Symmetry breaking Hopf bifurcations in equations with O(2) symmetry with application to the Kuramoto-Sivashinsky equation
- Univ. of Surrey, Guildford (United Kingdom)
- Heriot-Watt Univ., Edinburgh (United Kingdom)
In problems with O(2) symmetry, the Jacobian matrix at nontrivial steady state solutions with D{sub n} symmetry always has a zero eigenvalue due to the group orbit of solutions. We consider bifurcations which occur when complex eigenvalues also cross the imaginary axis and develop a numerical method which involves the addition of a new variable, namely the velocity of solutions drifting around the group orbit, and another equation, which has the form of a phase condition for isolating one solution on the group orbit. The bifurcating branch has a particular type of spatio-temporal symmetry which can be broken in a further bifurcation which gives rise to modulated travelling wave solutions which drift around the group orbit. Multiple Hopf bifurcations are also considered. The methods derived are applied to the Kuramoto-Sivashinsky equation and we give results at two different bifurcations, one of which is a multiple Hopf bifurcation. Our results give insight into the numerical results of Hyman, Nicolaenko, and Zaleski. 30 refs., 2 figs., 2 tabs.
- OSTI ID:
- 478409
- Journal Information:
- Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 1 Vol. 131; ISSN 0021-9991; ISSN JCTPAH
- Country of Publication:
- United States
- Language:
- English
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