Skip to main content Accessibility help
×
Home

Invariant algebraic curves and conditions for a centre

  • C. J. Christopher (a1)

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.

Copyright

References

Hide All
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.
2Chicone, C. and Jacobs, M.. Bifurcation of limit cycles from quadratic isochrones. J. Differential Equations 91 (1991), 268326.
3Christopher, Colin. Invariant Algebraic Curves in Polynomial Differential Systems (PhD Thesis, University College of Wales, Aberystwyth, 1990).
4Christopher, Colin. Quadratic systems having a parabola as an integral curves. Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 113134.
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.
6Coppel, W. A.. A survey of quadratic systems. J. Differential Equations 2 (1966), 293304.
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).
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.
9Dolov, M. V.. Limit cycles in the case of a centre. Differencialnye Uravneija 8 (1972), 16911692.
10Fulton, W.. Algebraic Curves (New York: W. A. Benjamin, 1969).
11Jin, X. and Wang, D.. On the conditions of Kukles for the existence of a centre. Bull. London Math. Soc. 22 (1990), 14.
12Kukles, I. S.. Sur quelques cas de distinction entre un foyer et un centre. Dokl. Akad. Nauk. SSSR 42 (1944), 208211.
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).
14Lloyd, N. G., Blows, T. R. and Kalenge, M. C.. Some cubic systems with several limit cycles. Nonlinearity 1 (1988), 653669.
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).
16Lloyd, N. G. and Pearson, J. M.. REDUCE and the bifurcation of limit cycles. J. Symbolic Comput. 9(1990), 215224.
17Lloyd, N. G. and Pearson, J. M.. Computing centre conditions for certain cubic systems. J. Comp. & Appl. Math. 40 (1992), 323336.
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.
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.
20Lynch, S.. Bifurcation of Limit Cycles in Systems of Lienard Type (PhD Thesis, University College of Wales, Aberystwyth, 1988).
21Prelle, M. J. and Singer, M. F.. Elementary first integrals of differential equations. Trans. Amer. Math. Soc. 279(1983), 215228.
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.
23Yasmin, N.. Closed Orbits of Certain Two-Dimensional Cubic Systems (PhD Thesis, University College of Wales, Aberystwyth, 1989).
24Yanqian, Ye and others. Theory of Limit Cycles, Translations of Mathematical Monographs 66 (Providence, R.I.: American Mathematical Society, 1986).

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed