Limit sets and global dynamic for 2-D divergence-free vector fields

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.

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.

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