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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

Abstract:

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
problem.

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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Technion Math. Net (TECHMATH)
Editor: Michael Cwikel   <techm@math.technion.ac.il>
Announcement from: Simeon Reich   <sreich@techunix.technion.ac.il>
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