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September  2014, 34(9): 3419-3435. doi: 10.3934/dcds.2014.34.3419

## Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension

 1 Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China

Received  October 2012 Revised  December 2013 Published  March 2014

Travelling wave solutions of a two-dimensional gaseous star with self-gravity and surface tension are considered. The star rotates along a clockwise direction with a travelling speed. The governing equations on a whole unknown domain are changed to ones on the boundary using the Dirichlet-Neumann operator. The problem of the existence of its periodic solutions is equivalent to one of a functional equation. After applying the method of Lyapunov-Schmidt reduction, the reduced equation of this functional equation has a generalized Pitchfork bifurcation with the bifurcation parameter being the travelling speed. This shows that there exist two nontrivial periodic solutions.
Citation: Shengfu Deng. Generalized pitchfork bifurcation on a two-dimensional gaseous star with self-gravity and surface tension. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3419-3435. doi: 10.3934/dcds.2014.34.3419
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