Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions

Authors:
Konstantin Mischaikow, Hal Smith and Horst R. Thieme

Journal:
Trans. Amer. Math. Soc. **347** (1995), 1669-1685

MSC:
Primary 34C35; Secondary 34D05, 34K20, 54H20

DOI:
https://doi.org/10.1090/S0002-9947-1995-1290727-7

MathSciNet review:
1290727

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Abstract: From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set. Here, we improve a result of L. Markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow. In the special case that there is a Lyapunov function for the limiting semiflow, sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the Lyapunov function.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1290727-7

Keywords:
Chain recurrence,
asymptotically autonomous semiflow,
Lyapunov function

Article copyright:
© Copyright 1995
American Mathematical Society