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This study explores Shilnikov attractors in 3D maps, revealing three distinct types: hyperchaotic, flow-like, and strongly dissipative, based on parameter variations. The findings highlight differences in orientation-reversing versus orientation-preserving systems.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • Shilnikov homoclinic attractors feature saddle-focus fixed points and homoclinic orbits.
  • Orientation-reversing diffeomorphisms introduce unique symmetries in stable manifolds compared to orientation-preserving ones.

Purpose of the Study:

  • To analyze the behavior of Shilnikov attractors in a 3D orientation-reversing map.
  • To classify different types of Shilnikov attractors based on Lyapunov exponents.
  • To investigate the formation of these attractors through parameter variations.

Main Methods:

  • Utilized the 3D Mirá map (x¯=y, y¯=z, z¯=Bx+Cy+Az-y²) with a negative Jacobian (B<0).
  • Analyzed the properties of Shilnikov attractors by varying parameters A, B, and C.
  • Calculated Lyapunov exponents to classify attractor types.

Main Results:

  • Identified three types of Shilnikov attractors: hyperchaotic (2 positive, 1 negative Lyapunov exponent), flow-like (1 positive, 1 near-zero, 1 negative Lyapunov exponent), and strongly dissipative (1 positive, 2 negative Lyapunov exponents).
  • Demonstrated that the type of attractor depends on the specific parameter values (A, B, C).
  • Studied the transition scenarios between these attractor types within one-parameter families.

Conclusions:

  • The orientation-reversing property significantly impacts Shilnikov attractor dynamics.
  • The Mirá map serves as a fundamental model for understanding diverse Shilnikov attractor behaviors.
  • Parameter control allows for the emergence of distinct chaotic and dissipative dynamics.