article
Unstable manifolds of relative periodic orbits in the symmetry reduced state space of the Kuramoto–Sivashinsky system
published
yes
Nazmi B
Budanur
author 3EA1010E-F248-11E8-B48F-1D18A9856A870000-0003-0423-5010
Predrag
Cvitanović
author
BjHo
department
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.
https://research-explorer.ista.ac.at/download/1211/5319/IST-2017-782-v1+1_BudCvi15.pdf
application/pdfno
Springer2017
eng
Journal of Statistical Physics10.1007/s10955-016-1672-z
1673-4636-655
Budanur, Nazmi B, and Predrag Cvitanović. “Unstable Manifolds of Relative Periodic Orbits in the Symmetry Reduced State Space of the Kuramoto–Sivashinsky System.” <i>Journal of Statistical Physics</i>. Springer, 2017. <a href="https://doi.org/10.1007/s10955-016-1672-z">https://doi.org/10.1007/s10955-016-1672-z</a>.
Budanur, N. B., & Cvitanović, P. (2017). Unstable manifolds of relative periodic orbits in the symmetry reduced state space of the Kuramoto–Sivashinsky system. <i>Journal of Statistical Physics</i>. Springer. <a href="https://doi.org/10.1007/s10955-016-1672-z">https://doi.org/10.1007/s10955-016-1672-z</a>
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 NB, Cvitanović P. Unstable manifolds of relative periodic orbits in the symmetry reduced state space of the Kuramoto–Sivashinsky system. <i>Journal of Statistical Physics</i>. 2017;167(3-4):636-655. doi:<a href="https://doi.org/10.1007/s10955-016-1672-z">10.1007/s10955-016-1672-z</a>
N.B. Budanur, P. Cvitanović, Journal of Statistical Physics 167 (2017) 636–655.
N. B. Budanur and P. Cvitanović, “Unstable manifolds of relative periodic orbits in the symmetry reduced state space of the Kuramoto–Sivashinsky system,” <i>Journal of Statistical Physics</i>, vol. 167, no. 3–4. Springer, pp. 636–655, 2017.
Budanur, Nazmi B., and Predrag Cvitanović. “Unstable Manifolds of Relative Periodic Orbits in the Symmetry Reduced State Space of the Kuramoto–Sivashinsky System.” <i>Journal of Statistical Physics</i>, vol. 167, no. 3–4, Springer, 2017, pp. 636–55, doi:<a href="https://doi.org/10.1007/s10955-016-1672-z">10.1007/s10955-016-1672-z</a>.
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