In Middle Volga Mathematical Society

On topological classification of gradient-like flows without heteroclinical intersection by means of energy function

E. Ya. Gurevich^a, A. N. Saharov^b, E. D. Tregubova<sup>b

a</sup> N. I. Lobachevski State University of Nizhni Novgorod
^b Nizhnii Novgorod State Agricultural Academy

Abstract: Presenting paper is an extension of paper [gurevich-GrPoSaRu] and devoted to topological classification of gradient-like flows on smooth closed orientable manifold $M^n$ of dimension $n\geq 3$ by means of energy function. We consider class $G(M^n)$ of gradient-like flows without heteroclinic intersection, all saddle equilibria of which have Morse index equal 1 or $(n-1)$. We show that necessary and sufficient condition of topological equivalence for flows from $G(M^n)$ is equivalence of corresponding energy functions and special condition of equivalence energy functions on some level surface. Also we define a class $G_0(M^n)$ of flows for which energy function is complete invariant. Obtained results may be applied for quantity studying of dynamics for structurally stable systems with known energy function from physical contest of the model (as, for instance, energy function of dissipative systems in mechanics, potential of electrostatic fields or potential of current free magnetic field

Keywords: Morse-Smale flows, energy function, topological equivalence, topological classification.

UDC: 517.938