ON THE MULTIPLICITY OF THE CIRCUMFERENCE IN PLANAR POLYNOMIAL VECTOR FIELDS

In this work we consider planar polynomial differential systems of the form: (x) over dot = P(x, y), (y) over dot = Q(x, y), where P(x, y) and Q(x, y) are polynomials with real coefficients whose maximum degree is d. We only consider systems of this form with the circumference x(2) + y(2) - 1 = 0 as a periodic orbit. These systems take the form: (x) over dot = -yc(x,y) + f(x,y)a(x,y), (y) over dot = xc(x,y) + f(x,y)b(x,y), where f(x,y) = (x(2) + y(2) -1)/2 and a, b and c are real polynomials. Our interest in this work is to study the multiplicity of the circumference as periodic orbit of the aforementioned system. This work contains some theorems that characterize when the circumference is a limit cycle of multiplicity m and when it belongs to a period annulus. Moreover, if we assume that the system is of a particular form, we will give an upper bound for the possible multiplicities that the circumference may have as a limit cycle. Finally, we apply our results to some examples.