dorsal/arxiv
View SchemaInstabilities of periodic orbits with spatio-temporal symmetries
| Authors | A. M. Rucklidge, Mary Silber |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9704002 |
| URL | https://arxiv.org/abs/patt-sol/9704002 |
| DOI | 10.1088/0951-7715/11/5/015 |
| Journal | Nonlinearity 11 (1998) 1435-1455 |
Abstract
Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare return map M to be the nth iterate of a map G: M=G^n. The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. When the bifurcation breaks these reflections, the map G has a "two-symmetry", as analysed by Lamb (1996). This leads to a doubling of the marginal Floquet multiplier and the possibility of bifurcation to two distinct types of drifting solutions.
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"abstract": "Motivated by recent analytical and numerical work on two- and\nthree-dimensional convection with imposed spatial periodicity, we analyse three\nexamples of bifurcations from a continuous group orbit of spatio-temporally\nsymmetric periodic solutions of partial differential equations. Our approach is\nbased on centre manifold reduction for maps, and is in the spirit of earlier\nwork by Iooss (1986) on bifurcations of group orbits of spatially symmetric\nequilibria. Two examples, two-dimensional pulsating waves (PW) and\nthree-dimensional alternating pulsating waves (APW), have discrete\nspatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and\nn=4 (APW). These symmetries force the Poincare return map M to be the nth\niterate of a map G: M=G^n. The group orbits of PW and APW are generated by\ntranslations in the horizontal directions and correspond to a circle and a\ntwo-torus, respectively. An instability of pulsating waves can lead to\nsolutions that drift along the group orbit, while bifurcations with Floquet\nmultiplier +1 of alternating pulsating waves do not lead to drifting solutions.\nThe third example we consider, alternating rolls, has the spatio-temporal\nsymmetry of alternating pulsating waves as well as being invariant under\nreflections in two vertical planes. When the bifurcation breaks these\nreflections, the map G has a \"two-symmetry\", as analysed by Lamb (1996). This\nleads to a doubling of the marginal Floquet multiplier and the possibility of\nbifurcation to two distinct types of drifting solutions.",
"arxiv_id": "patt-sol/9704002",
"authors": [
"A. M. Rucklidge",
"Mary Silber"
],
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"doi": "10.1088/0951-7715/11/5/015",
"journal_ref": "Nonlinearity 11 (1998) 1435-1455",
"title": "Instabilities of periodic orbits with spatio-temporal symmetries",
"url": "https://arxiv.org/abs/patt-sol/9704002"
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