History-dependent synchronization in the Burridge-Knopoff Model

Abstract

A three-blocks Burridge-Knopoff model is investigated. The dimensionless velocity-dependent friction force F (v) ∝ (1 + av) −1 is linearized around a = 0. In this way, the model is transformed into a six-dimensional mapping x(t n ) → x(t n+1 ), where t n are time moments when a block starts to move or stops. Between these moments, the equations of motion are integrable. For a < 0.1, the motion is quasiperiodic or periodic, depending on the initial conditions. For the periodic solution, we observe a synchronization of the motion of the lateral blocks. For a > 0.1, the motion becomes chaotic. These results are true for the linearized mapping, linearized numerical and non-linearized numerical solutions.

Dynamics of simple earthquake model with time delay and variation of friction strength

“…Moreover, the obtained results suggest that the system is fairly sensitive to perturbations, meaning that even the small stress changes could induce motion along the fault, and, consequently, the onset of earthquakes. One should emphasise that the sensitivity of the block motion on initial conditions was already observed in the work of Szkutnik et al (2003). In particular, the analysis there has shown that the character of the motion for the three-block model within a certain parameter range depends on the initial conditions.…”

A model for aperiodicity in earthquakes

“…Because it is possible to calculate the stiffness of system (2) at different times, a stiff solver seems to be the method of choice given high values of γ for which an extremely small time step is required. Rather than regularize the nonlinear friction term as done previously in Lapusta and Rice (2003) and Szkutnik et al (2003), it is possible to numerically integrate the system without an approximation to the friction term. Although we cannot yet study the bifurcations of solutions for higher values of γ , it is possible to calculate parameter spaces for which Hopf bifurcations occur and determine the chaotic regimes for this system.…”

“…In the past, the DieterichRuina friction term has been altered because of this problem. In Lapusta and Rice (2003) and Szkutnik et al (2003), the authors regularize the nonlinear friction term for values near zero. This can be done by either allowing rate values to be of either sign and taking absolute values or by linearizing the term giving only a close approximation in a small interval.…”

Study of a New Gliadin Capture Agent and Development of a Protein Microarray as a New Approach for Gliadin Detection

“…Furthermore, it will be important to see how this non-periodic behavior will affect the nucleation process in the numerical simulation of sequences of earthquakes. Rather than regularize the nonlinear friction term as done previously in Lapusta and Rice (2003) and Szkutnik et al (2003), it is possible to numerically integrate the system without an approximation to the friction term. There are also empirical results from the laboratory, obtained by Drummond (1999), whose friction measurements suggest the presence of a strange attractor.…”