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Efficient manifolds tracing for planar maps.

David Ciro1, Iberê L Caldas2, Ricardo L Viana1

  • 1Department of Physics, Federal University of Paraná, Curitiba, Paraná 81531-990, Brazil.

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We developed new methods for tracing invariant manifolds of unstable periodic orbits. These techniques offer precise calculations and efficient approximations for analyzing complex dynamical systems.

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Area of Science:

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

Background:

  • Unstable periodic orbits are crucial for understanding the behavior of dynamical systems.
  • Invariant manifolds dictate the global dynamics and chaotic behavior.
  • Existing methods for tracing these manifolds can be computationally intensive and redundant.

Purpose of the Study:

  • To introduce an exact calculation method for tracing invariant manifolds.
  • To develop an efficient approximation technique for invariant manifold tracing.
  • To analyze the transition from homoclinic to heteroclinic chaos using these methods.

Main Methods:

  • An exact method utilizing an adaptive refinement procedure to avoid redundant calculations.
  • An approximation technique based on a novel interpolation approach using normal displacement functions.
  • Application to the Chirikov-Taylor map to obtain invariant manifolds.

Main Results:

  • The exact method provides precise calculations of invariant manifolds.
  • The approximated method achieves high precision comparable to the exact method.
  • The computational cost of the approximation scales inversely with manifold length.
  • Demonstrated the transition from homoclinic to heteroclinic chaos in the Duffing oscillator.

Conclusions:

  • The developed methods offer accurate and efficient tools for analyzing invariant manifolds.
  • These techniques facilitate the study of complex transitions in chaotic systems.
  • The findings contribute to a deeper understanding of localized and global chaotic motion.