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Partial barriers to chaotic transport in 4D symplectic maps
Markus Firmbach1, Arnd Bäcker1, Roland Ketzmerick1
1Institut für Theoretische Physik and Center for Dynamics, Technische Universität Dresden, 01062 Dresden, Germany.
Chaos (Woodbury, N.Y.)
|February 1, 2023
Summary
Researchers studied chaotic transport in Hamiltonian systems, introducing a cantorus-normally hyperbolic invariant manifold (NHIM) as a partial barrier in 4D maps. They quantified flux across this barrier, relevant for understanding Arnold diffusion.
Area of Science:
- Dynamical systems
- Chaos theory
- Mathematical physics
Background:
- Chaotic transport in Hamiltonian systems is often limited by partial barriers, restricting phase space flux.
- In 2D maps, cantori (Cantor set remnants of broken tori) form the most restrictive partial barriers.
- Understanding these barriers is crucial for predicting long-term system behavior.
Purpose of the Study:
- To establish and analyze a novel partial barrier, the cantorus-normally hyperbolic invariant manifold (NHIM), in 4D symplectic maps.
- To quantify the global and local flux across this cantorus-NHIM barrier.
- To investigate the role of periodic NHIMs in approximating flux and understanding Arnold diffusion.
Main Methods:
- Establishing a cantorus-NHIM as a partial barrier in a 4D symplectic map.
- Utilizing a flux formula to determine global 4D flux by approximating the cantorus-NHIM with high-order periodic NHIMs.
- Introducing and analyzing a local 3D flux along resonance channels.
Main Results:
- The global 4D flux across the cantorus-NHIM was determined using periodic NHIM approximations.
- A local 3D flux, dependent on position along resonance channels, was introduced and analyzed.
- The flux across barriers formed by stable and unstable manifolds of a NHIM was quantified using periodic NHIMs.
Conclusions:
- The cantorus-NHIM serves as a significant partial barrier in 4D Hamiltonian systems.
- The developed flux formulas provide a method for quantifying transport across complex barriers.
- These findings offer insights into the dynamics of chaotic transport and slow Arnold diffusion.
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