In this paper, we investigate a quintic Liénard equation which has a center at the origin. We give the conditions for the parameters for the isochronous centers and weak centers of exact order. Then, we present the global phase portraits for the system having isochronous centers. Moreover, we prove that at most four critical periods can bifurcate and show with appropriate perturbations that local bifurcation of critical periods occur from the centers.

中文翻译：

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一类五次李纳德方程的临界期分岔

在本文中，我们研究了一个以原点为中心的五次 Liénard 方程。我们给出了等时中心和精确顺序弱中心的参数条件。然后，我们展示了具有等时中心的系统的全局相图。此外，我们证明了最多四个关键时期可以分叉，并表明在适当的扰动下，关键时期的局部分叉发生在中心。