Title: Periodic orbit analysis of a system with continuous symmetry—A tutorial

Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid in terms of a Fourier series truncated to a finite number of modes. Here, we study a 4-dimensional model with chaotic dynamics and SO(2) symmetry similar to those that appear in fluid dynamics problems. A crucial step in the analysis of such a system is symmetry reduction. We use the model to illustrate different symmetry-reduction techniques. The system's relative equilibria are conveniently determined by rewriting the dynamics in terms of a symmetry-invariant polynomial basis. However, for the analysis of its chaotic dynamics, the “method of slices,” which is applicable to very high-dimensional problems, is preferable. We show that a Poincaré section taken on the 'slice' can be used to further reduce this flow to what is for all practical purposes a unimodal map. This enables us to systematically determine all relative periodic orbits and their symbolic dynamics up to any desired period. We then present cycle averaging formulas adequate for systems with continuous symmetry and use them to compute dynamical averages using relative periodicmore » orbits. The convergence of such computations is discussed.« less

School of Physics and Center for Nonlinear Science, Georgia Institute of Technology, Atlanta, Georgia 30332 (United States)

(United States)

Publication Date:

OSTI Identifier:

22483217

Resource Type:

Journal Article

Resource Relation:

Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 25; Journal Issue: 7; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

Country of Publication:

United States

Language:

English

Subject:

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CALCULATION METHODS; CHAOS THEORY; FLUID MECHANICS; FLUIDS; NAVIER-STOKES EQUATIONS; ORBITS; PERIODICITY; POLYNOMIALS; SO-2 GROUPS