Abstract:
In this talk, we discuss some topics of flows on compact surfaces.
First we present a necessary and sufficient condition for the existence
of dense orbits of continuous flows on compact connected surfaces, which
is a generalization of a necessary and sufficient condition on
area-preserving flows obtained by H. Marzougui and G. Soler L'opez.
Second, we generalize the Poincare-Bendixson theorem for a flow with
arbitrarily many singular points on a compact surface. In fact, the
omega-limit set of any non-closed orbit is one of the following
exclusively: i) a nowhere dense subset of singular points; ii) a limit
cycle; iii) a limit ”quasi-circuit”; iv) a locally dense Q-set; v) a
”quasi-Q-set” which is not locally dense.
Third, we consider what class of flows on compact surfaces can be
characterized by finite labeled graphs. In particular, a class of
surface flows, up to topological conjugacy, which contains both the set
of Morse Smale flows and the set of area-preserving flows with finite
singular points is classified. In fact, although the set of topological
equivalent classes of minimal flows on a torus is uncountable, we
enumerate the set of topological equivalent classes of flows with
non-degenerate singular points and with at most finitely many limit
cycles but without non-closed recurrent orbits on a compact surface
using finite labeled graphs.
Finally, we introduce an implementation of the representation theorem
for Hamiltonian flows on punctured disks and apply to analysis on fluid
phenomena.
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