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The three-dimensional generalized Hénon map: Bifurcations and attractors.
Amanda E Hampton1, James D Meiss1
1Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309-0526, USA.
Chaos (Woodbury, N.Y.)
|December 1, 2022
Summary
We explored bifurcations in a 3D Hénon map generalization. Dissipative, orientation-preserving dynamics reveal Arnold tongues and diverse chaotic attractors, including Hénon-like and Lorenz-like types.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Mathematical Physics
Background:
- The Hénon map is a foundational model in studying chaotic dynamics.
- Understanding bifurcations is crucial for predicting complex system behavior.
- Generalizing low-dimensional maps to higher dimensions reveals new phenomena.
Purpose of the Study:
- To conduct a comprehensive parameter study of bifurcations in a 3D quadratic diffeomorphism.
- To investigate the emergence and characteristics of periodic orbits and aperiodic attractors.
- To classify different types of chaotic orbits, such as Hénon-like and Lorenz-like attractors.
Main Methods:
- Analysis of codimension-one and two bifurcations in a dissipative, orientation-preserving quadratic map.
- Identification of periodic orbits through resonant Neimark-Sacker bifurcations.
- Characterization of aperiodic attractors using rotation numbers and Lyapunov exponents.
Main Results:
- Periodic orbits form Arnold tongues in parameter space, originating from Neimark-Sacker bifurcations.
- Aperiodic attractors include invariant circles and chaotic orbits.
- Chaotic orbits exhibit Hénon-like and Lorenz-like behaviors, arising from period-doubling or invariant circle destruction.
Conclusions:
- The 3D quadratic diffeomorphism displays rich dynamics, generalizing features of the 2D Hénon map.
- Bifurcation analysis provides a framework for understanding the transition to chaos.
- The study classifies distinct routes to chaos and associated attractor structures.
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