Periodic orbit analysis of a system with continuous symmetry--A tutorial

Area of Science:

• Applied Mathematics
• Chaos Theory
• Fluid Dynamics

Background:

• Dynamical systems with continuous symmetries (translational/rotational) are common in physics, e.g., Navier-Stokes flows.
• Analyzing these systems often involves Fourier series truncation and symmetry reduction.
• A 4D model with SO(2) symmetry, relevant to fluid dynamics, is used for illustration.

Purpose of the Study:

• To illustrate and compare different symmetry-reduction techniques for chaotic dynamical systems.
• To analyze the chaotic dynamics of a model system with SO(2) symmetry.
• To systematically determine relative periodic orbits and compute dynamical averages.

Main Methods:

• Symmetry reduction using a symmetry-invariant polynomial basis for relative equilibria.
• Application of the 'method of slices' for analyzing chaotic dynamics in high-dimensional systems.
• Poincaré sections on slices to reduce the flow to a unimodal map for orbit analysis.

Main Results:

• Demonstrated effective symmetry reduction for analyzing chaotic dynamics.
• Systematically determined relative periodic orbits and their symbolic dynamics.
• Presented cycle averaging formulas for computing dynamical averages in systems with continuous symmetry.

Conclusions:

• The 'method of slices' combined with Poincaré sections is a powerful tool for analyzing chaotic dynamics and determining periodic orbits.
• Cycle averaging formulas provide a method for computing dynamical averages using relative periodic orbits.
• The study offers a systematic approach to understanding complex dynamical systems with continuous symmetry.