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

In this paper we investigate the existence of limit cycles of a system of the second-order differential equations with a vector parameter.
We propose a method for representing a solution as a sum of forms with respect to the initial value and the parameter; we call this technique the method of small forms. We establish the conditions under which a sufficiently small neighborhood of the equilibrium point contains no limit cycles. We construct a polynomial, whose positive roots of an odd multiplicity define the lower bound for the number of cycles, and prime positive roots (other positive roots do not exist) define the number of limit cycles in a sufficiently small neighborhood of the equilibrium point.
We prove theorems, whose conditions guarantee that a positive root of an odd multiplicity defines a unique limit cycle, but a positive root of an even multiplicity defines exactly two limit cycles. We propose a method for defining the type of the stability of limit cycles.