Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T02:32:15.750Z Has data issue: false hasContentIssue false

Uniqueness of limit cycles in polynomial systems with algebraic invariants

Published online by Cambridge University Press:  17 April 2009

André Zegeling
Affiliation:
Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Robert E. Kooij
Affiliation:
Faculty of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The uniqueness of limit cycles is proved for quadratic systems with an invariant parabola and for cubic systems with four real line invariants. Also a new, simple proof is given of the uniqueness of limit cycles occurring in unfoldings of certain vector fields with codimension two singularities.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Bautin, N.N., ‘On periodic solutions of a system of differential equations’, (Russian), Prikl. Mat. Mekh. 18 (1954), 128.Google Scholar
[2]Shuping, Chen, ‘Limit cycles of a real quadratic differential system having a parabola as an integral curve’, (Chinese), Kexue Tongbao 30 (1985), 401405.Google Scholar
[3]Cherkas, L.A. and Zhilevich, L.I., ‘Some tests for the absence or uniqueness of limit cycles’, (Russian), Differentsial'nye Uravneniya 6 (1970), 11701178. Translated in Differential Equations 6 (1970), 891897.Google Scholar
[4]Cherkas, L.A. and Zhilevich, L.I., ‘The limit cycles of certain differential equations’, (Russian), Differentsial'nye Uravneniya 8 (1972), 12071213. Translated in Differential Equations 8 (1972), 924929.Google Scholar
[5]Chow, Shui-Nee, Li, Chenghzi and Wang, Duo, ‘Uniqueness of periodic orbits of some vector fields with codimension two singularities’, J. Differential Equations 77 (1989), 231253.CrossRefGoogle Scholar
[6]Christopher, C.J., ‘Quadratic systems having a parabola as an integral curve’, Proceedings of the Royal Society of Edinburgh 112 A (1989), 113134.CrossRefGoogle Scholar
[7]Coppel, W.A., ‘Quadratic systems with a degenerate critical point’, Bull. Austral. Math. Soc. 38 (1988), 110.Google Scholar
[8]Coppel, W.A., ‘Some quadratic systems with at most one limit cycle’, Dynamics Reported 2 (1989), 6188.Google Scholar
[9]Coppel, W.A., ‘A new class of quadratic systems’, J. Differential Equations 92 (1991), 360372.Google Scholar
[10]Coppel, W.A., Private communication.Google Scholar
[11]Guoren, Dai and Songlin, Wo, ‘Closed orbits and straight line invariants in E 3 systems’, (Chinese), Acta Math. Sci. (Chinese) 9 (1989), 251261.Google Scholar
[12]Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (Springer-Verlag, Berlin, Heidelberg, New York, 1983).CrossRefGoogle Scholar
[13]Hilbert, D., ‘Mathematical problems’, Mary Newson Transl, Bull. Amer. Math. Soc. 8 (1902), 437479.Google Scholar
[14]Kooij, R.E., ‘Some new properties of cubic systems’, report 90–62, (Delft University of Technology, 1990).Google Scholar
[15] R.E. Kooij, ‘Existence and uniqueness of limit cycles for a special system of differential equations on the plane’, report 92–63, (Delft University of Technology, 1990).Google Scholar
[16]Jun, Liu, ‘Transformations and their applications in planar quadratic systems’, (Chinese), J. Wuhan Iron and Steel College 4 (1979), 1015.Google Scholar
[17]Rychkov, G.S., ‘The limit cycles of the equation u(x + 1)du = (-x + ax 2 + bxu + cu+ dx 2) dx’, (Russian), Differential'nye Uravneniya 8 (1972), 22572259. Translated in Differential Equations 8 (1972), 17481750.Google Scholar
[18]Xian, Wang, ‘On the uniqueness of limit cycles of the system = x(y) - F(x), = g(x)’, (Chinese), J. Nanjing Univ. Natur. Sci. 26 (1990), 363372.Google Scholar
[19]Yanqian, Ye, Theory of limit cycles, Transl. Math. Monographs 66 (American Mathematical Society, Providence, R.I., 1986).Google Scholar
[20]Pingguang, Zhang, ‘Study of non-existence of limit cycles around a weak focus of order two or three for quadratic systems’, Chinese Science Bull. 35 (1990), 11561161.Google Scholar
[21]Zhifen, Zhang, ‘On the uniqueness of limit cycles of certain equations of nonlinear oscillations’, (Russian), Dokl. Akad. Nauk SSSR 119 (1958), 659662.Google Scholar
[22]Zhifen, Zhang, ‘Proof of the uniqueness theorem of limit cycles of generalized Liénard equations’, Appl. Anal. 23 (1986), 6376.Google Scholar
[23]Zoladek, H., ‘On the versality of a family of symmetric vector fields in the plane’, Math. USSR-Sb 48 (1984), 463492.Google Scholar
[24]Zoladek, H., ‘Bifurcations of certain family of planar vector fields tangent to axes’, J. Differential Equations 67 (1987), 155.Google Scholar
[25]Zoladek, H., ‘Quadratic systems with center and their perturbations’. Preprint, (University of Warsaw, 1990).Google Scholar