Limit cycles of the polynomial Liénard system of degree 4 (Q2721696)

The author considers the polynomial Liénard system \(dx/dt=y-F(x)\), \(dy/dt=-x,\) where \(F(x)=a_{1}x+a_{2}x^{2}+\ldots+a_{n}x^{n}.\) \textit{A. Lins, W. de Melo} and \textit{C. C. Pugh} [Geom. Topol., III. Lat. Am. Sch. Math., Proc., Rio de Janeiro 1976, Lect. Notes Math. 597, 335-357 (1977; Zbl 0362.34022)] have conjectured that such a system has at most \(k\)~limit cycles, if the function \(F(x)\) is a polynomial of degree \(2k+1\) or \(2k+2.\) However, up to now, the problem that the system has at most one limit cycle is still not solved completely even for the case \(n=4.\) NEWLINENEWLINENEWLINEHere, the author studies the case \(n=4\) and obtains some results on the uniqueness of limit cycles under certain conditions.