TY  - Type of reference 
TI  - Limit sets and global dynamic for 2-D divergence-free vector fields 
AU  - Habib Marzougui 
AB  - The global structure of divergence-free vector fields on closed surfaces is investigated. We prove that if M is a closed surface and $$${V}$$$ is a divergence-free $$$C^{1}$$$-vector field with finitely many singularities on M then every orbit L of $$$\mathcal{V}$$$ is one of the following types: (i) a singular point, (ii) a periodic orbit, (iii) a closed (non periodic) orbit in M* = M - Sing($$$\mathcal{V}$$$), (iv) a locally dense orbit, where Sing($$$\mathcal{V}$$$) denotes the set of singular points of $$$\mathcal{V}$$$. On the other hand, we show that the complementary in M of periodic components and minimal components is a compact invariant subset consisting of singularities and closed (non compact) orbits in M*. These results extend those of T. Ma and S. Wang in [Discrete Contin. Dynam. Systems, 7 (2001), 431-445] established when the divergence-free vector field $$$\mathcal{V}$$$ is regular that is all its singular points are non-degenerate. 
DO  - 10.21494/ISTE.OP.2022.0837 
JF  - Advances in Pure and Applied Mathematics 
KW  - divergence free-vector -elds,  quasiminimal set,  !-limit set,  non-
trivial recurrent orbit,  Poincar-e-Bendixson Theorem,  minimal component,  closed surface, divergence free-vector -elds,  quasiminimal set,  !-limit set,  non-
trivial recurrent orbit,  Poincar-e-Bendixson Theorem,  minimal component,  closed surface,  
L1  - http://www.openscience.fr/IMG/pdf/iste_apam22v13n3_1.pdf 
LA  - en 
PB  - ISTE OpenScience 
DA  - 2022/06/1 
SN  - 1869-6090 
TT  - Ensembles limites et dynamique globale pour les 2-D champs de vecteurs sans divergence 
UR  - http://www.openscience.fr/Limit-sets-and-global-dynamic-for-2-d-divergence-free-vector-fields 
IS  - Issue 3 (June 2022) 
VL  - 13 
ER  -