Uniqueness of Limit Cycle For Liénard differential Equations of Degree Four

Agraïments: The first author is supported by NSFC-10831003 and by CICYT grant number 2009PIV00064.We prove that any classical Liénard differential equation of degree four has at most one limit cycle, and the limit cycle is hyperbolic if it exists. This result gives a positive answer to the conjecture by A. Lins, W. de Melo and C. C. Pugh [4] in 1977 about the number of limit cycles for polynomial Liénard differential equations for n = 4