Widening of the basins of attraction of a multistable switching dynamical system with the location of symmetric equilibria

Abstract

"A switching dynamical system by means of piecewise linear systems in that presents multistability is presented. The flow of the system displays multi-scroll attractors due to the unstable hyperbolic focus-saddle equilibria with stability index of type I, i.e., a negative real eigenvalue and a pair of complex conjugated eigenvalues with positive real part. This class of systems is constructed by a discrete control mode changing the equilibrium point regarding the location of their states. The scrolls appear when the stable and unstable eigenspaces of each adjacent equilibrium point generate the stretching and folding mechanisms needed in chaos, i.e., the unstable manifold in the first subsystem carries the trajectory towards the stable manifold of the immediate adjacent subsystem. The resulting attractors are located around four focus saddle equilibria. If the equilibria are located symmetrically to one of the axes and the distance between each equilibria is properly adjusted to generate two double-scroll chaotic attractors, the system can present from bistable to multistable parallel solutions regarding the position of their initial states. In addition the resulting basin of attraction presents a significatively widening when the distance between the equilibria of the parallel attractors is displaced."