Positive Topological Entropy of Chua's Circuit: A Computer Assisted Proof

Abstract

In this paper we describe a technique for proving that a particular system is chaotic in the topological sense, i.e. that it has positive topological entropy. This technique combines existence results based on the fixed point index theory and computer-assisted computations, necessary to verify the assumptions of the existence theorem. First we present an existence theorem for periodic points of maps, which could be appropriately homotoped with the deformed horseshoe map. As an example we consider Chua's circuit. We prove the existence of infinitely many periodic points of Poincaré map associated with Chua's Circuit. We also show how to use this result to prove that the topological entropy of the Poincaré map and also of the continuous system is positive.