Algebraic approach to chaos induced by snapback repeller

Abstract

In this poster we present the results of [1]. We consider the problem of detecting chaotic behaviors in discrete dynamical systems. We propose an algebraic criterion for determining whether all the zeros of a given polynomial are outside the unit circle in the complex plane. This criterion is used to deduce critical algebraic conditions for the occurrence of chaos in multi-dimensional discrete systems based on Marotto's theorem. Using these algebraic conditions we reduce the problem of analyzing chaos induced by snapback repeller to an algebraic problem, and introduce an algorithmic approach to solve this problem by means of symbolic computation. The proposed approach is effective as shown by several examples and can be used to determine the possibility that all the fixed points are snapback repellers.