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Published on: May 27, 2012
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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.
Chaos (Woodbury, N.Y.)
|October 4, 2018
Summary
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.
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.
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