A seminar at Center for Mathematical Sciences, July 19, Wednesday, at 2:00pm

Location: Center for Mathematical Sciences, Room 813（创新研究院恩明楼813室）

speaker: Prof. Claude Baesens, University of Warwick, FRS

Title: AN ALMOST SIMPLEST MONOTONE FAMILY OF VECTOR FIELDS ON A TORUS

Abstract: We analyse the bifurcation diagram for an explicit monotone family of vector fields on a torus and prove that it has at most two equilibria, precisely four Bogdanov-Takens points, no closed curves of centre nor closed curves of neutral saddle, at most two Reeb components, precisely four arcs of rotational homoclinic connection of “horizontal” homotopy type, eight horizontal saddle-node loop points, two necklace points, four points of neutral horizontal homoclinic connection, and two half-plane fan points, and there is no simultaneous existence of centre and neutral saddle, nor contractible homoclinic connection to a neutral saddle. Furthermore we prove that all saddle-nodes, Bogdanov-Takens points, non-neutral and neutral horizontal homoclinic bifurcations are non-degenerate and the Hopf condition is satisfied for all centres. We also find it has four points of degenerate Hopf bifurcation. Thus it provides an example of a family satisfying all the assumptions of our paper [BM1] on simplest bifurcation diagrams for such families except the one of at most one contractible periodic orbit.

Keywords: bifurcation, horizontal saddle-node, Hopf bifurcation