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)

 
 

 

Uniform persistence and repellors for maps


Authors: Josef Hofbauer and Joseph W.-H. So
Journal: Proc. Amer. Math. Soc. 107 (1989), 1137-1142
MSC: Primary 58F12; Secondary 58F40, 92A15
DOI: https://doi.org/10.1090/S0002-9939-1989-0984816-4
MathSciNet review: 984816
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We establish conditions for an isolated invariant set $ M$ of a map to be a repellor. The conditions are first formulated in terms of the stable set of $ M$. They are then refined in two ways by considering (i) a Morse decomposition for $ M$, and (ii) the invariantly connected components of the chain recurrent set of $ M$. These results generalize and unify earlier persistence results.


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

  • [1] E. Akin, Foundations for dynamical systems, part I, topological dynamics, preprint. MR 1441122 (98d:58108)
  • [2] N. P. Bhatia and O. Hajek, Local semi-dynamical systems, Lecture Notes in Math. Vol. 90, Springer-Verlag, New York, 1969. MR 0251328 (40:4559)
  • [3] R. Bowen, $ \omega $-limit sets for Axiom A diffeomorphisms, J. Diff. Eqns. 18 (1975), 333-339. MR 0413181 (54:1300)
  • [4] L. Block and J. E. Franke, The chain recurrent set, attractors and explosions, Ergod. Th. Dynam. Sys. 5 (1985), 321-327. MR 805832 (87i:58107)
  • [5] G. J. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), 425-426. MR 822433 (87d:58119)
  • [6] G. J. Butler and P. Waltman, Persistence in dynamical systems, J. Diff. Eqns. 63 (1986), 255-263. MR 848269 (87k:54058)
  • [7] C. Conley, Isolated invariant sets and the Morse index, CBMS Vol. 38, Amer. Math. Soc., Providence, 1978. MR 511133 (80c:58009)
  • [8] A. Fonda, Uniformly persistent semi-dynamical systems, Proc. Amer. Math. Soc. 104 (1988), 111-116. MR 958053 (90a:34094)
  • [9] H. I. Freedman and J. W.-H. So, Persistence in discrete models of a population which may be subjected to harvesting, Nat. Res. Modeling. 2 (1987), 135-145. MR 903009 (88h:92038)
  • [10] -, Persistence in semi-dynamical systems, SIAM J. Math. Anal. 20 (1989), 930-938.
  • [11] B. M. Garay, Uniform persistence and chain recurrence, J. Math. Anal. Appl. 139 (1989), 372-381. MR 996964 (90h:54049)
  • [12] J. K. Hale, Asymptotic behavior of dissipative systems, Math. Surveys Monographs 25, Amer. Math. Soc., Providence, 1988. MR 941371 (89g:58059)
  • [13] J. K. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal. 20 (1989), 388-395. MR 982666 (90b:58156)
  • [14] J. Hofbauer, A unified approach to persistence, Proc. Conf. Laxenburg 1987, Acta Applicandae Mathematicae, 14 (1989), 11-22. MR 990032 (90e:92064)
  • [15] J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol. 25 (1987) 553-570. MR 915090 (89a:92059)
  • [16] J. Hofbauer and K. Sigmund, The theory of evolution and dynamical systems, Cambridge University Press, 1988. MR 1071180 (91h:92019)
  • [17] V. Hutson and W. Moran, Persistence of species obeying difference equations, J. Math. Biol. 15 (1982) 203-213. MR 684934 (84e:92030)
  • [18] J. P. LaSalle, The stability of dynamical systems, Soc. Indust. Appl. Math., Philadelphia, 1976. MR 0481301 (58:1426)
  • [19] K. Mischaikow and R. D. Franzosa, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Diff. Eqns. 71 (1988) 270-287. MR 927003 (89c:54078)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F12, 58F40, 92A15

Retrieve articles in all journals with MSC: 58F12, 58F40, 92A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0984816-4
Keywords: Discrete semi-dynamical systems, permanence, uniform persistence, repellor, isolated invariant set, Morse decomposition, chain recurrent set, basic sets, average Lyapunov functions, global attractor
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society