Multistability and chaos in a spring-block model

A simple spring-block model with two degrees of freedom capable of producing chaotic dynamics is studied; The stability properties of fixed points and symmetry properties of solutions are analyzed. Frequency-entrained oscillations are investigated by means of an averaging procedure, in the vicinity of a principal resonance. An approximate analytical condition of asymmetric attractors arising, which can be used as a precursor of the appearance of chaotic motions, is derived from the analysis of the fixed points of averaged equations. Multistability of the averaged system is studied in detail and is shown to be typical for the present model. Extensive computer experiments carried out within a broad range of control parameters demonstrate statisfactory agreement between theoretical predictions and numerical simulations. The calculated results are given in the form of two-parameter bifurcation diagrams, along with phase portraits of different coexisting attractors. Many features, characteristics of nonlinear oscillators, were observed, including period-doubling cascades, hysteresis, intermittency, and crises of chaotic attractors. The system is strongly multistable in the sense that usually a regular attractor coexists with a chaotic or quasiperiodic one. In this work we use a modified velocity-weakening friction law that admits the presence of creeping motion. The introduction of a small region of creeping motion into the velocity-friction relation is demonstrated to bring no principally-new phenomena. Chaotic areas in control parameter space, however, broaden with the increase in the size of the creeping region.