Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T13:14:55.487Z Has data issue: false hasContentIssue false
  • English
  • Français

On the limit cycles of polynomial differential systems with homogeneous nonlinearities

Published online by Cambridge University Press:  20 January 2009

Chengzhi Li ,
Weigu Li ,
Jaume Llibre  and
Zhifen Zhang
Chengzhi Li
Affiliation:
Department of Mathematics, Peking University, Beijing 100871, People's Republic of China
Weigu Li
Affiliation:
Department of Mathematics, Peking University, Beijing 100871, People's Republic of China
Jaume Llibre
Affiliation:
Department de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
Zhifen Zhang
Affiliation:
Department of Mathematics, Peking University, Beijing 100871, People's Republic of China
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.

We consider three classes of polynomial differential equations of the form ẋ = y + establish Pn (x, y), ẏ = x + Qn (x, y), where establish Pn and Qn are homogeneous polynomials of degree n, having a non-Hamiltonian centre at the origin. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centres when we perturb them inside the class of all polynomial differential systems of the above form. A more detailed study is made for the particular cases of degree n = 2 and n = 3.

Keywords

limit cyclescentresbifurcation
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 3 , October 2000 , pp. 529 - 543
Copyright
Copyright © Edinburgh Mathematical Society 2000

References

1.Andronov, A. A., Les cycles limites de Poincaré et la théorie des oscillations autoentretenues, C. R. Acad. Sci. Paris 189 (1929), 559–561.Google Scholar
2.Arnold, V. I. and Ilyashenko, Y. S., Dynamical systems. I. Ordinary differential equations. Encyclopaedia of Mathematical Sciences, vol. 1 (Springer, New York, 1988).Google Scholar
3.Blows, T. R. and Perko, L. M., Bifurcation of limit cycles from centers and separatrix cycles of planes analytic systems, SIAM Rev. 36 (1994), 341–376.CrossRefGoogle Scholar
4.Carbonell, M. and Llibre, J., Limit cycles of a class of polynomial systems, Proc. R. Soc. Edinb. A 109 (1988), 187–199.CrossRefGoogle Scholar
5.Carbonell, M. and Llibre, J., Hopf bifurcation, averaging methods and Liapunov quantities for polynomial systems with homogeneous nonlinearities, in Proc. Eur. Conf. on Iteration Theory, ECIT87, pp. 145–160 (World Scientific, Singapore, 1989).Google Scholar
6.Carbonell, M., Coll, B. and Llibre, J., Limit cycles of polynomial systems with homogeneous nonlinearities, J. Math. Analysis Appl. 142 (1989), 573–590.CrossRefGoogle Scholar
7.Cherkas, L. A., Number of limit cycles of an autonomous second-order system, Diff. Eqns 5 (1976), 666–668.Google Scholar
8.Chicone, C., Limit cycles of a class of polynomial vector fields in the plane, J. Diff. Eqns 63 (1986), 68–87.CrossRefGoogle Scholar
9.Chicone, C. and Jacobs, M., Bifurcation of limit cycles from quadratic isochronous, J. Diff. Eqns 91 (1991), 268–326.CrossRefGoogle Scholar
10.Coddington, E. A. and Levinson, N., Theory of ordinary differential equations (McGraw-Hill, New York, 1955).Google Scholar
11.Coll, B., Gasull, A. and Prohens, R., Differential equations defined by the sum of two quasi homogeneous vector fields, Can. J. Math. 49 (1997), 212–231.CrossRefGoogle Scholar
12.Dumortier, F., Li, C. and Zhang, Z., Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops, J. Diff. Eqns 139 (1997), 146–193.CrossRefGoogle Scholar
13.Gasull, A. and Llibre, J., Limit cycles for a class of Abel equations, SIAM J. Math. Analysis 21 (1990), 1235–1244.CrossRefGoogle Scholar
14.Gasull, A., Llibre, J. and Sotomayor, J., Limit cycles of vector fields of the form X(v) = Av + f (v) Bv, J. Diff. Eqns 67 (1987), 90–110.CrossRefGoogle Scholar
