Centre Manifolds, Normal Forms and Elementary Bifurcations

Vanderbauwhede, André

doi:10.1007/978-3-322-96657-5_4

Cited by 250 publications

(205 citation statements)

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“…increasing in absolute value), and that we have continuous embeddings B k ρ → B kρ+µ k and B j ρ+µ j → B j ρ ; in the first embedding we reserved some spectral space to apply Corollary B.3. These choices-as well as more ideas in this section-are inspired by [Van89,Sec. 3], which is an interesting read for comparison in a simpler setting.…”

Section: Higher Order Derivativesmentioning

confidence: 99%

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

Eldering

2012

Comptes Rendus. Mathématique

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Section: Higher Order Derivativesmentioning

confidence: 99%

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

Eldering

2012

Comptes Rendus. Mathématique

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“…To mention only few references of the comprehensive amount of literature for autonomous differential equations, we refer to CARR [38] [179] and WIGGINS [182]. In the nonautonomous context, see AULBACH [13], AULBACH & WANNER [21], SELL [167], WANNER [180] and YI [183].…”

Section: Nonlinear Systemsmentioning

confidence: 99%

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

Rasmussen

2007

Lecture Notes in Mathematics

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“…If A has purely imaginary spectrum and is semisimple (i.e. there are no Jordan blocks of size 2 or bigger in its complex Jordan normal form), then we obtain a normal form transformation by averaging, see [Van89][Sec.2.4].…”

Section: Finite Order Estimates and Normal Formsmentioning

confidence: 99%

Exponential Estimates in Averaging and Homogenisation

Matthies

2006

Analysis, Modeling and Simulation of Multiscale Problems

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Abstract. Many partial differential equations with rapid spatial or temporal scales have effective descriptions which can be derived by homogenisation or averaging. In this article we deal with examples, where quantitative estimates of the error is possible for higher order homogenisation and averaging. In particular, we provide theorems, which allow homogenisation and averaging beyond all orders by giving exponential estimates of appropriately averaged and homogenised descriptions.Methods include iterated averaging transformations, optimal truncation of asymptotic expansions and highly regular solutions (Gevrey regularity). Prototypical examples are reaction-diffusion equations with heterogeneous reaction terms or rapid external forcing, nonlinear Schrödinger equations describing dispersion management, and second-order linear elliptic equations.

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Centre Manifolds, Normal Forms and Elementary Bifurcations

Cited by 250 publications

References 20 publications

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

Exponential Estimates in Averaging and Homogenisation

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