Friction memory effect in complex dynamics of earthquake model

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, 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.

Appendix

Starting from the system of equations:

$$\begin{array} {@{}l} \displaystyle\phi(\lambda) = \lambda^{3} + \lambda^{2} \biggl( \frac{\gamma^{2}}{\xi} + 1 \biggr) + \lambda \gamma^{2} \biggl( \frac{1}{\xi} + 1 \biggr) + \gamma ^{2} \\[4mm] \displaystyle \psi(\lambda) = - \lambda(1 + \varepsilon)\frac{\gamma^{2}}{\xi} e^{ - \lambda\tau} \end{array} $$

one could obtain:

Thus, we have:

$$\bigl| \phi(\lambda) \bigr| \ge| \lambda+ 1 |\gamma^{2} $$

While for the ψ(λ), we obtain:

assuming that |φ(λ)|>|ψ(λ)| on the contour C.

Thus, according to the previous: