Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions

Mischaikow, Konstantin; Smith, Hal; Thieme, Horst R.

doi:10.1090/S0002-9947-1995-1290727-7

Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions
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by Konstantin Mischaikow, Hal Smith and Horst R. Thieme PDF

Trans. Amer. Math. Soc. 347 (1995), 1669-1685 Request permission

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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