15.Gasull, A., Llibre, J. and Sotomayor, J., Further considerations on the number of limit cycles of vector fields of the form X(v) = Av + f (v) Bv, J. Diff. Eqns 68 (1987), 36–40.CrossRefGoogle Scholar
16.Giacomini, H., Llibre, J. and Viano, M., On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity 9 (1996), 501–516.CrossRefGoogle Scholar
17.Giacomini, H., Llibre, J. and Viano, M., On the shape of limit cycles that bifurcate from Hamiltonian centers, Nonlinear Analysis Theory Methods Appl. (In the press.)Google Scholar
18.Giacomini, H., Llibre, J. and Viano, M., The shape of limit cycles that bifurcate from non-Hamiltonian centers, Nonlinear Analysis Theory Methods Appl. (Submitted.)Google Scholar
19.Gonzales, E. A. V., Generic properties of polynomial vector fields at infinity, Trans. Am. Math. Soc. 143 (1969), 201–222.CrossRefGoogle Scholar
20.Guckenheimer, J. and Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Sci. 42, Second Printing (Springer, New York, 1986).Google Scholar
21.Horozov, E. and Iliev, I. D., On the number of limit cycles in perturbations of quadratic Hamiltonian systems, Proc. Land. Math. Soc. 69 (1994), 198–224.CrossRefGoogle Scholar
22.Kamke, E., Differentialgleichungen ‘losungsmethoden und losungen’, Col. Mathematik und ihre anwendungen, 18, Akademische Verlagsgesellschaft Becker und Erler Kom-Ges., Leipzig (1943).Google Scholar
23.Li, B. and Zhang, Z., A note on a result of G. S. Petrov about the weakened 16th Hilbert problem, J. Math. Analysis Appl. 190 (1995), 489–516.CrossRefGoogle Scholar
24.Liénard, A., Etude des oscillations entretenues, Rev. Générale de l'Electricité 23 (1928), 901–912.Google Scholar
25.Lloyd, N. G., Limit cycles of certain polynomial differential systems, in Nonlinear functional analysis and its applications (ed. Singh, S. P.), pp. 317–326, NATO AS1, series C, vol. 173 (Reidel, Dordrecht, 1986).CrossRefGoogle Scholar
26.Mardesic, P., The number of limit cycles of polynomials deformations of a Hamiltonian vector field, Ergodic Theory Dynam. Syst. 10 (1990), 523–529.CrossRefGoogle Scholar
27.Neto, A. L., On the number of solutions of the equation , for which x(0) = x(1), Inventiones Math. 59 (1980), 67–76.CrossRefGoogle Scholar
28.Petrov, G. S., Number of zeros of complete elliptic integrals, Funct. Analysis Appl. 18 (1988), 148–149.CrossRefGoogle Scholar
29.Petrov, G. S., The Chebyshev property of elliptic integrals, Funct. Analysis Appl. 22 (1988), 72–73.CrossRefGoogle Scholar
30.Poincaré, H., Mémoire sur les courbes définies par une équation differentielle, I, II, J. Math. Pures Appl. 7 (1881), 375–422; 8 (1882), 251–296; Sur les courbes définies pas les équations differentielles, III, IV, J. Math. Pures Appl. 1 (1885), 167–244; 2 (1886), 155–217.Google Scholar
31.Pontrjagin, L. S., Über autoschwingungs systeme, die den Hamiltonschen nahe liegen, Phyrikalische Zeit. Sowjetunion 6 (1934), 25–28.Google Scholar
32.Reyn, J. W., A bibliography of the qualitative theory of quadratic systems of differential equations in the plane, in Report of the Faculty of Technical Mathematics and Information, Delft, 3rd edn (1994), pp. 94–02.Google Scholar
33.Shaker, D. S. and Zegeling, A., Bifurcation of limit cycles from quadratic centers, J. Dig. Eqns 122 (1995), 48–70.Google Scholar
34.Van Der Pol, B., On relaxation-oscillations, Phil. Mag. 2 (1926), 978–992.CrossRefGoogle Scholar
35.Vulpe, N. I., Affine-invariant condition for the topological discrimination of quadratic systems with a center, Diff. Eqns 19 (1983), 273–280.Google Scholar
36.Zoladek, H., Quadratic systems with centers and their perturbations, J. Diff. Eqns 109 (1994), 223–273CrossRefGoogle Scholar