Smooth Linearization for a Saddle on Banach Spaces

Rodrigues, Hildebrando M.; Sol, Júlia Borràs

doi:10.1007/s10884-004-6116-9

Cited by 30 publications

(25 citation statements)

References 8 publications

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“…Recently, we have been working in C 1 -linearization in infinite dimensions, in the works Rodrigues and Solà-Morales [9], for the case of invertible contractions, and Rodrigues and Solà-Morales [10] where a case of a saddle point is studied. In both cases, applications to abstract wave equations have been presented.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

An invertible contraction that is not C1-linearizable

Rodrigues

Solà-Morales

2005

Comptes Rendus. Mathématique

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Section: Introduction and Main Resultsmentioning

confidence: 99%

An invertible contraction that is not C1-linearizable

Rodrigues

Solà-Morales

2005

Comptes Rendus. Mathématique

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“…In some papers of some of us we tried to extend or to analyse in infinite dimensions known results for finite dimensional problems. This was the case of Kloeden & Rodrigues [10], Rodrigues [11], Rodrigues & Ruas [16], Rodrigues & Solà-Morales [17,18,19,20], Rodrigues, Caraballo & Gameiro [14] and Rodrigues, Teixeira & Gameiro [15].…”

Section: Introductionmentioning

confidence: 99%

Stability problems in nonautonomous linear differential equations in infinite dimensions

Rodrigues¹,

Solà-Morales²,

Nakassima³

2020

Communications on Pure &Amp; Applied Analysis

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Dedicated to Tomás Caraballo for his 60th birthday.Abstract. One goal of this paper is to study robustness of stability of nonautonomous linear ordinary differential equations under integrally small perturbations in an infinite dimensional Banach space. Some applications are obtained to the case of rapid oscillatory perturbations, with arbitrary small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel [3]. Based in Rodrigues [11] and in Kloeden & Rodrigues [10] we introduce a class of functions that we call Generalized Almost Periodic Functions that extend the usual almost periodic functions and are suitable to deal with oscillatory perturbations. We also present an infinite dimensional example of the previous results. We show in another example that it is possible to stabilize an unstable system using a perturbation with large period and small mean value.Finally, we give an example where we stabilize an unstable linear ODE with small perturbation in infinite dimensions using some ideas developed in Rodrigues & Solà-Morales [21] and in an example of Kakutani, see [13].MSC: 37C75; 47A10, 43D20, 35B35.

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“…In 1970s Belitskii gave conditions on

C^{k}

linearization for

C^{k, 1}

(

k ⩾ 1

) diffeomorphisms, which implies that

C^{1, 1}

diffeomorphisms can be

C^{1}

linearized locally if the eigenvalues

λ_{1}, \dots, λ_{n}

satisfy a nonresonant condition that

\begin{matrix} true \\ right | λ_{i} | \cdot | λ_{j} | \neq | λ_{ι} | \end{matrix}

for all

ι = 1, \dots, n

false| λ_{i} false| < 1 < false| λ_{j} false|

. This result was partially generalized to infinite‐dimensional spaces in . Note that in the contractive (or expansive) case holds automatically and therefore

C^{1}

linearization can always be realized in

R^{n}

.…”

Section: Introductionmentioning

confidence: 99%

Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy

Dragičević

Zhang

2019

Proc. London Math. Soc.

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In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of a discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and Hölder continuous near 0. Restricted to the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and Hölder linearization of hyperbolic systems without any nonresonant conditions.

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Smooth Linearization for a Saddle on Banach Spaces

Cited by 30 publications

References 8 publications

An invertible contraction that is not C1-linearizable

An invertible contraction that is not C1-linearizable

Stability problems in nonautonomous linear differential equations in infinite dimensions

Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy

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