Deriving some new conditions on the existence of eight limit cycles for a cubic system

In James and Lloyd's paper, some conditions on the existence of eight limit cycles bifurcating from the origin for a cubic system were derived [1]. We have currently derived some new conditions for generating eight limit cycles from the origin for this cubic system. In fact, deriving such conditions is reduced to finding the symbolic real solutions of systems of algebraic equations and inequalities, but the involved computations are very complex. We combine the Ritt-Wu method with the Budan-Fourier theorem to solve systems of equations and inequalities. With this strategy, we derive these new conditions using MapleV 3 on SUN SPARC station 10 and SUN SERVER 1104. In addition, when verifying the conditions derived in [1] by solving the system of equations and inequalities arising in [1] in a different ordering, we found that one of the conditions a(7) not equal 0 should be a(7) > 0.