|Speaker :||Professor Claude Baesens (University of Warwick)|
|Title :||Simplest Bifurcation Diagrams for Families of Vector Fields on a Torus|
|Time :||2017-07-27 (Thu) 15:00 - 16:30|
|Place :||Seminar Room 722, Institute of Mathematics (NTU Campus)|
We prove that the bifurcation diagram for a monotone two-parameter family
of vector fields on a torus has to be at least as complicated as the conjectured
simplest one proposed in [BGKM]. To achieve this we define "simplest" by minimising
sequentially the numbers of equilibria, Bogdanov-Takens points, closed curves of centre
and of neutral saddle, intersections of curves of centre and neutral saddle,
Reeb components, other invariant annuli, arcs of rotational homoclinic bifurcation of
horizontal homotopy type, necklace points, contractible periodic orbits, points of neutral horizontal homoclinic bifurcation and half-plane fan points. We obtain two types of simplest case, including that initially proposed [BM1].
We analyse the bifurcation diagram of the explicit monotone family and prove it satisfies all the assumptions on simplest bifurcation diagram except the one of at most one contractible periodic orbit. Indeed we find that it has four points of degenerate Hopf bifurcation [BM2].