Abstract competitive systems and orbital stability in ℝ³

Ortega, Rafael; Sánchez, Luis

doi:10.1090/S0002-9939-00-05610-0

Abstract competitive systems and orbital stability in $\mathbf {{\mathbb R}^3}$
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by Rafael Ortega and Luis Ángel Sánchez

Proc. Amer. Math. Soc. 128 (2000), 2911-2919

DOI: https://doi.org/10.1090/S0002-9939-00-05610-0

Published electronically: April 7, 2000

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Abstract:

Competitive autonomous systems in ${\mathbb R}^3$ have the remarkable property of verifying an analogue of the Poincaré-Bendixon theorem for planar equations. This fact allows us to prove the existence of orbitally stable closed orbits for those systems under easily checkable hypothesis. Our aim is to introduce, by changing the ordering in ${\mathbb R}^3$, a new class of autonomous systems for which the preceding results directly extend. As a consequence we shall reinterpret some of the results of R. A. Smith in terms of the theory of monotone systems.

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