### On the existence of periodic orbits for continuous Morse-Smale flows

#### V. Z. Grines^{1}, E. V. Zhuzhoma^{2}, V. S. Medvedev^{3}, N. A. Tarasova^{4}

Annotation | We consider the class of continuous Morse-Smale flows defined on a topological closed manifold $M^n$ of dimension $n$ which is not less than three, and such that the stable and unstable manifolds of saddle equilibrium states do not have intersection. We establish a relationship between the existence of such flows and topology of closed trajectories and topology of ambient manifold. Namely, it is proved that if $f^t$ (that is a continuous Morse-Smale flow from considered class) has $mu$ sink and source equilibrium states and $\nu$ saddles of codimension one, and the fundamental group $\pi_1 (M ^ n)$ does not contain a subgroup isomorphic to the free product $g =\frac {1} {2} \left (\nu - \mu +2\right)$ copies of the group of integers $\mathbb {Z} $, then the flow $ f^t$ has at least one periodic trajectory. |
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Keywords | Morse-Smale flows, periodic orbits, heteroclinic orbits |

^{1}professor of Department of fundamental mathematics, Higher School of Economics, Nizhny Novgorod; vgrines@yandex.ru^{2}professor of Department of fundamental mathematics, Higher School of Economics, Nizhny Novgorod; zhuzhoma@mail.ru^{3}researcher TAPRADESS laboratory, Higher School of Economics, Nizhny Novgorod; vmedvedev@hse.ru^{4}associate professor of Department of IMD , Institute of food technology and design, Nizhny Novgorod; tarasova-na-an@rambler.ru

**Citation**: V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, N. A. Tarasova, "[On the existence of periodic orbits for continuous Morse-Smale flows]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,18:1 (2016) 12–16 (In Russian)