Melnikov-Type Chaos of Planar Systems with Two Discontinuities

Abstract

In this paper, the chaotic behavior of a driven planar system with two discontinuous terms and a pseudo-equilibrium point in the intersection of the discontinuity surfaces is analyzed. This scenario is not covered by smooth techniques of chaos analysis or other techniques like the extension of Melnikov's method for nonsmooth systems. In consequence, we propose to use an approximate model of the discontinuous system for which this technique can be applied, and compare the responses of both systems, the discontinuous and the approximate, when this last model is close, in a certain way, to the discontinuous system. One of the discontinuous terms, given by a sign function, is approximated by a saturation with high slope at the equilibrium point. Some conditions that determine the chaotic behavior of the approximate system are formally established, and the convergence of its chaotic orbits to some orbits of the discontinuous system, when the slope of the approximation is large enough, is shown. In particular, we show the similarity of the dynamical behavior of both systems where a chaotic behavior can be displayed, for a parameter region determined by the application of the Melnikov technique to nonsmooth systems. A comparison of the Feigenbaum diagrams, for a parameter range obtained from the application of this technique, shows the similarity of their dynamics and the chaotic nature of the discontinuous system.