Geodesic
On the existence of periodic orbits for continuous Morse-Smale flows
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 18 (2016) no. 1, pp. 12-16

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: Morse-Smale flows, periodic orbits, heteroclinic orbits. 
@article{SVMO_2016_18_1_a1,
     author = {V. Z. Grines and E. V. Zhuzhoma and S. V. Medvedev and N. A. Tarasova},
     title = {On the existence of periodic orbits for continuous {Morse-Smale} flows},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {12--16},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2016_18_1_a1/}
}
TY  - JOUR
AU  - V. Z. Grines
AU  - E. V. Zhuzhoma
AU  - S. V. Medvedev
AU  - N. A. Tarasova
TI  - On the existence of periodic orbits for continuous Morse-Smale flows
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2016
SP  - 12
EP  - 16
VL  - 18
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SVMO_2016_18_1_a1/
LA  - ru
ID  - SVMO_2016_18_1_a1
ER  - 
%0 Journal Article
%A V. Z. Grines
%A E. V. Zhuzhoma
%A S. V. Medvedev
%A N. A. Tarasova
%T On the existence of periodic orbits for continuous Morse-Smale flows
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2016
%P 12-16
%V 18
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SVMO_2016_18_1_a1/
%G ru
%F SVMO_2016_18_1_a1
V. Z. Grines; E. V. Zhuzhoma; S. V. Medvedev; N. A. Tarasova. On the existence of periodic orbits for continuous Morse-Smale flows. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 18 (2016) no. 1, pp. 12-16. http://geodesic.mathdoc.fr/item/SVMO_2016_18_1_a1/