Criterion for the Topological Conjugacy of Multi-Dimensional Gradient-Like Flows with No Heteroclinic Intersections on a Sphere

We study gradient-like flows with no heteroclinic intersections on an n -dimensional ( n ≥ 3) sphere from the point of view of topological conjugacy. We prove that the topological conjugacy class of such a flow is completely determined by the bicolor tree corresponding to the frame of separatrices of codimension 1. We show that for such flows the notions of topological equivalence and topological conjugacy coincide (which is not the case if there are limit cycles and connections.