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Abstract competitive systems and orbital stability in $\mathbf{{\mathbb R}^3}$


Authors: Rafael Ortega and Luis Ángel Sánchez
Journal: Proc. Amer. Math. Soc. 128 (2000), 2911-2919
MSC (2000): Primary 34C25, 34C12, 34D20
DOI: https://doi.org/10.1090/S0002-9939-00-05610-0
Published electronically: April 7, 2000
MathSciNet review: 1701688
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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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  • 1. A. Berman and R. J. Plemmons, Nonnegative matrices in mathematical sciences, SIAM, Philadelphia, 1994. MR 95e:15013
  • 2. W. Hahn, Stability of Motion, Springer-Verlag, Berlin, 1967. MR 36:6716
  • 3. M. W. Hirsch, Systems of differential equations which are competitive or cooperative. I: Limit sets, SIAM J. Math. Anal. 13 (1982), 167-179. MR 83i:58081
  • 4. M. W. Hirsch, Systems of differential equations which are competitive or cooperative. II: Convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), 423-439. MR 87a:58137
  • 5. M. W. Hirsch, Systems of differential equations which are competitive or cooperative. IV: Structural stability in three-dimensional systems., SIAM J. Math. Anal. 21 (1990), 1225-1234. MR 92h:58109
  • 6. H. Hofer and J. Toland, Homoclinic, heteroclinic, and periodic orbits for a class of indefinite hamiltonian systems, Math. Ann. 268 (1984), 387-403. MR 85j:58123
  • 7. M. A. Krasnoselskij, J. A. Lifshits, A. V. Sobolev, Positive Linear Systems, Heldermann Verlag, Berlin, 1989. MR 91f:47051
  • 8. W. Rudin, Functional Analysis, McGraw Hill, New York, 1973. MR 51:1315
  • 9. H. L. Smith, Monotone Dynamical Systems, American Mathematical Society, Providence, 1995. MR 96c:34002
  • 10. R. A. Smith, Existence of periodic orbits of autonomous ordinary differential equations, Proceedings of the Royal Society of Edinburgh 85A (1980), 153-172. MR 81e:34034
  • 11. R. A. Smith, Orbital Stability for Ordinary Differential Equations, Journal of Differential Equations 69 (1987), 265-287. MR 88g:34087
  • 12. R. A. Smith, Certain differential equations have only isolated periodic orbits, Ann. Mat. Pura Appl. 137 (1984), 217-244. MR 86d:34070
  • 13. H. R. Zhu and H. L. Smith, Stable Periodic Orbits for a Class of Three Dimensional Competitive Systems, Journal of Differential Equations 110 (1994), 143-156. MR 95e:34033

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Additional Information

Rafael Ortega
Affiliation: Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: rortega@goliat.ugr.es

Luis Ángel Sánchez
Affiliation: Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: lasperez@goliat.ugr.es

DOI: https://doi.org/10.1090/S0002-9939-00-05610-0
Keywords: Orbital stability, competitive systems, monotone systems
Received by editor(s): November 3, 1998
Published electronically: April 7, 2000
Additional Notes: This research was supported by DGES PB95-1203 (Spain)
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society

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