Abstract Li and Llibre in [J. Differential Equations 252 (2012) 3142–3162] proved that a Lienard system of degree four: d x d t = y − ( a x + b x 2 + c x 3 + x 4 ) , d y d t = − 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.

中文翻译：

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四阶 Liénard 微分方程的完整分岔图和全局相图

摘要 Li 和 Llibre [J. 微分方程 252 (2012) 3142–3162] 证明了四阶 Lienard 系统：dxdt = y − ( ax + bx 2 + cx 3 + x 4 ) ，dydt = − x 至多有一个极限环。此外，如果存在，极限环是稳定的和双曲线的。基于他们的工作，本文的目的是进一步给出该系统庞加莱圆盘中完整的分叉图和全局相图。首先，我们分析无穷大和无穷大的均衡。然后，通过旋转性质建立了分界环存在的充要条件。此外，还得到了极限环存在的充要条件。最后，我们展示了完整的分岔图包括一个Hopf分岔面和一个分界环的分岔面。