Uniform persistence and repellors for maps
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- by Josef Hofbauer and Joseph W.-H. So PDF
- Proc. Amer. Math. Soc. 107 (1989), 1137-1142 Request permission
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
- Ethan Akin, Dynamical systems: the topological foundations, Six lectures on dynamical systems (Augsburg, 1994) World Sci. Publ., River Edge, NJ, 1996, pp. 1–43. MR 1441122
- N. P. Bhatia and O. Hájek, Local semi-dynamical systems, Lecture Notes in Mathematics, Vol. 90, Springer-Verlag, Berlin-New York, 1969. MR 0251328
- Rufus Bowen, $\omega$-limit sets for axiom $\textrm {A}$ diffeomorphisms, J. Differential Equations 18 (1975), no. 2, 333–339. MR 413181, DOI 10.1016/0022-0396(75)90065-0
- Louis Block and John E. Franke, The chain recurrent set, attractors, and explosions, Ergodic Theory Dynam. Systems 5 (1985), no. 3, 321–327. MR 805832, DOI 10.1017/S0143385700002972
- Geoffrey Butler, H. I. Freedman, and Paul Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), no. 3, 425–430. MR 822433, DOI 10.1090/S0002-9939-1986-0822433-4
- Geoffrey Butler and Paul Waltman, Persistence in dynamical systems, J. Differential Equations 63 (1986), no. 2, 255–263. MR 848269, DOI 10.1016/0022-0396(86)90049-5
- Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
- Alessandro Fonda, Uniformly persistent semidynamical systems, Proc. Amer. Math. Soc. 104 (1988), no. 1, 111–116. MR 958053, DOI 10.1090/S0002-9939-1988-0958053-2
- H. I. Freedman and J. W.-H. So, Persistence in discrete models of a population which may be subjected to harvesting, Natur. Resource Modeling 2 (1987), no. 1, 135–145. MR 903009, DOI 10.1111/j.1939-7445.1987.tb00029.x —, Persistence in semi-dynamical systems, SIAM J. Math. Anal. 20 (1989), 930-938.
- Barnabas M. Garay, Uniform persistence and chain recurrence, J. Math. Anal. Appl. 139 (1989), no. 2, 372–381. MR 996964, DOI 10.1016/0022-247X(89)90114-5
- Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
- Jack K. Hale and Paul Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal. 20 (1989), no. 2, 388–395. MR 982666, DOI 10.1137/0520025
- Josef Hofbauer, A unified approach to persistence, Acta Appl. Math. 14 (1989), no. 1-2, 11–22. Evolution and control in biological systems (Laxenburg, 1987). MR 990032, DOI 10.1007/BF00046670
- J. Hofbauer, V. Hutson, and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol. 25 (1987), no. 5, 553–570. MR 915090, DOI 10.1007/BF00276199
- Josef Hofbauer and Karl Sigmund, The theory of evolution and dynamical systems, London Mathematical Society Student Texts, vol. 7, Cambridge University Press, Cambridge, 1988. Mathematical aspects of selection; Translated from the German. MR 1071180
- V. Hutson and W. Moran, Persistence of species obeying difference equations, J. Math. Biol. 15 (1982), no. 2, 203–213. MR 684934, DOI 10.1007/BF00275073
- J. P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein. MR 0481301
- Robert D. Franzosa and Konstantin Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Differential Equations 71 (1988), no. 2, 270–287. MR 927003, DOI 10.1016/0022-0396(88)90028-9
Additional Information
- © Copyright 1989 American Mathematical Society
- 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