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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bounding the number of cycles of O.D.E.s in ${\mathbf R}^n$


Authors: M. Farkas, P. van den Driessche and M. L. Zeeman
Journal: Proc. Amer. Math. Soc. 129 (2001), 443-449
MSC (2000): Primary 34A26, 34C05, 34C25, 37C27
Published electronically: July 27, 2000
MathSciNet review: 1800234
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Abstract:

Criteria are given under which the boundary of an oriented surface does not consist entirely of trajectories of the $C^1$ differential equation $\dot{x} = f(x)$ in ${\mathbf R}^n$. The special case of an annulus is further considered, and the criteria are used to deduce sufficient conditions for the differential equation to have at most one cycle. A bound on the number of cycles on surfaces of higher connectivity is given by similar conditions.


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Additional Information

M. Farkas
Affiliation: School of Mathematics, University of Technology, H-1521 Budapest, Hungary
Email: fm@math.bme.hu

P. van den Driessche
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada V8W 3P4
Email: pvdd@smart.math.uvic.ca

M. L. Zeeman
Affiliation: Division of Mathematics and Statistics, University of Texas at San Antonio, San Antonio, Texas 78249-0664
Email: zeeman@math.utsa.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05735-X
PII: S 0002-9939(00)05735-X
Keywords: Bendixson-Dulac, cycles, periodic orbit, genus, Stokes' Theorem
Received by editor(s): April 18, 1999
Published electronically: July 27, 2000
Additional Notes: The first author’s research was supported in part by the Hungarian Foundation for Scientific Research grant no. T029893
The second author’s research was supported in part by an NSERC Research Grant and the University of Victoria Committee on Faculty Research and Travel.
The third author’s research was supported in part by the University of Texas at San Antonio Office of Research Development.
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society