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Chain recurrence, semiflows, and gradients

  • Mike Hurley
Article

Abstract

This paper is a study of chain recurrence and attractors for maps and semiflows on arbitrary metric spaces. The main results are as follows. (i) C. Conley's characterization of chain recurrence in terms of attractors holds for maps and semiflows on any metric space. (ii) An alternative definition of chain recurrence for semiflows is given and is shown to be equivalent to the usual definition. The alternative definition uses chains formed of orbit segments whose lengths are at least 1, while in the usual definition these lengths are required to be arbitrarily long. (iii) The chain recurrent set of a continuous semiflow is the same as the chain recurrent set of its time-one map. (iv) Conditions on a real-valued function are given that ensure that the semiflow generated by its gradient has only equilibria in its chain recurrent set. An example is given (onR3) showing that a gradient flow may have nonequilibrium chain recurrent points if these conditions are violated.

Key words

Chain recurrent set attractor semiflow gradient flow 

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References

  1. Abraham, R., and Robbin, J. (1967).Transversal Mappings and Flows, W. A. Benjamin, New York.Google Scholar
  2. Chen, X.-Y., and Poláčik, P. (1993). Gradient-like structure and morse decompositions for time-periodic one-dimensional parabolic equations (preprint).Google Scholar
  3. Conley, C. (1978).Isolated Invariant Sets and the Morse Index, AMS, Providence.Google Scholar
  4. Franks, J. (1988). A variation of the poincaré-birkhoff theorem. InHamiltonian Dynamical Systems; Vol. 81, Contemporary Mathematics, AMS, Providence, pp. 111–117.Google Scholar
  5. Hurley, M. (1982). Attractors: Persistence, and density of their basins.Trans. AMS 269, 247–271.Google Scholar
  6. Hurley, M. (1991). Chain recurrence and attraction in non-compact spaces.Erg. Th. Dyn. Syst. 11, 709–729.Google Scholar
  7. Hurley, M. (1992). Noncompact chain recurrence and attraction.Proc. AMS 115, 1139–1148.Google Scholar
  8. Kelley, J. L. (1967).General Topology, Van Nostrand, Princeton.Google Scholar
  9. Whitney, H. (1934). Analytic extensions of differentiable functions defined in closed sets.Trans. AMS 36, 63–89.Google Scholar
  10. Whitney, H. (1935). A function not constant on a connected set of critical points.Duke Math. J. 1, 514–517.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Mike Hurley
    • 1
  1. 1.Department of MathematicsCase Western Reserve UniversityCleveland

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