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The geometry of limit cycle bifurcations in polynomial dynamical systems

Pages: 447 - 456, Issue Special, September 2011

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Valery A. Gaiko - United Institute of Informatics Problems, National Academy of Sciences of Belarus, L. Beda Str. 6-4, Minsk 220040, Belarus (email)

Abstract: 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 Lienard 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.

Keywords:  Planar polynomial dynamical system, classical Lienard polynomial system, Hilbert's sixteenth problem, Smale's thirteenth problem, eld rotation parameter, bifurcation; limit cycle
Mathematics Subject Classification:  Primary: 34C05, 34C07, 34C23; Secondary: 37G05, 37G10, 37G15

Received: July 2010;      Revised: January 2011;      Published: October 2011.