Multivalued perturbations of a saddle dynamics

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 e > 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.