On the Hartman–Grobman Theorem with Parameters

Abstract

The Hartman-Grobman Theorem of linearization is extended to families of dynamical systems in a Banach space X, depending continuously on parameters. We prove thattheconjugacyalsochangescontinuously.Thecasesofnonlinearmapsandflowsarecon- sidered, and both in global and local versions, but global in the parameters. To use a special version of the Banach-Caccioppoli Theorem we introduce equivalent norms on X depend- ing on the parameters. The functional setting is suitable for applications to some nonlinear evolution partial differential equations like the nonlinear beam equation.