Hopf Bifurcation in Symmetric Networks of 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.  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 or Central Manifold Theorem. However, this method  cannot be applied if the system in question is not smooth enough and/or the phase space does not admit a 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 the corresponding phase spaces do not admit a 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 ‚Äì 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.