Unstable manifolds of relative periodic orbits in the symmetry-reduced state space of the Kuramoto-Sivashinsky system

Abstract

Systems such as fluid flows in channels and pipes or the complex Ginzburg-Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto-Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.

23 pages, 9 figures

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• Symmetry-reduced Dynamic Mode Decomposition of Near-wall Turbulence
• Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states
• Heteroclinic path to spatially localized chaos in pipe flow
• Inferring symbolic dynamics of chaotic flows from persistence
• Symmetry reduction for deep reinforcement learning active control of chaotic spatiotemporal dynamics
• Dynamics-based machine learning of transitions in Couette flow
• Natural convection in a vertical channel. Part 3. Bifurcations of many (additional) unstable periodic orbits and their dynamical relevance