The limit cycles of a second-order system of differential equations: the method of small forms (Q1037082)

The paper deals with the problem of the existence of limit cycles for a system in the form \[ \dot{x}=-y+ \sum_{i+j=k_0}^{n} (a_{ij}+ \mu_{ij})x^i y^j,\quad \dot{y}=x+ \sum_{i+j=k_0}^{n} (b_{ij}+ \lambda_{ij})x^i y^j, \] where \(a_{ij},\) \(b_{ij}\) are real constants, \(\mu_{ij},\) \(\lambda_{ij}\) are parameters, \(k_0\geq 2.\) For this purpose, the method of small forms, based on the representation of a solution as a sum of forms with respect to the initial value and the parameter, is proposed. The author constructs a polynomial, whose positive roots allow to obtain the number of limit cycles in a sufficiently small neighborhood of the equilibrium point.