TY - JOUR

T1 - Asymptotically autonomous semiflows

T2 - Chain recurrence and lyapunov functions

AU - Mischaikow, Konstantin

AU - Smith, Hal

AU - Thieme, Horst

PY - 1995/5

Y1 - 1995/5

N2 - 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.

AB - 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.

KW - Asymptotically autonomous semiflow

KW - Chain recurrence

KW - Lyapunov function

UR - http://www.scopus.com/inward/record.url?scp=0000703838&partnerID=8YFLogxK

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U2 - 10.1090/S0002-9947-1995-1290727-7

DO - 10.1090/S0002-9947-1995-1290727-7

M3 - Article

AN - SCOPUS:0000703838

VL - 347

SP - 1669

EP - 1685

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -