The sufficient center conditions for a class of bidimensional polynomial systems of differential equations with nonlinearities of the fourth degree

Abstract

In this paper we consider the class of systems (1) (or (2)) with the conditions I3 = 0, I1 =0, I2 > 0. The conditions I1 = 0, I2 > 0 mean that the eigenvalues of the Jacobian matrix at the singular point (0, 0) are pure imaginary, i.e., the system has the center or a weak focus at (0, 0). The system (2) with I1 = 0, I2 > 0 and I3 = 0 can be reduced by a centeraffine transformation and time scaling to the form. In this paper the sufficient center conditions for the origin of coordinates of the phase plane for the bidimensional polynomial systems of differential equations with nonlinearities of the fourth degree with I1 = 0, I2 > 0, I3 = 0 were established.