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Integrable approximation of regular regions with a nonlinear resonance chain
Julius Kullig1, Clemens Löbner2, Normann Mertig3
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 and Institut für Theoretische Physik, Universität Magdeburg, Postfach 4120, 39016 Magdeburg, Germany.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 11, 2014
Summary
Researchers developed a new method to approximate regular regions in mixed Hamiltonian systems. This technique, using iterative canonical transformations, accurately models nonlinear resonance chains and regular tori shapes.
Area of Science:
- Nonlinear dynamics
- Hamiltonian systems
- Mathematical physics
Background:
- Generic Hamiltonian systems exhibit mixed phase space, featuring coexisting regions of regular and chaotic motion.
- Understanding and approximating these regular regions is crucial for analyzing complex dynamical systems.
Purpose of the Study:
- To present a generalized method for constructing an integrable approximation of regular phase-space regions in generic Hamiltonian systems.
- To incorporate nonlinear resonance chains into the integrable approximation.
Main Methods:
- The method generalizes the iterative canonical transformation technique.
- Step 1: Adapt a normal-form Hamiltonian with a resonance chain to match actions and frequencies of the nonintegrable system.
- Step 2: Apply a sequence of canonical transformations to the integrable approximation to precisely match the shape of regular tori.
Main Results:
- The proposed method successfully constructs integrable approximations for regular phase-space regions.
- Demonstrated effectiveness on the generic standard map across various parameters.
- The method accurately captures the dynamics of nonlinear resonance chains and the geometry of regular tori.
Conclusions:
- The developed method provides a powerful tool for analyzing and approximating regular dynamics in mixed Hamiltonian systems.
- This approach offers a systematic way to simplify complex systems by approximating their regular behaviors.
- The generalization of iterative canonical transformations enhances the study of nonlinear resonance phenomena.
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