Bounding the number of limit cycles for a polynomial Liénard system by using regular chains

In this paper, we study the bound of the number of limit cycles by Poincaré bifurcation for a Liénard system of type (4,3). An automatic algorithm is constructed based on the Chebyshev criteria and the tools of regular chain theory in polynomial algebra. We prove the system can bifurcate at most 6 limit cycles from the periodic annulus by this algorithm and at least 4 limit cycles by asymptotic expansions of the related Melnikov functions.