Singular limit analysis of a model for earthquake faulting

TL;DR

This paper analyzes a spring-block model for earthquake faulting using advanced mathematical techniques, revealing the nature of periodic earthquake episodes and their behavior at infinity.

Contribution

It provides a detailed singular perturbation analysis of the model, identifying the origin of limit cycles and proposing a conjecture on their behavior as parameters vary.

Findings

Limit cycles originate from a degenerate Hopf bifurcation due to Hamiltonian structure.

The system's behavior at infinity involves loss of hyperbolicity, which is recovered using a blow-up method.

Numerical illustrations support the theoretical analysis and conjecture.

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 illustrate our findings with numerics and suggest an outline of the proof of this conjecture.