an:05368378
Zbl 1163.34349
Zou, Lan; Chen, Xingwu; Zhang, Weinian
Local bifurcations of critical periods for cubic Li??nard equations with cubic damping
EN
J. Comput. Appl. Math. 222, No. 2, 404-410 (2008).
00233463
2008
j
34C23 34C05 34C25
Li??nard system; weak center; isochronous center; bifurcation
The paper is devoted to the local bifurcation of critical periods near a nondegenerate center \(O(0,0)\) of the Li??nard 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 \textit{C. Christopher} and \textit{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 Li??nard 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.
Alexander Grin (Grodno)
Zbl 1059.34020