Seminar:   Nonlinear Analysis and Optimization
Speaker:   Zalman Balanov
            University of Texas at Dallas
Title:     Equivariant Hopf bifurcation in van der Pol oscillators
            with hysteretic memory
Time:      Sunday, December 26, 2010, at 14:30
Place:     Room 814, Amado Mathematics Building

Hopf Bifurcation in Hysteresis Models for Symmetric Networks of Coupled 
Identical van der Pol  Oscillators with  Ferromagnetic Core: Twisted 
Equivariant Degree Approach

The van der Pol oscillator (VDPO) consists of an LCR-contour and a 
negative feedback loop which can be implemented on the basis of a triode 
or two diodes. The corresponding second order (autonomous)  van der Pol 
equation is the simplest nonlinear mathematical model widely used in 
electrical engineering.

In practice, one is usually dealing with networks  of  VDPOs coupled 
symmetrically. Studying the impact of symmetries of a system to the 
actual dynamics (in particular, symmetric properties and minimal number 
of periodic regimes) constitutes a problem of great importance and 
complexity which can be traced back to the classical 16-th Hilbert 

Periodic solutions to symmetric autonomous systems very often are 
studied via the so-called equivariant Hopf bifurcation in a 
parameterized system (i.e. a phenomenon occurring when the parameter 
crosses some "critical" value causing a change of stability of the 
"trivial solution", which results in appearance of small amplitude 
non-constant periodic solutions near the trivial one). The commonly used 
method (M. Golubitsky et al.) is based on the equivariant singularity 
theory (EST) combined with a Lyapunov-Schmidt reduction /Central 
Manifold Theorem. However, this method meets serious difficulties when 
applied to a system with lack of smoothness and/or having a phase space 
without local linear structure.   

On the other hand, if an inductance element in the van der Pol 
oscillator contains a ferromagnetic core, the ferromagnetic material can 
introduce a hysteresis relation between the magnetic induction and 
magnetic field. In many cases (for example, in the presence of the 
ferroresonance phenomenon), the hysteresis effect cannot be  neglected. 
As a matter of fact, systems with hysteresis almost always are 
non-smooth and their phase spaces do not admit local linear structure, 
therefore, the EST based method cannot be applied to them. In my talk, I 
will show how an alternative method based on the usage of the new 
invariant  twisted equivariant degree  (introduced by Z. Balanov and W. 
Krawcewicz)  -- can be effectively applied to symmetric networks of VDPO 
with hysteresis. In particular, a direct link between physics, topology, 
algebra and analysis underlying the VDPOs will be established. 

This talk is based on a joint work with W. Krawcewicz, D. Rachinskii and 
A. Zhezherun.  

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