### Abstract

In this paper we consider the one dimensional spring-block model describing earthquake faulting. By using geometric singular perturbation theory and the blow-up method we provide a detailed description of the periodicity of the earthquake episodes. In particular, the limit cycles arise from a degenerate Hopf bifurcation whose degeneracy is due to an underlying Hamiltonian structure that leads to large amplitude oscillations. We use a Poincar\'e compactification to study the system near infinity. At infinity the critical manifold loses hyperbolicity with an exponential rate. We use an adaptation of the blow-up method to recover the hyperbolicity. This enables the identification of a new attracting manifold that organises the dynamics at infinity. This in turn leads to the formulation of a conjecture on the behaviour of the limit cycles as the time-scale separation increases. We provide the basic foundation for the proof of this conjecture and illustrate our findings with numerics.

Original language | English |
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Journal | Nonlinearity |

Volume | 30 |

Issue number | 7 |

Pages (from-to) | 2805-34 |

ISSN | 0951-7715 |

Publication status | Published - 2017 |

### Keywords

- Singular perturbation
- Hamiltonian systems
- Rate and state friction
- Blowup
- Earthquake dynamics
- Poincaré compactification

## Cite this

Bossolini, E., Brøns, M., & Kristiansen, K. U. (2017). Singular limit analysis of a model for earthquake faulting.

*Nonlinearity*,*30*(7), 2805-34. http://arxiv.org/abs/1603.02448