Singular limit analysis of a model for earthquake faulting

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

In: Nonlinearity, Vol. 30, No. 7, 2017, p. 2805-34.

Research output: Contribution to journalJournal article – Annual report year: 2016Researchpeer-review

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@article{29f916300bea482e959bc250c5c1402a,
title = "Singular limit analysis of a model for earthquake faulting",
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.",
keywords = "Singular perturbation, Hamiltonian systems, Rate and state friction, Blowup, Earthquake dynamics, Poincar{\'e} compactification",
author = "Elena Bossolini and Morten Br{\o}ns and Kristiansen, {Kristian Uldall}",
year = "2017",
language = "English",
volume = "30",
pages = "2805--34",
journal = "Nonlinearity",
issn = "0951-7715",
publisher = "IOP Publishing",
number = "7",

}

RIS

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 -