University of Missouri-Rolla
Department of Mathematics and
Statistics
Seminars and Colloquia
Week of May 9 - 13, 2005
Colloquium
schedule for Spring 2005
Monday, May 9
Colloquium 4:00 - 5:15 PM, G-5
Valery Gaiko, Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus and The University of Memphis, Memphis, TN.
Global Bifurcation Theory
and Applications of
Polynomial Dynamical Systems.
Abstract: Two-dimensional polynomial dynamical systems are
considered. The main problem of qualitative theory of such systems is Hilbert's
Sixteenth Problem on the maximum number and relative position of limit cycles.
There are three local bifurcations of limit cycles: 1) Andronov-Hopf
bifurcation (from a singular point of center or focus type); 2) separatrix cycle bifurcation (from a homoclinic
or heteroclinic orbit); 3) multiple limit cycle
bifurcation (from a multiple limit cycle of even or odd multiplicity). We
connect all these local bifurcations by means of the Wintner-Perko
termination principle and develop a new global approach to the complete
solution of the Problem in the quadratic case of polynomial systems [1].
Using Erugin's two-isocline method, we construct also a canonical Kukles-type cubic system with field-rotation parameters and
apply it for studying limit cycle bifurcations. In particular, we consider a
special case of the Kukles system which corresponds
to a generalized Liénard equation and is very
important for applications, classify separatrix
cycles and study global bifurcations of limit cycles.
Finally, we carry out the global qualitative analysis
of a cubic centrally symmetric dynamical system which can be used as a learning
model of planar neural networks.
References:
[1] V.A. Gaiko, Global Bifurcation Theory and Hilbert's
Sixteenth Problem,
Tuesday, May 10
Topology Seminar, 4:00 – 5:30 PM, G-4
We
will continue the seminar in June and July.
Wednesday, May 11
None announced.
Thursday, May 12
None announced.
Friday, May 13
None announced.
To place your seminar
announcement on this sheet, send email to wjcharat@umr.edu.