Effect of stochastically moving border on basins of attraction in a class of piecewise smooth maps

Abstract

Determination of the basin of attraction of an attractor plays an important role in understanding the dynamics of a system. In all the existing literature, the basin of attraction of any attractor has been described to be deterministic. In this paper we show the existence of a non-deterministic basin of attraction of an attractor. We have considered a piecewise smooth (PWS) one-dimensional map, having stochastically varying border, which is allowed to move in a small bounded region of the phase space while retaining the deterministic dynamics on each compartment of the phase space. In case of this type of systems there exists a region in the phase space with the property that orbits starting from a single point lying inside this region do not display the same property of convergence or divergence, i.e., one may converge while another may diverge. In other words, the convergence or divergence of an orbit starting from a point inside this region is a probabilistic event, and the probabilities of convergence and divergence are both non-zero. We also derive the upper and lower bounds of the corresponding probability curves. Since all physical systems contain noise, the occurrence of such non-deterministic basin of attraction is a definite possibility, if the noise affects the position of the border. This may lead to dangerous consequences, as a region of the basin of attraction of an attractor may become non-deterministic, with a non-zero probability of divergence of orbits starting inside it.