A cellular-automata, slider-block model for earthquakes I. Demonstration of chaotic behaviour for a low-order system

Abstract

SUMMARY We consider a pair of slider blocks connected to each other and to a constant velocity driver by springs. Stick-slip behaviour is obtained using static and dynamic coefficients of friction. With any asymmetry, numerical calculations show that this system can exhibit classical chaotic behaviour. We simplify the behaviour of the system by assuming that the two blocks never slide together. One block may trigger the slip of the other, but its slip is delayed until the first block sticks. The orbits in phase space are periodic in the symmetrical case but the orbit the system evolves to depends on the initial conditions. If the friction coefficients of the blocks are different, we can have chaotic behaviour with positive Lyapunov exponents for some parameter values, and periodic or limit cycles orbits for other values. In some cases we have limit cycle orbits for one interval of initial conditions and chaotic behaviour for others. The Poincark map for the model is a piecewise linear function and is strongly dependent on the parameters of the model. The behaviour of the simplified single slip model is quite similar to the original multiple slip model. The advantage of the simplified model is that the solution is algebraic and it can form the basis for a cellular-automata model which is completely deterministic.