Nonlinear dynamics behind the seismic cycle: One-dimensional phenomenological modeling

In present paper, authors examine the dynamics of a spring-slider model, considered as a phenomenological setup of a geological fault motion. Research is based on an assumption of delayed interaction between the two blocks, which is an idea that dates back to original Burridge-Knopoff model. In contrast to this first model, group of blocks on each side of transmission zone (with delayed interaction) is replaced by a single block. Results obtained indicate predominant impact of the introduced time delay, whose decrease leads to transition from steady state or aseismic creep to seismic regime, where each part of the seismic cycle (co-seismic, post-seismic and inter-seismic) could be recognized. In particular, for coupling strength of order 102 observed system exhibit inverse Andronov-Hopf bifurcation for very small value of time delay, τ≈0.01, when long-period (T = 12) and high-amplitude oscillations occur. Further increase of time delay, of order 10−1, induces an occurrence of a direct Andronov-Hopf bifurcation, with short-period (T = 0.5) oscillations of approximately ten times smaller amplitude. This reduction in time delay could be the consequence of the increase of temperature due to frictional heating, or due to decrease of pressure which follows the sudden movement along the fault. Analysis is conducted for the parameter values consistent with previous laboratory findings and geological observations relevant from the seismological viewpoint.