No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

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

### Abstract

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 periodic orbits. The convergence of such computations is discussed.

© 2015 AIP Publishing LLC

Received 11 February 2015
Accepted 22 June 2015
Published online 14 July 2015

Lead Paragraph:
Periodic orbit theory provides a way to compute dynamical averages for chaotic flows by means of cycle averaging formulas that relate the time averages of observables to the spectra of unstable periodic orbits. Standard cycle averaging formulas are valid under the assumption that the stability multipliers of all periodic orbits have a single marginal direction corresponding to time evolution and are hyperbolic in all other directions. However, if a dynamical system has *N* continuous symmetries, periodic orbits are replaced by relative periodic orbits, invariant (*N* + 1)-dimensional tori with marginal stability in (*N* + 1) directions. Such exact invariant solutions arise in studies of turbulent flows, such as pipe flow or plane Couette flow, which have continuous symmetries. In practice, the translational invariance of these flows is approximated in numerical simulations by using periodic domains so that the state of the fluid is conveniently expressed as a Fourier series, truncated to a large but finite number (from tens to thousands) of Fourier modes. This paper is a tutorial on how such problems can be analyzed using periodic orbit theory. We illustrate all the necessary steps using a simple “two-mode” model as an example.

Acknowledgments:
We are grateful to Evangelos Siminos for his contributions to this project and Mohammad M. Farazmand for a critical reading of the manuscript. We acknowledge stimulating discussion with Xiong Ding, Ruslan L. Davidchack, Ashley P. Willis, Al Shapere, and Francesco Fedele. We are indebted to the 2012 ChaosBook.org class, in particular, to Keith M. Carroll, Sarah Flynn, Bryce Robbins, and Lei Zhang, for the initial fearless fishing expeditions into the enormous sea of parameter values of the two-mode model. P.C. thanks the family of late G. Robinson, Jr. and NSF DMS-1211827 for support. D.B. thanks M. F. Schatz for support during the early stages of this work under NSF CBET-0853691.

Article outline:

I. INTRODUCTION
II. CONTINUOUS SYMMETRIES
A. Method of slices
B. First Fourier mode slice
C. Geometric interpretation of the first Fourier mode slice
III. TWO-MODE SO(2)-EQUIVARIANT FLOW
A. Invariant polynomial bases
B. Equilibria of the symmetry-reduced dynamics
C. No chaos when the reflection symmetry is restored
D. Visualizing two-mode dynamics
IV. PERIODIC ORBITS
V. CYCLE AVERAGES
A. Classical trace formula
B. Decomposition of the trace formula over irreducible representations
C. Cycle expansions
D. Numerical results
VI. CONCLUSIONS AND DISCUSSION

/content/aip/journal/chaos/25/7/10.1063/1.4923742

http://aip.metastore.ingenta.com/content/aip/journal/chaos/25/7/10.1063/1.4923742

Article metrics loading...

/content/aip/journal/chaos/25/7/10.1063/1.4923742

2015-07-14

2016-10-25

Full text loading...

###
Most read this month

Article

content/aip/journal/chaos

Journal

5

3

Commenting has been disabled for this content