Hostname: page-component-7c8c6479df-24hb2 Total loading time: 0 Render date: 2024-03-29T12:13:48.151Z Has data issue: false hasContentIssue false

Invariant algebraic curves and conditions for a centre

Published online by Cambridge University Press:  14 November 2011

C. J. Christopher
Affiliation:
Department of Mathematics, University of Wales, Aberystwyth, Dyfed, Wales SY23 3BZ, U.K.

Abstract

Conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux. We show how these methods can be used in the search for maximal numbers of bifurcating limit cycles. We also extend the method to include more degenerate cases such as are encountered in less generic systems. These lead to new classes of integrals. In particular, the Kukles system is considered, and new centre conditions for this system are obtained.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alwash, M. A. M. and Lloyd, N. G.. Non-autonomous equations related to polynomial twodimensional systems. Proc. Roy. Soc. Edinburgh Sect. A 1054 (1987), 129152.CrossRefGoogle Scholar
2Chicone, C. and Jacobs, M.. Bifurcation of limit cycles from quadratic isochrones. J. Differential Equations 91 (1991), 268326.CrossRefGoogle Scholar
3Christopher, Colin. Invariant Algebraic Curves in Polynomial Differential Systems (PhD Thesis, University College of Wales, Aberystwyth, 1990).Google Scholar
4Christopher, Colin. Quadratic systems having a parabola as an integral curves. Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 113134.CrossRefGoogle Scholar
5Christopher, C. J. and Lloyd, N. G.. On the paper of Jin and Wang concerning the conditions for a centre in certain cubic systems. Bull. London Math. Soc. 22 (1990), 512.CrossRefGoogle Scholar
6Coppel, W. A.. A survey of quadratic systems. J. Differential Equations 2 (1966), 293304.CrossRefGoogle Scholar
7Coppel, W. A.. The limit cycle configurations of quadratic systems. In Proceedings of the Ninth Conference on Ordinary and Partial Differential Equations, University of Dundee, 1986, Pitman Research Notes in Mathematical Sciences (Harlow: Longman, 1987).Google Scholar
8Darboux, G.. Memoire sur les equations differentielles algebriques du premier ordre et du premier degre. Bull. Sci. Math. Serie 22 (1878), 6096; 123-144; 151-200.Google Scholar
9Dolov, M. V.. Limit cycles in the case of a centre. Differencialnye Uravneija 8 (1972), 16911692.Google Scholar
10Fulton, W.. Algebraic Curves (New York: W. A. Benjamin, 1969).Google Scholar
11Jin, X. and Wang, D.. On the conditions of Kukles for the existence of a centre. Bull. London Math. Soc. 22 (1990), 14.CrossRefGoogle Scholar
12Kukles, I. S.. Sur quelques cas de distinction entre un foyer et un centre. Dokl. Akad. Nauk. SSSR 42 (1944), 208211.Google Scholar
13Lloyd, N. G.. Limit cycles of polynomial systems–some recent developments. In New Directions in Dynamical Systems, eds Bedford, T. and Swift, J., Lecture, L. M. S. Note Series 127, 192234 (Cambridge: Cambridge University Press, 1988).CrossRefGoogle Scholar
14Lloyd, N. G., Blows, T. R. and Kalenge, M. C.. Some cubic systems with several limit cycles. Nonlinearity 1 (1988), 653669.CrossRefGoogle Scholar
15Lloyd, N. G. and Pearson, J. M.. Conditions for a centre and the bifurcation of limit cycles in a class of cubic systems. In Bifurcations and Periodic Orbits of Planar Vector Fields, eds Francoise, J. P. and Roussarie, R., Lecture Notes in Mathematics 1455 (Berlin: Springer, 1990).Google Scholar
16Lloyd, N. G. and Pearson, J. M.. REDUCE and the bifurcation of limit cycles. J. Symbolic Comput. 9(1990), 215224.CrossRefGoogle Scholar
17Lloyd, N. G. and Pearson, J. M.. Computing centre conditions for certain cubic systems. J. Comp. & Appl. Math. 40 (1992), 323336.CrossRefGoogle Scholar
18Lunkevich, V. A. and Sibirskii, K. S.. Integrals of a general quadratic differential system in cases of a centre. Differencial'nye Uravnenija 18 (1982), 786792.Google Scholar
19Lunkevich, V. A. and Sibirskii, K. S.. Integrals of a system with a homogeneous third degree nonlinearity in the case of a centre. Differencial'nye Uravnenija 20 (1984), 13601365.Google Scholar
20Lynch, S.. Bifurcation of Limit Cycles in Systems of Lienard Type (PhD Thesis, University College of Wales, Aberystwyth, 1988).Google Scholar
21Prelle, M. J. and Singer, M. F.. Elementary first integrals of differential equations. Trans. Amer. Math. Soc. 279(1983), 215228.CrossRefGoogle Scholar
22Shube, A. S.. Sufficient conditions for the centre of a two-dimensional autonomous system with cubic right hand side. Differencial'nye Uravnenija 25 (1989), 20142016.Google Scholar
23Yasmin, N.. Closed Orbits of Certain Two-Dimensional Cubic Systems (PhD Thesis, University College of Wales, Aberystwyth, 1989).Google Scholar
24Yanqian, Ye and others. Theory of Limit Cycles, Translations of Mathematical Monographs 66 (Providence, R.I.: American Mathematical Society, 1986).Google Scholar