Existence proofs for strange attractors in piecewise-smooth nonlinear Lozi-Hénon and Belykh maps

Abstract

ABSTRACT Extensive research has previously been carried out on the existence of strange attractors in 2D piecewise linear maps, including the renowned Lozi map. However, the rigorous analysis of strange attractors in 2D nonlinear maps remains an underdeveloped area of study. In this paper, we introduce a 2D map with a single piecewise-smooth nonlinear function that is suitable for analytic studies of its strange attractor. This 2D piecewise-smooth nonlinear map represents a broad range of chaotic maps, including a hybrid Lozi-Hénon map and the Belykh map. To prove the existence of a strange attractor in the 2D map, we (i) construct its trapping region that contains all limit sets and (ii) demonstrate that the invariant set's trajectories have negative and positive Lyapunov exponents. We develop an invariant cone approach to establish the latter property, which involves constructing expanding and contracting cones as bounds for the eigenvectors of the variational equations along the chaotic trajectories. We apply our approach to analyze the chaotic hybrid Lozi-Hénon map, the original Lozi map, and the Belykh map.