Valery Romanovski
Center for Applied Mathematics and Theoretical Physics
University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia and
Department of Mathematics
University of North Carolina at Charlotte
Charlotte, NC 28223, USA
Abstract.
Consider systems of the form
| dx/dt=P_n(x,y), | dy/dt=Q_n(x,y), | (1) |
where P_n(x,y) and Q_n(x,y) are polynomials of degree n with coefficients from a parameter space E. Following Bautin [1] we say that the singular point (x_0, y_0) of the system E_0 in E has cyclicity k with respect to E if and only if any perturbation of E_0 in E has at most k limit cycles in a neighborhood of (x_0, y_0) and k is the minimal number with this property. The problem of cyclicity is often called the local 16th Hilbert problem.
We apply algorithms of computational algebra to investigate the cyclicity problem and the similar problem of critical period bifurcations for system (1). We show that both of the problems can be reduced to the algebraic problem of computing a basis of the ideal generated by the coefficients of the Poincaré map and the period function, correspondingly, and demonstrate how to find such a basis for some systems of the form (1) finding, therefore, a bound for the bifurcating limit cycles and critical periods in these systems.
We also discuss the problem of computing invariants of a rotation group for system (1).
References