title: Reduction of SO(2) symmetry for spatially extended dynamical systems.
abstract: Spatially extended systems, such as channel or pipe flows, are often equivariant under continuous symmetry transformations, with each state of the flow having an infinite number of equivalent solutions obtained from it by a translation or a rotation. This multitude of equivalent solutions tends to obscure the dynamics of turbulence. Here we describe the "first Fourier mode slice," a very simple, easy to implement reduction of SO(2) symmetry. While the method exhibits rapid variations in phase velocity whenever the magnitude of the first Fourier mode is nearly vanishing, these near singularities can be regularized by a time-scaling transformation. We show that after application of the method, hitherto unseen global structures, for example, Kuramoto-Sivashinsky relative periodic orbits and unstable manifolds of traveling waves, are uncovered.
authors.affiliation: Center for Nonlinear Science, School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA.
authors.forename: Nazmi Burak
authors.initials: NB
authors.lastname: Budanur
authors.affiliation: Center for Nonlinear Science, School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA.
authors.forename: Predrag
authors.initials: P
authors.lastname: Cvitanović
authors.affiliation: Department of Mathematics, University of Leicester, Leicester LE1 7RH, United Kingdom.
authors.forename: Ruslan L
authors.initials: RL
authors.lastname: Davidchack
authors.affiliation: Max-Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187 Dresden, Germany.
authors.forename: Evangelos
authors.initials: E
authors.lastname: Siminos