Solitary waves in the excitable Burridge–Knopoff model

Abstract

The Burridge-Knopoff model is a lattice differential equation describing a chain of blocks connected by springs and pulled over a surface. This model was originally introduced to investigate nonlinear effects arising in the dynamics of earthquake faults. One of the main ingredients of the model is a nonlinear velocity-dependent friction force between the blocks and the fixed surface. For some classes of non-monotonic friction forces, the system displays a large response to perturbations above a threshold, which is characteristic of excitable dynamics. Using extensive numerical simulations, we show that this response corresponds to the propagation of a solitary wave for a broad range of friction laws (smooth or nonsmooth) and parameter values. These solitary waves develop shock-like profiles at large coupling (a phenomenon connected with the existence of weak solutions in a formal continuum limit) and propagation failure occurs at low coupling. We introduce a simplified piecewise linear friction law (reminiscent of the McKean nonlinearity for excitable cells) which allows us to obtain an analytical expression of solitary waves and study some of their qualitative properties, such as wave speed and propagation failure. We propose a possible physical realization of this system as a chain of impulsively forced mechanical oscillators. In certain parameter regimes, non-monotonic friction forces can also give rise to bistability between the ground state and limit-cycle oscillations and allow for the propagation of fronts connecting these two stable states.

Traveling waves in a spring-block chain sliding down a slope

“…In the case of the piecewiselinear friction law F 0 , the computation of the front velocity is much simpler than for the solitary wave of the Burridge-Knopoff model. A detailed study of solitary waves in the excitable Burridge-Knopoff model is presented in [7].…”

“…We consider here the piecewise linear force F 0 , and we assume that each block is in a bistable regime, i.e. we have V ∈ (a, a(1 + α)) and U 1 = −αa as in (7). We assume that the traveling front solution crosses the threshold (6) for only one value of ξ.…”

“…We consider here a friction force of spinodal type: the steady-state kinetic friction coefficient has a nonmonotonic profile versus sliding velocity such as the one depicted in Fig.2. Such friction laws have been reported to induce excitable dynamics (see [7] and references therein) reminiscent of neural excitability [8,9], i.e., a perturbation above a certain threshold produces a large excursion in the phase space before returning to an equilibrium state. In biology, it is well documented that a large class of excitable media is able to support nonlinear solitary waves [10].…”

Numerical evidences of almost convergence of wave speeds for the Burridge–Knopoff model

“…However, it is important to notice that the overall strategy is totally explicit because the predicted value is employed within the implicit method where the dependence on the incoming time instant appears. The second original contribution has been inspired by the fact that some continuous versions of the Burridge-Knopoff model exhibit propagating fronts (see [26,31]). Hence, we wanted to explore if there is some numerical evidence of similar kind of structures for the original model, where it is known that the analysis is much more intricated due to the inherent discrete structure.…”

A phase-plane analysis of localized frictional waves

“…We finally note that a variety of solitary waves in the form of slip pulses or fronts were also numerically found within the one-dimensional continuum limit of the Burridge-Knoppoff model [51], either with velocity-dependent non-smooth friction laws [52][53][54][55][56] or the Ruina rate-and-state law [31].…”