. 1-2, pp. 151-199. https://doi.org/10.1007/BF01010405

}

TY - JOUR

T1 - Homoclinic orbits and mixed-mode oscillations in far-from-equilibrium systems

AU - Gaspard, P.

AU - Wang, Xiao-Jing

PY - 1987/7

Y1 - 1987/7

N2 - 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.

AB - 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.

KW - bifurcation theory

KW - chaos

KW - chemical thermokinetics

KW - cool flame-ignition oscillations

KW - Homoclinic tangency

KW - hyperbolic repellor

KW - periodic attractors

KW - symbolic dynamics

UR - http://www.scopus.com/inward/record.url?scp=0001527863&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001527863&partnerID=8YFLogxK

U2 - 10.1007/BF01010405

DO - 10.1007/BF01010405

M3 - Article

AN - SCOPUS:0001527863

VL - 48

SP - 151

EP - 199

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -