Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T09:37:39.809Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  20 November 2018

B. Coll ,
A. Gasull  and
R. Prohens
B. Coll
Affiliation:
Departament de Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa, Km7.5, 07071 Palma de Mallorca, Spain
A. Gasull
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Spain e-mail: GASULL@mat.uab.es
R. Prohens
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
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.

In this paper we prove, that under certain hypotheses, the planar differential equation: ˙x = X1(x, y) + X2(x, y), ˙y = Y1(x, y) + Y2(x, y), where (Xi, Yi), 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.

Keywords

34C0558F21
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 2 , 01 April 1997 , pp. 212 - 231
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Anosov, D.V. and Arnold, V.I., Dynamical Systems I, Encyclopaedia of Mathematical Sciences 1, Springer- Verlag, Berlin, Heidelberg, New York, 1988.Google Scholar
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. 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. 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. Cherkas, L.A., Number of limit cycles of an autonomous second-order system,, Differential Equations 5(1976), 666–668.Google Scholar
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. Carbonell, M. and Llibre, J., Limit cycles of a class of polynomial systems,, Proc. Roy. Soc. Edinburgh 109A(1988), 187–199.Google Scholar
8. Dumortier, F. and Rousseau, C., Cubic Liénard equations with linear damping,, Nonlinearity 3(1990), 1015– 1039.Google Scholar
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. 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. Isaacson, E. and Keller, H.B., Analysis of numerical methods, Wiley and Sons, 1966.Google Scholar
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. 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. Lyapunov, A.M., Stability of motion, Mathematics in Science and Engineering, 30, Academic Press, New York, London, 1966.Google Scholar
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. Pliss, V.A., Non-local problems of the Theory of Oscillations, Academic Press, New York, 1966.Google Scholar
17. Poincaré, H.,Mémoire sur les courbes définies par une équation différentielle, Oeuvres T.1, J. Math. Pures Appl.Google Scholar
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