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Bifurcation to Quasi-Periodic Tori in the Interaction of Steady State and Hopf Bifurcations

Scheurle, Jürgen and Marsden, Jerrold (1984) Bifurcation to Quasi-Periodic Tori in the Interaction of Steady State and Hopf Bifurcations. SIAM Journal on Mathematical Analysis, 15 (6). pp. 1055-1074. ISSN 0036-1410. doi:10.1137/0515082.

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Bifurcations to quasi-periodic tori in a two parameter family of vector fields are studied. At criticality, the vector field has an equilibrium point with a zero eigenvalue and a pair of complex conjugate eigenvalues. This situation has been studied by Langford, Iooss, Holmes and Guckenheimer. Here we provide explicitly computed conditions under which the stability of the secondary branch of tori, and whether the flow on them is quasiperiodic, can be determined. The results are applied to "Brusselator" system of reaction diffusion equations.

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Additional Information:©1984 Society for Industrial and Applied Mathematics. Received by the editors December 13, 1982, and in revised form September 26, 1983. We thank John David Crawford, Marty Golubitsky, John Guckenheimer and Gerard Iooss for useful discussions. We also thank Henk Broer for making his preprints available to us. The research of the author [J.S.] was supported by Deutsche Forschungsgemeinschaft contract 13 Sche 233/2-1 while he was visiting the University of California, Berkeley. The research of this author [J.M.] was partially supported by the U.S. Department of Energy under contract DE-AT03-82ERI2097.
Issue or Number:6
Record Number:CaltechAUTHORS:SCHEsiamjma84
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10017
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Deposited On:05 Apr 2008
Last Modified:08 Nov 2021 21:04

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