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

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.

KEYWORDS:

• Dynamical systems
• nonlinear maps
• attractors
• hyperbolicity
• bifurcations

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 34C37
• 34C23
• 34D45
• 34C25
• 34C28

Acknowledgments

We are thankful to René Lozi for the invitation to contribute to this special issue. We are also grateful to Igor Belykh for helpful discussions and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation under Project No. 0729-2020-0036 and the Russian Science Foundation under Grant No. 22-21-00553.