Number and location of limit cycles in a class of perturbed polynomial systems

被引：0

作者：

Yang C.-X. ^{[1
]}

Wang R.-Q. ^{[2
]}

机构：

[1] Department of Mathematics, Yuxi Normal College

[2] Academy of Mathematics and Systems Science, Academia Sinica, Beijing

来源：

Acta Mathematicae Applicatae Sinica | 2004年 / 20卷 / 1期

基金：

中国国家自然科学基金;

关键词：

Bifurcation; Limit cycles; Polynomial system; Stability;

D O I：

10.1007/s10255-004-0158-y

中图分类号：

学科分类号：

摘要：

In this paper, we investigate the number, location and stability of limit cycles in a class of perturbed polynomial systems with (2n + 1) or (2n + 2)-degree by constructing detection function and using qualitative analysis. We show that there are at most n limit cycles in the perturbed polynomial system, which is similar to the result of Perko in [8] by using Melnikov method. For n = 2, we establish the general conditions depending on polynomial's coefficients for the bifurcation, location and stability of limit cycles. The bifurcation parameter value of limit cycles in [5] is also improved by us. When n = 3 the sufficient and necessary conditions for the appearance of 3 limit cycles are given. Two numerical examples for the location and stability of limit cycles are used to demonstrate our theoretical results. © Springer-Verlag 2004.

引用

页码：155 / 166

页数：11

共 11 条

[1]

Wang Y.Q., Jing Z.J., Cubic Lienard equations with quadratic damping (II), English Series, 18, 1, pp. 103-116, (2002)

[2]

Han M., On the number and distributions of limit cycles in a cubic system, Chin. Ann. of Math, 23 A, 2, pp. 143-152, (2002)

[3]

Ilieve I.D., Perko I.M., Higher order bifurcations of limit cycles, J. Diff. Eqns, 154, pp. 339-363, (1999)

[4]

Rychkov G.S., The maximum number of limit cycles of the system x′ = y - a0x - a1x 3 - a2x5, y′ = -x is two, Diff. Eqns, 11, pp. 380-391, (1973)

[5]

Li J.B., Huang Q.M., Bifurcations of limit cycles forming compound eyes in the cubic system, Chin. Ann. Math, 8 B, pp. 391-403, (1987)

[6]

Liu Z.R., Yang Z.Y., Jiang T., The same distribution of limit cycles in five perturbed cubic Hamiltonian systems, International Joural of Bifurcation and Chaos, 13, 1, pp. 243-249, (2003)

[7]

Perko L., Differential equations and dynamical system, (1996)

[8]

Melnikov V.K., On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc, 12, pp. 1-57, (1963)

[9]

Arnold V.I., Loss of stability of self-oscillation close to resonance and versal deformations of equivariant vector fields, Funct. Anal. Appl, 11, pp. 1-10, (1977)

[10]

Li C.Z., Li W.G., Llibre J., Et al., Polynomial systems: A lower bound for the weakened 16th Hilbert problem, Extracta Mathematicae, 16, 3, pp. 441-447, (2001)

← 1 2 →