Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generalization of Bendixson's criterion

Author: Michal Feckan
Journal: Proc. Amer. Math. Soc. 129 (2001), 3395-3399
MSC (2000): Primary 34A34, 34C40, 37C10
Published electronically: April 25, 2001
MathSciNet review: 1845018
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


Bendixson's condition on the nonexistence of periodic solutions for planar ordinary differential equations is extended to higher dimensional ordinary differential equations with first integrals to preclude the existence of certain invariant Lipschitz compact submanifolds for those equations.

References [Enhancements On Off] (What's this?)

  • 1. I. BENDIXSON, Sur les curbes définiés par des équations différentielles, Acta Math. 24 (1901), 1-88.
  • 2. Stavros Busenberg and P. van den Driessche, A method for proving the nonexistence of limit cycles, J. Math. Anal. Appl. 172 (1993), no. 2, 463–479. MR 1200999, 10.1006/jmaa.1993.1037
  • 3. Geoffrey Butler, Rudolf Schmid, and Paul Waltman, Limiting the complexity of limit sets in self-regulating systems, J. Math. Anal. Appl. 147 (1990), no. 1, 63–68. MR 1044686, 10.1016/0022-247X(90)90384-R
  • 4. W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965. MR 0190463
  • 5. W.B. DEMIDOWITSCH, Eine verallgemeinerung des kriteriums von Bendixson, Z. angew. Math. Mech. (ZAMM) 46 (1966), 145-146.
  • 6. H. DULAC, Recherche des cycles limites, C. R. Acad. Sci. Paris 204 (1937), 1703-1706.
  • 7. M. FECKAN, Criteria on the nonexistence of invariant Lipschitz submanifolds for dynamical systems, J. Differential Equations (to appear).
  • 8. M.Y. LI, Geometrical Studies on the Global Asymptotic Behaviour of Dissipative Dynamical Systems, Ph.D. Thesis, University of Alberta, 1993.
  • 9. Yi Li and James S. Muldowney, On Bendixson’s criterion, J. Differential Equations 106 (1993), no. 1, 27–39. MR 1249175, 10.1006/jdeq.1993.1097
  • 10. -, Evolution of surface functionals and differential equations, in ``Ordinary and Delay Diff. Equations'', J.K. Hale & J. Wiener (Eds.), Pitman Res. Not. Math. Ser. Vol. 272, Longman, Harlow (1992), 144-148. CMP 97:08
  • 11. Michael Y. Li and James S. Muldowney, Lower bounds for the Hausdorff dimension of attractors, J. Dynam. Differential Equations 7 (1995), no. 3, 457–469. MR 1348736, 10.1007/BF02219372
  • 12. -, Dynamics of differential equations on invariant manifolds, J. Differential Equations (to appear).
  • 13. C. Connell McCluskey and James S. Muldowney, Bendixson-Dulac criteria for difference equations, J. Dynam. Differential Equations 10 (1998), no. 4, 567–575. MR 1659945, 10.1023/A:1022677008393
  • 14. James S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 (1990), no. 4, 857–872. Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1988). MR 1096556, 10.1216/rmjm/1181073047
  • 15. Russell A. Smith, An index theorem and Bendixson’s negative criterion for certain differential equations of higher dimension, Proc. Roy. Soc. Edinburgh Sect. A 91 (1981/82), no. 1-2, 63–77. MR 648917, 10.1017/S0308210500012634
  • 16. Russell A. Smith, Some applications of Hausdorff dimension inequalities for ordinary differential equations, Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), no. 3-4, 235–259. MR 877904, 10.1017/S030821050001920X
  • 17. Michael Spivak, Calculus on manifolds. A modern approach to classical theorems of advanced calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965. MR 0209411

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34A34, 34C40, 37C10

Retrieve articles in all journals with MSC (2000): 34A34, 34C40, 37C10

Additional Information

Michal Feckan
Affiliation: Department of Mathematical Analysis, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia

Keywords: Invariant submanifolds, first integrals, flows
Received by editor(s): April 10, 2000
Published electronically: April 25, 2001
Additional Notes: This work was supported by Grant GA-MS 1/6179/00.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2001 American Mathematical Society