Algorithm for finding horseshoes in three-dimensional hyperchaotic maps and its application

Abstract

Topological horseshoe theory is fundamental for studying chaos rigorously, which, however, has rarely applied to hyperchaos. The reason is that it is too hard to find a topological horseshoe in a hyperchaotic system, due to the high dimension of the system and the multiple expansion directions in the state space. Therefore, in this paper a practical algorithm for three-dimensional (3D) hyperchaotic maps is proposed. Usually, a hyperchaotic system has a large negative Lyapunov exponent, its attractor is often contracted closely to a certain surface. Based on this feature, the algorithm first deducts the dimension along the direction of contraction to obtain a 2D projective system; then it detects a projective horseshoe with 2D expansion; finally, it constructs a 3D horseshoe for the original system. In order to verify the validity of the algorithm, it is applied to the classic hyperchaotic Lorenz system and the famous Saito hyperchaotic circuit, and their horseshoes with 2D expansion are successfully found from the Poincaré mapping.