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Integrable approximation of regular islands: the iterative canonical transformation method.
Clemens Löbner1, Steffen Löck2, Arnd Bäcker1
1Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany and Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 4, 2014
Summary
Researchers developed an iterative method to approximate the regular dynamics of Hamiltonian systems. This technique extends regular motion into chaotic regions, applicable to complex systems with multiple degrees of freedom.
Area of Science:
- Mathematical Physics
- Dynamical Systems Theory
- Nonlinear Dynamics
Background:
- Generic Hamiltonian systems exhibit mixed phase spaces, featuring coexisting regular and chaotic dynamics.
- Understanding the interplay between regular and chaotic motion is crucial in various physics domains.
Purpose of the Study:
- To develop an iterative method for constructing an integrable approximation of generic Hamiltonian systems.
- To extend the regular dynamics of a given system into its classically chaotic regions.
Main Methods:
- Construction of an integrable approximation in action representation.
- Iterative refinement of the approximation in phase space using canonical transformations.
- Application to systems with arbitrary degrees of freedom, including strongly perturbed ones.
Main Results:
- Successfully developed an iterative method (H(reg)) that approximates regular dynamics.
- The method effectively extends regular motion into chaotic regions of the phase space.
- Demonstrated applicability to the standard map and the cosine billiard.
Conclusions:
- The proposed iterative method provides a powerful tool for analyzing mixed Hamiltonian systems.
- This approach offers insights into the dynamics of systems previously intractable due to strong perturbations or complexity.
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