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Local bifurcations of critical periods for cubic Liénard equations with cubic damping. (English) Zbl 1163.34349
The paper is devoted to the local bifurcation of critical periods near a nondegenerate center $$O(0,0)$$ of the Liénard system
$\dot{x}=y,\quad \dot{y}=-g(x)y - f(x)$
with $$f(x)=a_1x+a_2x^2+a_3x^3$$ and $$g(x)=x+b_2x^2+b_3x^3,$$ where $$a_1, a_2, a_3, b_2, b_3\in{{\mathbb{R}}}$$. The authors first apply the results from C. Christopher and J. Devlin [J. Differ. Equations 200, No. 1, 1–17 (2004; Zbl 1059.34020)] to give a necessary and sufficient condition for the coefficients under which the cubic Liénard system with cubic damping has a center at $$O$$ and finding the set of coefficients in which the center is isochronous. It is proved that at most $$2$$ local critical periods can be produced from either a weak center of finite order or the linear isochronous center and that at most one local critical period can be produced from nonlinear isochronous centers.

##### MSC:
 34C23 Bifurcation theory for ordinary differential equations 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
##### Keywords:
Liénard system; weak center; isochronous center; bifurcation
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