Center manifolds and contractions on a scale of Banach spaces

Abstract. We study the behaviour of solutions to nonlinear autonomous functional differential equations of mixed type in the neighbourhood of an equilibrium. We show that all solutions that remain sufficiently close to an equilibrium can be captured on a finite dimensional invariant center manifold, that inherits the smoothness of the nonlinearity. In addition, we provide a Hopf bifurcation theorem for such equations. We illustrate the application range of our results by discussing an economic life-cycle model that gives rise to functional differential equations of mixed type.

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“…The presentation given here contains slight adapations of results given in [42]. w w n n n n n n n n n n n n n n n…”

Section: A Embedded Contractionsmentioning

confidence: 99%

Center Manifold Theory for Functional Differential Equations of Mixed Type

Hupkes

Lunel

2006

J Dyn Diff Equat

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“…The technique of using a scale of Banach spaces as developed by Vanderbauwhede and Van Gils [VG87], and the fiber contraction theorem of Hirsch and Pugh [HP70] can be applied, and we obtain each D k Θ as a fixed point in the appropriate space. The final conclusion h ∈ C k follows by evaluating D k Θ at t = 0.…”

Section: Outline Of the Proofmentioning

confidence: 99%

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

Eldering

2012

Comptes Rendus. Mathématique

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“…Namely, we first show that M cu (Y ) is C 1 following the arguments of [18] and then use the C r Section Theorem in [26], [47] to improve the smoothness inductively. For other approaches to prove the smoothness of an invariant manifold (for example, the application of Henry's Lemma and the use of a scale of Banach spaces), we refer the readers to [13], [41], [53], [51], [56].…”

Section: Shui-nee Chow Weishi Liu and Yingfei Yimentioning

confidence: 99%

“…[7], [26], [48], [51], [53] etc. ), has become an important subject and found tremendous applications in the study of flows and diffeomorphisms (see [7], [11], [12], [17], [21], [24] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Center manifolds for smooth invariant manifolds

Chow

Liu

2000

Trans. Amer. Math. Soc.

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Abstract. We study dynamics of flows generated by smooth vector fields in R n in the vicinity of an invariant and closed smooth manifold Y . By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of Y ) based on the information of the linearization along Y , which contains every locally bounded solution and is persistent under small perturbations.

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Center manifolds and contractions on a scale of Banach spaces

Cited by 154 publications

References 3 publications

Center Manifold Theory for Functional Differential Equations of Mixed Type

Center Manifold Theory for Functional Differential Equations of Mixed Type

Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

Center manifolds for smooth invariant manifolds

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