Differential Equations Defined by the Sum of two Quasi-Homogeneous Vector Fields

Coll, B. ; Gasull, A. ; Prohens, R.

Canadian journal of mathematics, Tome 49 (1997) no. 2, pp. 212-231

Résumé

In this paper we prove, that under certain hypotheses, the planar differential equation: ̇x = X 1(x, y) + X 2(x, y), ̇y = Y 1(x, y) + Y 2(x, y), where (X i , Y i ), i = 1, 2, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar ́e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

Détail

DOI : 10.4153/CJM-1997-011-0

Mots-clés : 34C05, 58F21

Bibliographie

[1] 1. Anosov, D.V. and Arnold, V.I., Dynamical Systems I, Encyclopaedia of Mathematical Sciences , Springer- Verlag, Berlin, Heidelberg, New York, 1988. Google Scholar

[2] 2. Broer, H.W., Dumortier, F., Strien, S.J.van and Takens, F., Structures in Dynamics, Studies Math. Phys. 2, (ed. de Jager, E. M.), North-Holland, 1981. Google Scholar

[3] 3. Blows, T.R. and Lloyd, N.G., The number of limit cycles of certain polynomial differential equations,, Proc. Roy. Soc. Edinburgh 98A(1984), 215–239. Google Scholar

[4] 4. Brunella, M. and Miari, M., Topological equivalence of a plane vector field with its principal part defined, through Newton polyhedra, J. Differential Equations 85(1990), 338–366. Google Scholar

[5] 5. Cherkas, L.A., Number of limit cycles of an autonomous second-order system,, Differential Equations 5(1976), 666–668. Google Scholar

[6] 6. Chicone, C., Limit cycles of a class of polynomial vector fields in the plane,, J. Differential Equations 63(1986), 68–87. Google Scholar

[7] 7. Carbonell, M. and Llibre, J., Limit cycles of a class of polynomial systems,, Proc. Roy. Soc. Edinburgh 109A(1988), 187–199. Google Scholar

[8] 8. Dumortier, F. and Rousseau, C., Cubic Liénard equations with linear damping,, Nonlinearity 3(1990), 1015– 1039. Google Scholar

[9] 9. Gasull, A. and Llibre, J., Limit cycles for a class of Abel equations,, SIAM J. Math. Anal. 21(1990), 1235– 1244. Google Scholar

[10] 10. Gasull, A., Llibre, J. and Sotomayor, J., Limit cycles of a vector field of the form: X(v) ¬ Av + f (v)Bv,, J. Differential Equations 167(1987), 90–110. Google Scholar

[11] 11. Isaacson, E. and Keller, H.B., Analysis of numerical methods, Wiley and Sons, 1966. Google Scholar

[12] 12. Koditschek, D.E. and Narendra, K.S., Limit cycles of planar quadratic differential equations,, J. Differential Equations 54(1984), 181–195. Google Scholar

[13] 13. Lins, A., On the number of solutions of the equation dx/dt=Σ aj (t)xj, 0 ≤ °t ≤ 1, for which x(0) = x(1),, Invent. Math. 59(1980), 67–76. Google Scholar

[14] 14. Lyapunov, A.M., Stability of motion, Mathematics in Science and Engineering, 30, Academic Press, New York, London, 1966. Google Scholar

[15] 15. Lloyd, N.G., A note on the number of limit cycles in certain two-dimensional systems,, J. London Math. Soc. 20(1979), 277–286. Google Scholar

[16] 16. Pliss, V.A., Non-local problems of the Theory of Oscillations, Academic Press, New York, 1966. Google Scholar

[17] 17. Poincaré, H.,Mémoire sur les courbes définies par une équation différentielle, Oeuvres T.1, J. Math. Pures Appl. Google Scholar

[18] 18. Perko, L.M. and Shü Shih-Lung, Existence, uniqueness, and nonexistence of limit cycles for a class of, quadratic systems in the plane, J. Differential Equations 53(1984), 146–171. Google Scholar