Colloquium Mathematics, Robert Roussarie

Join us for coffee and tea at 15.45 p.m.

 

 

Date: Thursday, November 10^th 2011

Speaker: Robert Roussarie

Room: 5161.0289 (Bernoulliborg),

Time:16.15

 

 

Title: Canard cycles of finite codimension with two breaking parameters

 Abstract:

We consider 2-dimensional slow-fast systems with canard cycles occuring in
the layer equation obtained for " = 0. The canard cycles under consideration
may be broken by two independent mechanisms : either a turning point or
jump between two contact points. Each of these mechanisms is associated to
a parameter permitting to break it in a generic way. These canard cycles pass
through two independent horizontal levels parametrized by u, v. This gives a
whole 2-parameter family uv of such canard cycles.
The properties of the system depend on four slow-fast divergence integrals,
which are functions of u, v and also of a parameter _. First, in terms of these
integrals we obtain an upper bound for the total number of limit cycles bifurcating
from the region containing the canard cycles. Next, the codimension of
each canard cycle is defined in terms of these integrals. The cyclicity of a canard
cycle of finite codimension c is bounded by c+1. We also give conditions on the
four slow divergence integrals in order to have a versal c-unfolding. If c = 2 the
versal parameters are just the two breaking parameters, but if c > 2, we need
the (c − 2)-dimensional parameter _ to obtain a versal unfolding. Finally, for
any finite c _ 2, we give an example of polynomial Li´enard family exhibiting a
versal unfolding of such a canard cycle of codimension c.
This talk is based on a work in collaboration with Lylia Mamouhdi, University
of Annaba, Algeria.
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Colloquium coordinators are Prof.dr. A.C.D. van Enter (e-mail : A.C.D.van.Enter@rug.nl) and
Dr. A.V. Kiselev (e-mail: a.v.kiselev@rug.nl)