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arXiv · 2206.07855

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

Amanda E Hampton·James D Meiss
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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.

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Amanda E Hampton, James D Meiss. 2022-06-16. Visualizing Attractors of the Three-Dimensional Generalized H\'{e}non Map. https://doi.org/10.1063/5.0103436

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