Bounding the number of cycles of O.D.E.s in

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Criteria are given under which the boundary of an oriented surface does not consist entirely of trajectories of the differential equation in . 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.

**1.**William M. Boothby,*An introduction to differentiable manifolds and Riemannian geometry*, 2nd ed., Pure and Applied Mathematics, vol. 120, Academic Press, Inc., Orlando, FL, 1986. MR**861409****2.**S. Busenberg and P. van den Driessche,*Analysis of a disease transmission model in a population with varying size*, J. Math. Biol.**28**(1990), no. 3, 257–270. MR**1047163**, 10.1007/BF00178776**3.**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**4.**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**5.**W. B. Demidowitsch,*Eine Verallgemeinerung des Kriteriums von Bendixson*, Z. Angew. Math. Mech.**46**(1966), 145–146 (German). MR**0200532****6.**Miklós Farkas,*Periodic motions*, Applied Mathematical Sciences, vol. 104, Springer-Verlag, New York, 1994. MR**1299528****7.**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**8.**Michael Y. Li and James S. Muldowney,*Phase asymptotic semiflows, Poincaré’s condition, and the existence of stable limit cycles*, J. Differential Equations**124**(1996), no. 2, 425–448. MR**1370150**, 10.1006/jdeq.1996.0018**9.**N. G. Lloyd,*A note on the number of limit cycles in certain two-dimensional systems*, J. London Math. Soc. (2)**20**(1979), no. 2, 277–286. MR**551455**, 10.1112/jlms/s2-20.2.277**10.**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**11.**-,*Solution to problem 828,*Nieuw Arch. Wisk.**10**(1992), 150-152.**12.**J. A. Pace and M. L. Zeeman,*A bridge between the Bendixson-Dulac criterion in 𝑅² and Liapunov functions in 𝑅ⁿ*, Canad. Appl. Math. Quart.**6**(1998), no. 3, 189–193. Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology (Edmonton, AB, 1996). MR**1662901****13.**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**14.**Michael Spivak,*Calculus on manifolds. A modern approach to classical theorems of advanced calculus*, W. A. Benjamin, Inc., New York-Amsterdam, 1965. MR**0209411****15.**P. van den Driessche and M. L. Zeeman,*Three-dimensional competitive Lotka-Volterra systems with no periodic orbits*, SIAM J. Appl. Math.**58**(1998), no. 1, 227–234. MR**1610080**, 10.1137/S0036139995294767

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
34A26,
34C05,
34C25,
37C27

Retrieve articles in all journals with MSC (2000): 34A26, 34C05, 34C25, 37C27

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

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