Pitchfork bifurcations of invariant manifolds |
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Authors: | Jyoti Champanerkar Denis Blackmore |
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Affiliation: | a Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA b Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA |
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Abstract: | A pitchfork bifurcation of an (m−1)-dimensional invariant submanifold of a dynamical system in Rm is defined analogous to that in R. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds M+ and M−, after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the theorem involve differential topology and analysis. The theorem is illustrated by means of a canonical example. |
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Keywords: | Bifurcation Pitchfork bifurcation Invariant manifolds |
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