Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 636–655 | Cite as

Unstable Manifolds of Relative Periodic Orbits in the Symmetry-Reduced State Space of the Kuramoto–Sivashinsky System

  • Nazmi Burak Budanur
  • Predrag Cvitanović


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.


Kuramoto–Sivashinsky equation Equivariant systems Relative periodic orbits Unstable manifolds Chaos Symmetries 



This work was supported by the family of late G. Robinson, Jr. and NSF Grant DMS-1211827. We are grateful to Xiong Ding, Evangelos Siminos, Simon Berman, and Mohammad Farazmand for many fruitful discussions.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Science and Technology (IST)KlosterneuburgAustria
  2. 2.School of Physics, Center for Nonlinear ScienceGeorgia Institute of TechnologyAtlantaUSA

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