Dynamics of simple earthquake model with time delay and variation of friction strength
Abstract. We examine the dynamical behaviour of the phenomenological Burridge–Knopoff-like model with one and two blocks, where the friction term is supplemented by the time delay τ and the variable friction strength c. Time delay is assumed to reflect the initial quiescent period of the fault healing, considered to be a function of history of sliding. Friction strength parameter is proposed to mimic the impact of fault gouge thickness on the rock friction. For the single-block model, interplay of the introduced parameters c and τ is found to give rise to oscillation death, which corresponds to aseismic creeping along the fault. In the case of two blocks, the action of c1, c2, τ1 and τ1 may result in several effects. If both blocks exhibit oscillatory motion without the included time delay and frictional strength parameter, the model undergoes transition to quasiperiodic motion if only c1 and c2 are introduced. The same type of behaviour is observed when τ1 and τ2 are varied under the condition c1 = c2. However, if c1, and τ1 are fixed such that the given block would lie at the equilibrium while c2 and τ2 are varied, the (c2, τ2) domains supporting quasiperiodic motion are interspersed with multiple domains admitting the stationary solution. On the other hand, if (c1, τ1) warrant oscillatory behaviour of one block, under variation of c2 and τ2 the system's dynamics is predominantly quasiperiodic, with only small pockets of (c2, τ2) parameter space admitting the periodic motion or equilibrium state. For this setup, one may also find a transient chaos-like behaviour, a point corroborated by the positive value of the maximal Lyapunov exponent for the corresponding time series.
