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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.

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
Full Text: DOI
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