On a cubic system with eight limit cycles

For a famous cubic system given by James and Lloyd, there exist some sufficient conditions such that the system has eight limit cycles. In this paper, we try to derive by computers the necessary and sufficient conditions for this system to have eight limit cycles. In order to find the symbolic real solutions to semi-algebraic systems where polynomials are Lyapunov quantities, we transform the equations into triangular systems by pseudo-division, locate the real solutions of the last equation and verify the inequalities by the Budan-Fourier theorem. The necessary and sufficient conditions for the system to have eight limit cycles are given under a reasonable limitation.