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.

Appendix: Computational Details

Throughout this paper, we used the 16 Fourier mode truncation of Kuramoto–Sivashinsky equation (2), which renders the state space 30-dimensional. Sufficiency of this truncation was demonstrated for $L=22$ in Ref. [17]. In all our computations, we integrate (12) and its gradient system numerically, using a general purpose adaptive integrator odeint from scipy.integrate [35], which is a wrapper of lsoda from ODEPACK library [32]. Note that (12) is singular if $\hat{b}_1 = 0$, i.e., whenever the first Fourier mode vanishes. This singularity can be regularized by a time-rescaling if a fixed time step integrator is desired [11].

Transformation of trajectories and tangent vectors to the fully symmetry-reduced state space (17) is applied as post-processing. For a trajectory $\hat{a}(\tau )$, we simply apply the reflection reducing transformation to obtain the trajectory as $\tilde{a}(\tau ) = \tilde{a}(\hat{a}(\tau ))$. Velocity field (12) transforms to (17) by acting with the Jacobian matrix

$$\begin{aligned} \tilde{v}(\tilde{a}) = \frac{d \tilde{a}(\hat{a})}{d \hat{a}}\hat{v}(\hat{a}) \,. \end{aligned}$$

Floquet vectors transform to the fully symmetry-reduced state space similarly, however, their computations in the first Fourier mode slice requires some care. Remember that the reflection symmetry remains after the continuous symmetry reduction, and its action is represented by (13). Thus, denoting finite time flow induced by (12) by ${\hat{f}^{\tau }(\hat{a})}$, pre-periodic orbit within the slice satisfies

$$\begin{aligned} \hat{a}_{pp}= \hat{{D}}(\sigma ) {\hat{f}^{{T_{p}}}(\hat{a}_{pp})} \,, \end{aligned}$$

with its linear stability given by the spectrum of the Jacobian matrix

$$\begin{aligned} \hat{J}_{\!_{pp}} = \hat{{D}}(\sigma ) \hat{J}^{{T_{p}}} (\hat{a}_{pp}) \,, \end{aligned}$$

where $\hat{J}^{{T_{p}}} (\hat{a}_{pp})$ is the Jacobian matrix of the flow function ${\hat{f}^{{T_{p}}}(\hat{a}_{pp})}$. Thus, in order to find the Floquet vectors in fully symmetry-reduced representation, we first find the eigenvectors $\hat{V}$ of the Jacobian matrix $\hat{J}_{\!_{pp}}$ and then transform them as $\tilde{V}(\tilde{a}) = {d \tilde{a}(\hat{a}_{pp})}/{d \hat{a}}\,\hat{V}(\hat{a}) \,.$