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The three-dimensional generalized Hénon map: Bifurcations and attractors.

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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.

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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.