Global dynamics of a Wilson polynomial Liénard equation

Abstract:

Gasull and Sabatini in [Ann. Mat. Pura Appl. 198 (2019), pp. 1985–2006] studied limit cycles of a Liénard system which has a fixed invariant curve, i.e., a Wilson polynomial Liénard system. The Liénard system can be changed into $\dot x=y-(x^2-1)(x^3-bx), ~ \dot y=-x(1+y(x^3-bx))$. For $b\leq 0.7$ and $b\geq 0.76$, limit cycles of the system are studied completely. But for $0.7<b<0.76$, the exact number of limit cycles is still unknown, and Gasull and Sabatini conjectured that the exact number of limit cycles is two (including multiplicities). In this paper, we give a positive answer to this conjecture and study all bifurcations of the system. Finally, we show the expanding of the moving limit cycle as $b>0$ increases and give all global phase portraits on the Poincaré disk of the system completely.