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Parameter Dependence of Stable Manifolds for Delay Equations with Polynomial Dichotomies

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Abstract

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x′ = L(t)x t + f (t, x t , λ), assuming that the linear equation x′ = L(t)x t admits a polynomial dichotomy and that f is a sufficiently small Lipschitz perturbation. Moreover, we show that the stable invariant manifolds are Lipschitz in the parameter λ. We also consider the general case of nonuniform polynomial dichotomies.

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Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, 1049-001, Lisboa, Portugal

    Luis Barreira & Claudia Valls

  2. School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, 130024, Jilin, People’s Republic of China

    Meng Fan

  3. School of Mathematical Sciences, Heilongjiang University, 74 Xuefu Street, Harbin, 150080, Heilongjiang, People’s Republic of China

    Jimin Zhang

Authors
  1. Luis Barreira
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  2. Meng Fan
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  3. Claudia Valls
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  4. Jimin Zhang
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Correspondence to Luis Barreira.

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Barreira, L., Fan, M., Valls, C. et al. Parameter Dependence of Stable Manifolds for Delay Equations with Polynomial Dichotomies. J Dyn Diff Equat 24, 101–118 (2012). https://doi.org/10.1007/s10884-011-9232-3

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  • DOI: https://doi.org/10.1007/s10884-011-9232-3

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