Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T19:00:25.227Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometrical conditions for the stability of orbits in planar systems

Published online by Cambridge University Press:  24 October 2008

R. A. Garcia ,
A. Gasull  and
A. Guillamon
R. A. Garcia
Affiliation:
Departamento de Matemática, Universidade Federal de Goiás Cx. Postal 131, Goiânia, Brazil
A. Gasull
Affiliation:
Departament de Matemàtiques, Universitat Autonòma de Barcelona, 08193 Bellaterra, Catalonia, Spain e-mail address: [email protected].
A. Guillamon
Affiliation:
Departament de Matemàtiques, Universitat Autonòma de Barcelona, 08193 Bellaterra, Catalonia, Spain e-mail address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a vector field X on the real plane, we study the influence of the curvature of the orbits of ẋ = X┴(x) in the stability of those of the system x˙ = X(x). We pay special attention to the case in which this curvature is negative in the whole plane. Under this assumption, we classify the possible critical points and give a criterion for a point to be globally asymptotically stable. In the general case, we also provide expressions for the first three derivatives of the Poincaré map associated to a periodic orbit in terms of geometrical quantities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 3 , October 1996 , pp. 499 - 519
Copyright
Copyright © Cambridge Philosophical Society 1996

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

REFERENCES

[1]Berger, M. and Gostiaux, B.. Differential geometry: manifolds, curves and surfaces (Springer-Verlag, 1988).CrossRefGoogle Scholar
[2]Chicone, C.. Bifurcations of nonlinear oscillations and frequency entrainment near resonance. SIAM J. Math. Anal. 23 (1992), 1577–1608.CrossRefGoogle Scholar
[3]Diliberto, S.. On systems of ordinary differential equations. Contributions to the Theory of Nonlinear Oscillators 1 (1950), 1–38.Google Scholar
[4]Fessler, R.. A proof of the two-dimensional Markus-Yamabe conjecture, Annales Polonici Mathematici 62 (1995), 45–75.CrossRefGoogle Scholar
[5]Gutiérrez, C.. A solution to the bidimensional global asymptotic stability conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (6) (1995), 627–671.CrossRefGoogle Scholar
[6]Gasull, A., Llibre, J. and Sotomayor, J.. Global asymptotic stability of differential equations in the plane. J. Differential Equations 91 (1991), 327–335.CrossRefGoogle Scholar
[7]Lloyd, N. G.. A note on the number of limit cycles in certain two-dimensional systems. J. London Math. Soc. (2) 20 (1979), 277–286.CrossRefGoogle Scholar
[8]Olech, C.. On the global stability of an autonomous system on the plane. Contributions to Differential Equations 1 (1963), 389–400.Google Scholar
[9]Pogorelov, A. V.. Differential geometry (Noordhoff N.V., 1956).Google Scholar
[10]Wang, Chengwen. Generalized geographical family and the stability of limit cycles. J. Math. Annal. Appl. 185 (1994), 175–188.CrossRefGoogle Scholar
[11]Yamato, K.. An effective method of counting the number of limit cycles. NagoyaMath. J. 76 (1979), 35–114.Google Scholar
[12]Yan-Qian, Ye et al. Theory of limit cycles. Transl. of Math. Monographs 66 (Amer. Math. Soc., 1986).Google Scholar