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. Its 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.

18 pages, 7 figures

References in corpus (5)

• Relative periodic orbits form the backbone of turbulent pipe flow
• Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics
• Reduction of SO(2) symmetry for spatially extended dynamical systems
• Continuous symmetry reduction and return maps for high-dimensional flows
• Periodic eigendecomposition and its application to Kuramoto-Sivashinsky system

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• Data-driven state-space and Koopman operator models of coherent state dynamics on invariant manifolds
• Complexity of the laminar-turbulent boundary in pipe flow
• Inferring symbolic dynamics of chaotic flows from persistence
• Intermittent direction reversals of moving spatially-localized turbulence observed in two-dimensional Kolmogorov flow
• Origin of Jumping Oscillons in an Excitable Reaction-Diffusion System
• Predicting chaotic statistics with unstable invariant tori
• Stochastic Latent Transformer: Efficient Modelling of Stochastically Forced Zonal Jets
• Computing Chaotic Time-Averages from Few Periodic or Non-Periodic Orbits
• Dynamical relevance of periodic orbits under increasing Reynolds number and connections to inviscid dynamics