Visualizing Attractors of the Three-Dimensional Generalized H\'{e}non Map

TL;DR

This paper explores the complex dynamics of a 3D quadratic map, revealing various attractors, bifurcations, and the structure of chaotic and periodic behaviors through a detailed parameter study.

Contribution

It provides a comprehensive analysis of bifurcations, attractors, and chaotic regimes in a 3D generalization of the Hénon map, including new insights into invariant circles and attractor structures.

Findings

Identification of Arnold tongues in parameter space

Classification of chaotic attractors as Hénon-like and Lorenz-like

Description of bifurcation scenarios leading to chaos

Abstract

We study dynamics of a generic quadratic diffeomorphism, a 3D generalization of the planar H\'{e}non map. Focusing on the dissipative, orientation preserving case, we give a comprehensive parameter study of codimension-one and two bifurcations. Periodic orbits, born at resonant, Neimark-Sacker bifurcations, give rise to Arnold tongues in parameter space. Aperiodic attractors include invariant circles and chaotic orbits; these are distinguished by rotation number and Lyapunov exponents. Chaotic orbits include H\'{e}non-like and Lorenz-like attractors, which can arise from period-doubling cascades, and those born from the destruction of invariant circles. The latter lie on paraboloids near the local unstable manifold of a fixed point.