### Article

## Construction of the Morse –Bott Energy Function for Regular Topological Flows

In this paper, we consider regular topological flows on closed n-manifolds. Such

flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number

of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale

flows, which are closely related to the topology of the supporting manifold. This connection is

provided by the existence of the Morse –Bott energy function for the Morse – Smale flows. It

is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds,

on which dynamical systems can be considered only in a continuous category. The existence of

continuous analogs of regular flows on any topological manifolds is an open question, as is the

existence of energy functions for such flows. In this paper, we study the dynamics of regular

topological flows, investigate the topology of the embedding and the asymptotic behavior of

invariant manifolds of fixed points and periodic orbits. The main result is the construction of

the Morse –Bott energy function for such flows, which ensures their close connection with the

topology of the ambient manifold.