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

N.B. Budanur, P. Cvitanović, Journal of Statistical Physics 167 (2017) 636–655.

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Journal Article | Published | English

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Author
Budanur, Nazmi BIST Austria; Cvitanović, Predrag
Department
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.
Publishing Year
Date Published
2017-05-01
Journal Title
Journal of Statistical Physics
Acknowledgement
This work was supported by the family of late G. Robinson, Jr. and NSF Grant DMS-1211827.
Volume
167
Issue
3-4
Page
636-655
IST-REx-ID

Cite this

Budanur NB, Cvitanović P. Unstable manifolds of relative periodic orbits in the symmetry reduced state space of the Kuramoto–Sivashinsky system. Journal of Statistical Physics. 2017;167(3-4):636-655. doi:10.1007/s10955-016-1672-z
Budanur, N. B., & Cvitanović, P. (2017). Unstable manifolds of relative periodic orbits in the symmetry reduced state space of the Kuramoto–Sivashinsky system. Journal of Statistical Physics, 167(3–4), 636–655. https://doi.org/10.1007/s10955-016-1672-z
Budanur, Nazmi B, and Predrag Cvitanović. “Unstable Manifolds of Relative Periodic Orbits in the Symmetry Reduced State Space of the Kuramoto–Sivashinsky System.” Journal of Statistical Physics 167, no. 3–4 (2017): 636–55. https://doi.org/10.1007/s10955-016-1672-z.
N. B. Budanur and P. Cvitanović, “Unstable manifolds of relative periodic orbits in the symmetry reduced state space of the Kuramoto–Sivashinsky system,” Journal of Statistical Physics, vol. 167, no. 3–4, pp. 636–655, 2017.
Budanur NB, Cvitanović P. 2017. Unstable manifolds of relative periodic orbits in the symmetry reduced state space of the Kuramoto–Sivashinsky system. Journal of Statistical Physics. 167(3–4), 636–655.
Budanur, Nazmi B., and Predrag Cvitanović. “Unstable Manifolds of Relative Periodic Orbits in the Symmetry Reduced State Space of the Kuramoto–Sivashinsky System.” Journal of Statistical Physics, vol. 167, no. 3–4, Springer, 2017, pp. 636–55, doi:10.1007/s10955-016-1672-z.
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