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Multivalued perturbations of a saddle dynamics

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Abstract

We consider multivalued perturbations both of the discrete and of the continuous time hyperbolic dynamical system of the type

$$ x_{k + 1} \in Xx_k + f\left( {x_k } \right) + G\left( {x_k } \right), \dot x \in Ax + f\left( x \right) + G\left( x \right) $$

, where G is a parameterized multivalued map, i.e., \( G\left( x \right) = g\left( {x,\varepsilon \mathcal{B}_\mathcal{X} } \right) \) with \( \mathcal{B}_\mathcal{X} \) denoting the closed unit ball of a Banach space X and ε > 0. Under the assumptions that f and g are Lipschitz, with small Lipschitz constant, we prove that the saddle-type dynamics persists under the multivalued perturbation. More precisely, we construct analogues of the stable and unstable manifolds, which are typical of a single-valued hyperbolic dynamics and remain graphs of Lipschitz maps in the multivalued setting. Under more stringent assumptions on g, we prove some further topological properties. Also the maximal bounded invariant set is investigated.

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, Padova University, via Trieste 63, 35121, Padova, Italy

    Giovanni Colombo

  2. Department of Mathematical Analysis and Numerical Mathematics, Comenius University, Mlynská dolina, 842 48, Bratislava, Slovakia

    Michal Fečkan

  3. Faculty of Information Technology, PPKE and SZTAKI, 1083, Budapest, Hungary

    Barnabas M. Garay

Authors
  1. Giovanni Colombo
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  2. Michal Fečkan
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  3. Barnabas M. Garay
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Corresponding author

Correspondence to Giovanni Colombo.

Additional information

Dedicated to the memory of Bernd Aulbach.

Partially supported by grants VEGA-MS 1/2001/05 and OTKA 81403.

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Colombo, G., Fečkan, M. & Garay, B.M. Multivalued perturbations of a saddle dynamics. Differ Equ Dyn Syst 18, 29–56 (2010). https://doi.org/10.1007/s12591-010-0008-8

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