transactions of the american mathematical society Volume 347, Number 5, May 1995 ASYMPTOTICALLYAUTONOMOUS SEMIFLOWS: CHAIN RECURRENCE AND LYAPUNOVFUNCTIONS KONSTANTIN MISCHAIKOW,HAL SMITH, AND HORST R. THIEME 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. Introduction In the well-known paper of Markus [Ma], a fundamental result concerning the large time behavior of solutions of asymptotically autonomous ordinary differential equations is obtained. Recall that the nonautonomous system of differential equations in R" (0.1) x' = f(t,x), is said to be asymptotically autonomous—with limit equation (0.2) y' = g{y), if f(t,x)->g(x), /-too, where the convergence is uniform on each compact subset of R" . For simplicity, we assume in this introduction that / and g are continuous functions and that they are locally Lipschitz on R" . The colimit set, co(to, Xq), of a bounded solution x(t) of (0.1) on t > to satisfying x(to) = xq is defined in the usual way: (¿>(to,Xq) = \ y : y = lim x(t¡), for some sequence t,■ —> oo >. I >-°° J The result of Markus is the following. Received by the editors March 29, 1994 and, in revised form, July 5, 1994. 1991MathematicsSubjectClassification. Primary 34C35, 35B40,58F35. Key words and phrases. Chain recurrence, asymptotically autonomous semiflow, Lyapunov function. Research of K. M. funded in part by NSF grant DMS 9101412. Research of H. S. funded in part by NSF grant DMS 9300974. Research of H. R. T. funded in part by NSF grant DMS 9101979. © 1995 American Mathematical Society 0002-9947/95 $1.00 + $.25 per page 1669 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use