Abstract
The main aim of this paper is localization of the chain recurrent set in shape theoretical framework. Namely, using the intrinsic approach to shape from [1] we present a result which claims that under certain conditions the chain recurrent set preserves local shape properties. We proved this result in [2] using the notion of a proper covering. Here we give a new proof using the Lebesque number for a covering and verify this result by investigating the symbolical image of a couple of systems of differential equations following [3].
References
[1] N. Shekutkovski, Intrinsic definition of strong shape for compact metric spaces, Topology Proceedings, 39 (2012), 27-39.Search in Google Scholar
[2] N. Shekutkovski, M. Shoptrajanov, Intrinsic Shape of the Chain recurrent set, Topology and its Applications, 202 (2016), 117-126.10.1016/j.topol.2015.12.062Search in Google Scholar
[3] G. Osipenko, Symbolical analysis of the chain recurrent trajectories of dynamical systems, Differential Equations and Control Processes, (1998), No.4.Search in Google Scholar
[4] H. M. Hastings, A higher dimensional Poincare-Bendixon theorem, Glasnik Matematicki, 34 (1979), 263-268.Search in Google Scholar
[5] B. A. Bogatyi, V. I. Gutsu, On the structure of attracting compacta, Differentsial’nye Uravneniya, 25 (1989), 907-909.Search in Google Scholar
[6] J. B. Gunther, J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. AMS, 119 (1993), 321-329.10.1090/S0002-9939-1993-1170545-4Search in Google Scholar
[7] J. M. Sanjurjo, On the structure of uniform attractors, J.Math. Anal. Appl, 192 (1995), 519-528.10.1006/jmaa.1995.1186Search in Google Scholar
[8] I. Rodnianski, L. Kapitanski, Shape and Morse theory of attractors, Communications On Pure and Applied Mathematics, 53 (2000), no.2, 218-242.Search in Google Scholar
[9] J. M. Sanjurjo, A non continuous description of the shape category of compacta, Quart.J.Math.Oxford, 40 (1989), 351-359.10.1093/qmath/40.3.351Search in Google Scholar
[10] M. Shoptrajanov,N. Shekutkovski, Characterization of topologically tranzitive atrractors for analitic plane flows, Filomat, 29:1 (2015), 89-97.10.2298/FIL1501089SSearch in Google Scholar
[11] C.C. Conley, The gradient structure of the flow, I.Ergodic Th.Dynam. Systems, 8 (1988), 11-26.10.1017/S0143385700009305Search in Google Scholar
[12] J. E. Felt, ε-continuity and shape, Procedings of the American Mathematical Society, 46 (1974), no.3, 426-430.Search in Google Scholar
[13] C. Zuo, X. Wang, Attractors and Quasi-Attractors of a Flow, J.Appl.Math and Computing, Vol.23 (2007), no.1-2, 411-417.Search in Google Scholar
[14] C.C. Conley, Isolated invariant sets and the Morse index, AMS Providence R.I., 38 (1978), 234-240.Search in Google Scholar
[15] S. Gabites, Dynamical systems and shape, Revista de la Real Academia de Ciencias Exactas, Fizicas y Naturales. Serie A, Matematicas, 102 Vol. 102 (1) (2008), 127-159.10.1007/BF03191815Search in Google Scholar
[16] J. Keesling, Locally compact full homeomorphism group are zero-dimensional, Proc. Amer.Math Soc. 29 (1971), 390-396.Search in Google Scholar
[17] G. Osipenko, Dynamical systems, graphs and algorithms, Springer-Verlag, New York, (2000).Search in Google Scholar
© 2019 M. Shoptrajanov, published by De Gruyter
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