Complete bifurcation diagram and global phase portraits of Lienard differential equations of degree four

Li and Llibre in [J. Differential Equations 252 (2012) 3142-3162] proved that a Lienard system of degree four: dx/dt = y - (ax + bx(2) + cx(3) + x(4) ), dy/dt = -x has at most one limit cycle. Moreover, the limit cycle is stable and hyperbolic if it exists. Based on their works, the aim of this paper is to give the complete bifurcation diagram and global phase portraits in the Poincare disc of this system further. First we analyze the equilibria at both finity and infinity. Then, a necessary and sufficient condition for existence of separatrix loop is founded by the rotation property. Moreover, a necessary and sufficient condition of the existence of limit cycles is also obtained. Finally, we show that the complete bifurcation diagram includes one Hopf bifurcation surface and one bifurcation surface of separatrix loop. (C) 2019 Elsevier Inc. All rights reserved.