### Abstract

Original language | English |
---|---|

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

*Nonlinearity*,

*30*(7), 2805-34.

}

*Nonlinearity*, vol. 30, no. 7, pp. 2805-34.

**Singular limit analysis of a model for earthquake faulting.** / Bossolini, Elena; Brøns, Morten; Kristiansen, Kristian Uldall.

Research output: Contribution to journal › Journal article › Research › peer-review

TY - JOUR

T1 - Singular limit analysis of a model for earthquake faulting

AU - Bossolini, Elena

AU - Brøns, Morten

AU - Kristiansen, Kristian Uldall

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Singular perturbation

KW - Hamiltonian systems

KW - Rate and state friction

KW - Blowup

KW - Earthquake dynamics

KW - Poincaré compactification

M3 - Journal article

VL - 30

SP - 2805

EP - 2834

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 7

ER -