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Working paper

On Embedding of Multidimensional Morse-Smale Diffeomorphisms into Topological Flows

A numerical solution  of differential equation is  adequate if, in particular,   the obtained discrete model is topologically conjugate to the time one map of the original flow.  B.~Garay showed  that the Runge-Kutta discretization of a gradient-like flow $(n > 2)$  on the $n$-disk is topologically conjugate to the time one map  for a sufficiently small step size.  J.~Palis found necessary  conditions for  a Morse-Smale diffeomorphism on a closed $n$-dimensional manifold $M^n$  to  embed into a topological  flow and proved that these conditions are also sufficient for $n=2$. We find sufficient conditions for a Morse-Smale diffeomorphism to embed in a topological flow for the  case when $M^n$ is  the sphere $S^n, \,n\geq 4$.