地震断层模型的奇异极限分析
Singular limit analysis of a model for earthquake faulting
摘要：
本文考虑描述地震断层作用的一维弹簧-块体模型。利用几何奇异扰动理论和blow-up方法，对地震幕的周期性进行了详细的描述。特别是，极限环产生于退化的Hopf分岔，其退化是由于一个潜在的哈密顿结构，导致大振幅振荡。我们使用庞加莱紧化来研究系统在无穷远点附近。在无穷远处，临界流形以指数速率失去双曲性。我们使用一个适应的爆破方法来恢复双曲。这使得能够识别一个新的吸引流形，组织在无穷大的动态。这反过来又导致制定一个猜想的行为的极限环的时间尺度分离增加。我们说明我们的研究结果与数值，并提出了一个大纲证明这一猜想。
英文摘要：
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 Poincaré 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.
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