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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].
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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

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