NDLI: The Existence of Bifurcating Invariant Tori in a Spatially Extended Reaction-Diffusion-Convection System with Spatially Localized Amplification

Content Provider	Springer Nature Link
Author	Kirchhoff, Andreas Schneider, Guido Düll, Wolf Patrick
Copyright Year	2013
Abstract	We consider a spatially extended reaction-diffusion-convection system with a marginally stable ground state and a spatially localized amplification. We are interested in solutions bifurcating from the spatially homogeneous ground state in the case when pairs of imaginary eigenvalues simultaneously cross the imaginary axis. For this system we prove the bifurcation of a family of invariant tori which may contain quasiperiodic solutions. There is a serious difficulty in obtaining this result, because the linearization at the ground state possesses an essential spectrum up to the imaginary axis for all values of the bifurcation parameter. To construct the invariant tori, we use their invariance under the flow which manifests in a condition in PDE form. The nonlinear terms of this resulting PDE exhibit a loss of regularity. Since the linear part of this PDE is not smoothing, an adaption of the hard implicit function theorem (or Nash-Moser scheme) and energy estimates will be used to prove our result.
Ending Page	358
Page Count	54
Starting Page	305
File Format	PDF
ISSN	09388974
e-ISSN	14321467
Journal	Journal of Nonlinear Science
Issue Number	2
Volume Number	24
Language	English
Publisher	Springer US
Publisher Date	2013-10-29
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	ApplicationMathematics/Computational Methods of Engineering Invariant tori Analysis Theoretical, Mathematical and Computational Physics Bifurcation Mechanics Economic Theory Normal forms, center manifold theory, bifurcation theory Reaction-diffusion equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Engineering Modeling and Simulation

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