Alexandr Prishlyak, Illia Ovtsynov
abstract:
We describe all possible topological structures of Morse flows and codimension
one gradient flows generated by vector fields emerging in typical one-parameter
bifurcations. Research is held on a 2-sphere with holes in the case where the
number of singular points of flows is at most six.
For that purpose, we construct topological invariants of these flows which are
actually the graphs endowed with certain information. In this case, they are
separatrix skeletons and distinguished graphs of flows. For example, the
saddle-node singularity is specified by selecting a separatrix in the skeleton
of the flow before the bifurcation, whereas the saddle connection is specified
by a separatrix which connects two saddles. Apart from that, we construct
special codes for these graphs so that we can regain a skeleton and a graph from
the code and thus recover the whole flow on a certain surface.
Mathematics Subject Classification: 37C10, 37C15, 37C20
Key words and phrases: Bifurcation, topological equivalence, structural stability, separatrix skeleton, saddle-node, saddle connection