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

Shengfu Deng

doi:10.3934/dcds.2014.34.3419

Article Contents

2014, Volume 34, Issue 9: 3419-3435. Doi: 10.3934/dcds.2014.34.3419

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

Shengfu Deng^1,

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

Received: September 30, 2012

Revised: November 30, 2013

Published: August 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B10, 35B32; Secondary: 76B15.

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