Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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CHAIN RECURRENCE AND ATTRACTORS IN GENERAL DYNAMICAL SYSTEMS

  • Published : 2007.10.31
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Abstract

We introduce here notions of chain recurrent sets, attractors and its basins for general dynamical systems and prove important properties including (i) the chain recurrent set CR(f) of f on a metric space (X, d) is the complement of the union of sets B(A) -A as A varies over the collection of attractors and (ii) genericity of general dynamical systems.

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References

  1. E. Akin, The general topology of dynamical syatems, Graduate Studies in Mathematics, Amer. Math. Providence 71, 1993
  2. N. Aoki and K. Hiradie, Topological theory of dynamical systems, North-Holland, New York, 1994
  3. N. P. Bhatia and G. P. Szego, Stability theory of dynamical systems, Springer Verlag, New York, 1970
  4. L. Block and J. E. Franke, The chain recurrence, attractors and explosions, Ergodic Th. & Dynamical Systems 5 (1985), 321-327 https://doi.org/10.1017/S0143385700002972
  5. C. Conley, Isolated invarant sets and the morse index, CBMS Publication, A. M. S. providence 38, 1978
  6. S. Elaydi and S. K. Kaul, Asmptotic stablity in general dynamical systems, Nonlineer Analysis Theory, Methods and Applications 13 (1998), 657-669 https://doi.org/10.1016/0362-546X(89)90085-0
  7. M. Hurley, Chain recurrence and attraction in noncompacr spaces, Ergodic theory dynamical systems 11 (1991), 709-729 https://doi.org/10.1017/S014338570000643X
  8. M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. of the A. M. S. 126 (1998), 245-256 https://doi.org/10.1090/S0002-9939-98-04500-6
  9. M. Hurley, Weak attractors from Lyapunov functions, Topology and Applications 109 (2001), 201-210 https://doi.org/10.1016/S0166-8641(99)00158-3
  10. Kuratowski, Topology 2, Academic Press-PWN-Polish Sci. Pub. Warszawa, 1968
  11. D. Li, On dynamical stability in general dynamical systems, J. math. Annal. Appl. 263 (2001), 455-478 https://doi.org/10.1006/jmaa.2001.7620
  12. S. Y. Pilyugin, The space of dynamical systems with the CO-topology, Lecture Notes in Mathematics, Springer-Verlag, 1571, 1994
  13. E. Roxin, Stability in general control systems, J. differential equations (1965), 115-156 https://doi.org/10.1016/0022-0396(65)90015-X
  14. K. S. Sibirsky, Introduction to topological dynamics, Noordhoff International Pub. Leyden, 1975

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  2. Random chain recurrent sets for random dynamical systems vol.24, pp.4, 2009, https://doi.org/10.1080/14689360903164173