### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1669-1685 |

Number of pages | 17 |

Journal | Transactions of the American Mathematical Society |

Volume | 347 |

Issue number | 5 |

DOIs | |

State | Published - 1995 |

### Fingerprint

### Keywords

- Asymptotically autonomous semiflow
- Chain recurrence
- Lyapunov function

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Asymptotically autonomous semiflows : Chain recurrence and lyapunov functions.** / Mischaikow, Konstantin; Smith, Hal; Thieme, Horst.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 347, no. 5, pp. 1669-1685. https://doi.org/10.1090/S0002-9947-1995-1290727-7

}

TY - JOUR

T1 - Asymptotically autonomous semiflows

T2 - Chain recurrence and lyapunov functions

AU - Mischaikow, Konstantin

AU - Smith, Hal

AU - Thieme, Horst

PY - 1995

Y1 - 1995

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

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

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 -