Topological conjugacy of discrete time-map and Euler discrete dynamical systems generated by a gradient flow on a two-dimensional compact manifold
The paper deals with qualitative properties of discretized gradient systems. The following main theorem is proved: Let \({\mathcal M}\) be a 2-dimensional compact, connected Riemannian \(C^j\)-manifold, \(j\geq 2\) in \(\mathbb{R}^k\) and let \(\phi \: {\mathcal M}\times \mathbb{R}\to {\mathcal M}\) be a dynamical system generated by a differential equation \(\dot x=-\text{ grad} E(x)\) for \(E\in C^2({\mathcal M},\mathbb{R})\), having a finite number of singularities, all hyperbolic. Let furthermore, \(\phi\) have no saddle-saddle connections or each saddle-saddle connections be a geodesic line. Let \(\phi_h(x)=\phi (x,h)\) and \(\psi_h\) be the Euler discretization of the system. Then for \(h>0\) sufficiently small, there is a homeomorphism \(\alpha_h \: {\mathcal M}\to {\mathcal M}\) such that \(\psi_h\circ \alpha_h =\alpha_h \circ \phi_h\).
scientific article; zbMATH DE number 1836647
scientific article