## Abstract

The key of Marotto's theorem on chaos for multi-dimensional maps is the existence of snapback repeller. For practical application of the theory, locating a computable repelling neighborhood of the repelling fixed point has thus become the key issue. For some multi-dimensional maps F, basic informa- tion of F is not sufficient to indicate the existence of snapback repeller for F. In this investigation, for a repeller z of F, we start from estimating the repelling neighborhood of z under F ^{k} for some k ≥ 2, by a theory built on the first or second derivative of F ^{k} . By employing the Interval Arithmetic computation, we locate a snapback point z _{0} in this repelling neighborhood and examine the nonzero determinant condition for the Jacobian of F along the orbit through z _{0} . With this new approach, we are able to conclude the existence of snapback repellers under the valid definition, hence chaotic behaviors, in a discrete-time predator-prey model, a population model, and the FitzHugh nerve model.

Original language | English |
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Pages (from-to) | 81-92 |

Number of pages | 12 |

Journal | Journal of Computational Dynamics |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - 2018 |

## Keywords

- Chaos
- Homoclinic orbit
- Interval arithmetic
- Snapback repeller