Coexistence of Thread and Sheet Chaotic Attractors for Three-Dimensional Lozi Map

Abstract

Since its original publication in 1978, Lozi’s chaotic map has been thoroughly explored and continues to be. Hundreds of publications have analyzed its particular structure or applied its properties in many fields (electronic devices such as memristors, A.I. with swarm intelligence, etc.). Several generalizations have been proposed, transforming the initial two-dimensional map into a multidimensional one. However, they do not respect the original constraint that allows this map to be one of the few strictly hyperbolic: a constant Jacobian. In this paper, we introduce a three-dimensional piece-wise linear extension respecting this constraint and we explore a special property never highlighted for chaotic mappings: the coexistence of thread chaotic attractors (i.e., attractors that are formed by a collection of lines) and sheet chaotic attractors (i.e., attractors that are formed by a collection of planes). This new three-dimensional mapping can generate a large variety of chaotic and hyperchaotic attractors. We give five examples of such behavior in this article. In the first three examples, there is the coexistence of thread and sheet chaotic attractors. However, their shapes are different and they are constituted by a different number of pieces. In the last two examples, the blow up of the attractors with respect to parameter a and b is highlighted.