DOI: 10.1017/s0308210500024173. Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry

Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry

10.1017/s0308210500024173

Crossref journal-article

Cambridge University Press (CUP)

Proceedings of the Royal Society of Edinburgh: Section A Mathematics (56)

Abstract

SynopsisIn this paper we show that in O(2) symmetric systems, structurally stable, asymptoticallystable, heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions. These cycles are found in the third-order truncated normal forms of specific codimension two steady-state/Hopf and Hopf/Hopf mode interactions.We find these cycles using group-theoretic techniques; in particular, we look for certainpatterns in the lattice of isotropy subgroups. Once the pattern has been identified, the heteroclinic cycle can be constructed by decomposing the vector field on fixed-point subspaces into phase/amplitude equations (it is here that we use the assumption of normal form). The final proof of existence (and stability) relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied.

Bibliography

Melbourne, I., Chossat, P., & Golubitsky, M. (1989). Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 113(3â4), 315â345.

Authors 3

I. Melbourne (first)
P. Chossat (additional)
M. Golubitsky (additional)

References 26 Referenced 74

10.2307/2372437
19 Krupa M. . Bifurcations of critical group orbits (Thesis, University of Houston, 1988). Cf. Bifurcations of relative equilibria. SIAM J. Appl. Math, (submitted)
10.1017/S0305004100064732
10.1007/978-1-4612-4574-2
10.1016/0167-2789(88)90063-2
10.1007/BF00251598
8 Field M. . Equivariant bifurcation theory and symmetry breaking. Dyn. Diff. Eqn. (to appear).
{'key': 'S0308210500024173_ref006', 'first-page': '255', 'article-title': 'Hopf—Hopf mode interactions with O(2) symmetry', 'volume': '1', 'author': 'Chossat', 'year': '1986', 'journal-title': 'Dyn. Stab. Syst.'} / Dyn. Stab. Syst. / Hopf—Hopf mode interactions with O(2) symmetry by Chossat (1986)
10.1007/BF00284507
10.1126/science.208.4440.173
10.1016/0167-2789(88)90032-2
10.1090/conm/028/752001
10.2307/2372774
10.1016/0375-9601(87)90008-9
10.1007/BF00280698
10.1137/0129022
10.1007/BF01048454
10.1017/S0022112088000746
10 Field M. and Richardson R. W. . New examples of symmetry breaking bifurcations and the distribution of symmetry breaking isotropy types (in preparation).
10.1137/0517023
18 Kevrekidis I. G. , Nicolaencko B. and Scovel J. C. . Back in the saddle again: a computer assisted study of the Kuramoto-Shivashinsky equation. SIAM J. Appl. Math, (to appear).
{'key': 'S0308210500024173_ref001', 'volume-title': 'More on structurally stable H-orbits', 'author': 'Armbruster', 'year': '1989'} / More on structurally stable H-orbits by Armbruster (1989)
{'key': 'S0308210500024173_ref020', 'first-page': '3', 'article-title': 'Lie dynamical systems', 'volume': '109–110', 'author': 'Markus', 'year': '1983', 'journal-title': 'Asterisque'} / Asterisque / Lie dynamical systems by Markus (1983)
10.1007/978-3-642-67220-0_39
{'key': 'S0308210500024173_ref016', 'volume-title': 'Differential Equations, Dynamical Systems, and Linear Algebra', 'author': 'Hirsch', 'year': '1974'} / Differential Equations, Dynamical Systems, and Linear Algebra by Hirsch (1974)
10.1090/S0002-9947-1980-0561832-4

Dates

Type	When
Created	13 years, 9 months ago (Nov. 14, 2011, 8:19 a.m.)
Deposited	6 years, 3 months ago (May 19, 2019, 5:14 p.m.)
Indexed	1 month ago (July 30, 2025, 10:38 a.m.)
Issued	36 years, 7 months ago (Jan. 1, 1989)
Published	36 years, 7 months ago (Jan. 1, 1989)
Published Online	13 years, 9 months ago (Nov. 14, 2011)
Published Print	36 years, 7 months ago (Jan. 1, 1989)

Funders 0

None

BibTeX

@article{Melbourne_1989, title={Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry}, volume={113}, ISSN={1473-7124}, url={http://dx.doi.org/10.1017/s0308210500024173}, DOI={10.1017/s0308210500024173}, number={3–4}, journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, publisher={Cambridge University Press (CUP)}, author={Melbourne, I. and Chossat, P. and Golubitsky, M.}, year={1989}, pages={315–345} }

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