# American Institute of Mathematical Sciences

2011, 2011(Special): 447-456. doi: 10.3934/proc.2011.2011.447

## The geometry of limit cycle bifurcations in polynomial dynamical systems

 1 United Institute of Informatics Problems, National Academy of Sciences of Belarus, L. Beda Str. 6-4, Minsk 220040, Belarus

Received  July 2010 Revised  January 2011 Published  October 2011

In this paper, applying a canonical system with eld rotation parameters and using geometric properties of the spirals lling the interior and exterior domains of limit cycles, we solve the problem on the maximum number of limit cycles for the classical Lienard polynomial system which is related to the solution of Smale's thirteenth problem. By means of the same geometric approach, we generalize the obtained results and solve the problem on the maximum number of limit cycles surrounding a unique singular point for an arbitrary polynomial system which is related to the solution of Hilbert's sixteenth problem on the maximum number and relative position of limit cycles for planar polynomial dynamical systems.
Citation: Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447
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