Friction memory effect in complex dynamics of earthquake model
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O R I G I N A L PA P E R
Friction memory effect in complex dynamics of earthquake model Srdan ¯ Kosti´c · Igor Franovi´c · Kristina Todorovi´c · Nebojša Vasovi´c
Received: 20 December 2012 / Accepted: 15 April 2013 / Published online: 4 May 2013 © Springer Science+Business Media Dordrecht 2013
Abstract In present paper, an effect of delayed frictional healing on complex dynamics of simple model of earthquake nucleation is analyzed, following the commonly accepted assumption that frictional healing represents the main mechanism for fault restrengthening. The studied model represents a generalization of Burridge–Knopoff single-block model with Dieterich–Ruina’s rate and state dependent friction law. The time-dependent character of the frictional healing process is modeled by introducing time delay τ in the friction term. Standard local bifurcation analysis of the obtained delay-differential equations demonstrates that the observed model exhibits Ruelle–Takens–Newhouse route to chaos. Domain in parameters space where the solutions are stable for all values of time delay is determined by applying the Rouché theorem. The obtained results are corroborated by Fourier power spectra and largest Lyapunov exponents techniques. In contrast to previous research, S. Kosti´c () · N. Vasovi´c University of Belgrade Faculty of Mining and Geology, Ðušina 7, 11000 Belgrade, Serbia e-mail: [email protected] I. Franovi´c University of Belgrade Faculty of Physics, Studentski trg 12, 11000 Belgrade, Serbia K. Todorovi´c University of Belgrade Faculty of Pharmacy, Vojvode Stepe 450, 11000 Belgrade, Serbia
the performed analysis reveals that even the small perturbations of the control parameters could lead to deterministic chaos, and, thus, to instabilities and earthquakes. The obtained results further imply the necessity of taking into account this delayed character of frictional healing, which renders complex behavior of the model, already captured in the case of more than one block. Keywords Spring-block model · State variable · Time delay · Quasiperiodicity · Deterministic chaos
1 Introduction There is a general consensus about the nucleation mechanism of tectonic earthquakes—these are produced when rock breaks suddenly in response to various geological forces [1]. The main role in this process is ascribed to rock strength and the size of accumulated strain energy. However, after a crack propagation, once a fault has been formed, its further motion is controlled by the friction between the interacting parts of Earth’s crust [2]. It is this friction which determines whether the fault motion would be seismic or aseismic. In other words, depending on the frictional stability, the fault motion could eventually lead to earthquake. The usual way of studying rock friction is by introducing friction constitutive laws, which emphasize the frictional instability as a possible mechanism for repetitive stick-slip failure and the seismic cycle [3].
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Fig. 1 The Burridge–Knopoff block and spring model, represented by a slider coup
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