zbMATH — the first resource for mathematics

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
Full Text:
References:
 [1] Abhyankar, S.S., Local analytic geometry, (1964), Academic Press New York · Zbl 0146.17202 [2] Cherkas, L.A., Conditions for a center for the equation $$P_3(x) y y^\prime = \sum_{i = 0}^2 P_i(x) y^i$$, Differ. equ., 10, 2, 367-368, (1974), (in Russian) · Zbl 0296.34020 [3] Chicone, C.; Jacobs, M., Bifurcation of critical periods for plane vector fields, Trans. amer. math. soc., 312, 433-486, (1989) · Zbl 0678.58027 [4] Christopher, C., An algebraic approach to the classification of centers in polynomial Liénard systems, J. math. anal. appl., 229, 319-329, (1999) · Zbl 0921.34033 [5] Christopher, C.; Devlin, J., On the classification of Liénard systems with amplitude-independent periods, J. differential equations, 200, 1-17, (2004) · Zbl 1059.34020 [6] Dumortier, F.; Li, C., Quadratic Liénard equations with quadratic damping, J. differential equations, 139, 41-59, (1997) · Zbl 0881.34046 [7] Gelfand, I.M.; Kapranov, M.M.; Zelevinsky, A.V., Discriminants, resultants and multidimensional determinants, (1994), Birkhäuser Boston · Zbl 0827.14036 [8] Hartshorne, R., Algebraic geometry, (1977), Springer New York, Heidelberg, Berlin · Zbl 0367.14001 [9] Knuth, D.E., () [10] Ritt, J.F., Differential algebra, (1950), Amer. Math. Soc. New York · Zbl 0037.18501 [11] Romanovski, V.G.; Han, M., Critical period bifurcations of a cubic system, J. phys. A: math. gen., 36, 5011-5022, (2003) · Zbl 1037.34034 [12] Rousseau, C.; Toni, B., Local bifurcation of critical periods in vector fields with homogenous nonlinearities of the third degree, Canad. math. bull., 36, 473-484, (1993) · Zbl 0792.58030 [13] Rousseau, C.; Toni, B., Local bifurcation of critical periods in the reduced kukles system, Canad. math. bull., 49, 338-358, (1997) · Zbl 0885.34033 [14] Zhang, W.; Hou, X.-R.; Zeng, Z.-B., Weak center and bifurcation of critical periods in reversible cubic systems, Comput. math. appl., 40, 771-782, (2000) · Zbl 0962.34025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.