On topological classification of gradient-like flows without heteroclinical intersection by means of energy function
E. Ya. Gurevich1, A. N. SakharovE.D. Tregubova.2
| Annotation | Presenting paper is an extension of paper \cite{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 |
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| Keywords | Morse-Smale flows, energy function, topological equivalence, topological classification |
1Associated Professor of Chair of Theory of Control and Dynamics of Machines, Lobachevsky State University, Nizhny Novgorod, elena$_$gurevich@list.ru
2Assistant professor of department of higher mathematic, Nizhny Novgorod State Agricultural Academy, Nizhny Novgorod; ansakharov2008@yandex.ru.
3Assistant professor of department of higher mathematic, Nizhny Novgorod State Agricultural Academy, Nizhny Novgorod; math-ngaa@yandex.ru.
Citation: E. Ya. Gurevich, A. N. SakharovE.D. Tregubova., "[On topological classification of gradient-like flows without heteroclinical intersection by means of energy function]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,15:4 (2013) 91–100 (In Russian)
