Wang, Xiaoxia; Blackmore, Denis; Wang, Chengwen The \(\omega \)-limit sets of a flow and periodic orbits. (English) Zbl 1198.37034 Chaos Solitons Fractals 41, No. 5, 2690-2696 (2009). Summary: We discuss the \(\omega \)-limit sets of a flow using the Conley theory, chain recurrence and Morse decompositions. Our results generalize and improve the related result of J. Schropp [Z. Angew. Math. Mech. 76, No. 6, 349–356 (1996; Zbl 0879.34049)], and we also show how they can be used as a basis for some new criteria for the existence of periodic orbits.Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control. MSC: 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37B35 Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems 34C25 Periodic solutions to ordinary differential equations 37B25 Stability of topological dynamical systems 37B30 Index theory for dynamical systems, Morse-Conley indices 37C27 Periodic orbits of vector fields and flows Citations:Zbl 0879.34049 PDFBibTeX XMLCite \textit{X. Wang} et al., Chaos Solitons Fractals 41, No. 5, 2690--2696 (2009; Zbl 1198.37034) Full Text: DOI References: [1] Conley, C., Isolated invariant sets and the Morse index, CBMS regional conference series in mathematics, vol. 38 (1978), AMS: AMS Providence, RI · Zbl 0397.34056 [2] Conley, C., The gradient structure of a flow: I, Ergod Theor Dyn Syst, 8, 11-26 (1988) · Zbl 0687.58033 [3] Schropp, J., A reduction principle for \(\omega \)-limit sets, Z Angew Math Meth, 76, 6, 349-356 (1996) · Zbl 0879.34049 [4] Bhatia, N. P.; Szegö, G., Stability theory of dynamical systems (1970), Springer: Springer Berlin · Zbl 0213.10904 [5] Butler, G.; Waltman, P., Persistence in dynamical systems, J Diff Equ, 63, 255-263 (1986) · Zbl 0603.58033 [6] Hale JK. Ordinary differential equations. Wiley Interscience, 1969; Kreiger Publ. Co., 1980.; Hale JK. Ordinary differential equations. Wiley Interscience, 1969; Kreiger Publ. Co., 1980. [7] Conley, C.; Easton, R., Isolated invariant sets and the isolating blocks, Trans Am Math Soc, 158, 35-61 (1971) · Zbl 0223.58011 [8] Mischaikow, K.; Mrozek, M., Conley index theory, (Fiedler, B., Dynamical systems II: towards applications (2002), North-Holland: North-Holland Amsterdam) · Zbl 1035.37010 [9] Spanier, E., Algebraic topology (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0145.43303 [10] Huang, T., Decompositions and Liapounov functions, Chaos, Solitons & Fractals, 13, 209-214 (2002) · Zbl 1007.37004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.