Chain recurrence, semiflows, and gradients

  • Mike Hurley

DOI: 10.1007/BF02219371

Cite this article as:
Hurley, M. J Dyn Diff Equat (1995) 7: 437. doi:10.1007/BF02219371


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 

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

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