The train model of two blocks with stick-slip dynamics (M. de Sousa Vieira, 1995) is believed to be the simplest spring-block system, which displays chaos. Here we simplify it even more by linearizing the velocity dependence of the friction force. In this way, the nonlinearity of the equations of motion is reduced to the time moments, when a block starts to move or stops, and when the analytical solutions are to be sewn together. We demonstrate, that for small values of the velocity of blocks, the character of motion is not changed. This is observed on the bifurcation diagrams, the Lyapunov exponents, the phase portraits and the power spectra.

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.

Recently, synchronization in the Burridge-Knopoff model has been found to depend on the initial conditions. Here we report the existence of three modes of oscillations of the system of three blocks. In one of the modes, two lateral blocks are synchronized. In the second mode, the central block moves with almost constant velocity, i.e., it does not stick. Two lateral blocks do stick and they move in opposite phases. In the third mode, the blocks oscillate with aperiodic amplitude. The lateral blocks move in opposite phases and their frequency is lower than the one for the central block. The mode selected by the system depends on the initial conditions. Numerical results indicate that there is no modes in the phase space.