# Article

Full entry | PDF   (0.3 MB)
Keywords:
limit set of a set; attractor; quasi-attractor; hyperspace
Summary:
In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the $\omega$-limit set $\omega (Y)$ of $Y$ is the limit point of the sequence $\lbrace (\mathop {\mathrm Cl}Y)\cdot [i,\infty )\rbrace _{i=1}^{\infty }$ in $2^X$ and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.
References:
[1] N. P. Bhatia and G. P. Szegö: Stability Theory of Dynamical Systems. Springer-Verlag, Berlin, 1970. MR 0289890
[2] G. Butler and P. Waltmann: Persistence in dynamical systems. J.  Differential Equations 63 (1986), 255–263. DOI 10.1016/0022-0396(86)90049-5 | MR 0848269
[3] C. C. Conley: The gradient structure of a flow: I. Ergod. Th. & Dynam. Sys. $8^*$ (1988), 11–26. DOI 10.1017/S0143385700009305 | MR 0967626 | Zbl 0687.58033
[4] C. C. Conley: Isolated invariant sets and Morse index. Conf. Board Math. Sci., No  38, Amer. Math. Sci., Providence, 1978. MR 0511133
[5] C. C. Conley: Some abstract properties of the set of invariant sets of a flow. Illinois J.  Math. 16 (1972), 663–668. DOI 10.1215/ijm/1256065549 | MR 0315686 | Zbl 0241.54037
[6] J. K. Hale and P. Waltmann: Persistence in infinite-dimensional systems. SIAM J.  Math. Anal. 20 (1989), 388–395. DOI 10.1137/0520025 | MR 0982666
[7] R. Moeckel: Some comments on “The gradient structure of a flow: I”. vol. $8^*$, Ergod. Th. & Dynam. Sys., 1988. MR 0967626
[8] S. B. Nadler, Jr.: Continuum Theory: An Introduction. Marcel Dekker, New York-Basel-Hong Kong, 1992. MR 1192552 | Zbl 0757.54009
[9] T. Huang: Some global properties in dynamical systems. PhD. thesis, Inst. of Math., Academia Sinica, , 1998.

Partner of