Geometrical conditions for the stability of orbits in planar systems

R. A. Garcia; A. Gasull; A. Guillamon

Geometrical conditions for the stability of orbits in planar systems

R. A. Garcia, A. Gasull, A. Guillamon

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

3 Citations (Scopus)

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 ẋ = 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.

Original language	English
Pages (from-to)	499-519
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Volume	120
Issue number	3
Publication status	Published - 1 Oct 1996

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title = "Geometrical conditions for the stability of orbits in planar systems",

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 ẋ = 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 Poincar{\'e} map associated to a periodic orbit in terms of geometrical quantities.",

author = "Garcia, {R. A.} and A. Gasull and A. Guillamon",

year = "1996",

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T1 - Geometrical conditions for the stability of orbits in planar systems

AU - Garcia, R. A.

AU - Gasull, A.

AU - Guillamon, A.

PY - 1996/10/1

Y1 - 1996/10/1

N2 - 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 ẋ = 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.

AB - 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 ẋ = 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.

M3 - Article

VL - 120

SP - 499

EP - 519

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -