Relative Periodic Solutions of the Complex Ginzburg--Landau Equation

Abstract

A method of finding relative periodic orbits for differential equations with continuous symmetries is described and its utility demonstrated by computing relative periodic solutions for the one-dimensional complex Ginzburg-Landau equation (CGLE) with periodic boundary conditions. A relative periodic solution is a solution that is periodic in time, up to a transformation by an element of the equation's symmetry group. With the method used, relative periodic solutions are represented by a space-time Fourier series modified to include the symmetry group element and are sought as solutions to a system of nonlinear algebraic equations for the Fourier coefficients, group element, and time period. The 77 relative periodic solutions found for the CGLE exhibit a wide variety of temporal dynamics, with the sum of their positive Lyapunov exponents varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8. Preliminary work indicates that weighted averages over the collection of relative periodic solutions accurately approximate the value of several functionals on typical trajectories.

On the use of stabilizing transformations for detecting unstable periodic orbits in high-dimensional flows

“…Note that lmder is an implementation of the Levenburg-Marquardt algorithm [16]. Both methods have been successfully applied to spatially extended systems in order to detect periodic orbits in the past, in particular, in [26] the NA algorithm was able to detect many distinct UPOs of the KSE, whilst more recently, lmder has been used to determine many UPOs of the closely related complex Ginzburg-Landau equation [14].…”

“…Crucially, the number of searches which failed increases considerably as we look at larger system size, therefore, as we move to more complicated systems we would expect to see this difference in performance further increase. For example, in [14] Lopez et al use lmder to search for relative periodic orbits in the complex Ginzburg-Landau equation, where they have augmented the system with three additional equations. Our results suggest that this search would have benefited, not only in performance and numbers of UPOs detected, but by the savings in both time and effort required to construct additional equations and the resulting Jacobian, by setting all additional equations identically equal to zero.…”

Canonical methods of constructing invariant tori by phase–space sampling

“…The same partial derivatives are also needed when solving the model angles from Eq. (8). We obtain ∂u/∂J by inverting ∂(ϑ, J )/∂u.…”

“…Lan et al [6] provide a useful list of references related to the topic. Numerical methods based on Fourier series are used to study particle physics [7], superconductivity [8], quantum dynam-ics [9], celestial mechanics [10], etc. Often, these kinds of methods are general in nature, and many are also introduced as such [6,11].…”

Numerical continuation of invariant solutions of the complex Ginzburg–Landau equation

“…(For details see, for example, [2,21,25,34] and references therein.) Following [23], we consider here the CGLE with cubic nonlinearity in one spatial dimension,…”