On the discretization of a partial differential equation in the neighborhood of a periodic orbit
This article is concerned with the comparison of the dynamic of a partial differential equation and its time discretization. We restrict our attention to the neighborhood of a hyperbolic periodic orbit. We show that the discretization possesses an invariant closed curve near the periodic orbit and that the trajectories of the semigroups defined by the partial differential equation and its approximation are closed in a certain sense, provided that different initial data are allowed. This answers partly an open problem posed by \textit{W.-J. Beyn} [ibid. 51, 103- 122 (1987; Zbl 0617.65082)]. Examples of application to dissipative partial differential equations are provided.
scientific article; zbMATH DE number 569152
scientific article