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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
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