Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions
Authors:
Konstantin Mischaikow, Hal Smith and Horst R. Thieme
Journal:
Trans. Amer. Math. Soc. 347 (1995), 16691685
MSC:
Primary 34C35; Secondary 34D05, 34K20, 54H20
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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 [H2]
 , Asymptotic behavior of dissipative systems, Math. Surveys Monographs, vol. 25, Amer. Math. Soc., Providence, RI, 1988. MR 941371 (89g:58059)
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 [MS1]
 R. K. Miller and G. Sell, A note on Volterra integral equations and topological dynamics, Bull. Amer. Math. Soc. 74 (1968), 904908. MR 0233160 (38:1483)
 [MS2]
 , Volterra integral equations and topological dynamics, Mem. Amer. Math. Soc., No. 102 (1970). MR 0288381 (44:5579)
 [NS]
 V. V. Nemytskii and V. V. Stepanov, Qualitative theory of differential equations, Princeton Univ. Press, Princeton, NJ, 1960. MR 0121520 (22:12258)
 [R]
 C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), 425437. MR 0494300 (58:13200)
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 S. H. Saperstone, Semidynamical systems in infinite dimensional spaces, SpringerVerlag, New York, 1981. MR 638477 (84m:58024)
 [Se1]
 G. Sell, Nonautonomous differential equations and topological dynamics. I. The basic theory, Trans. Amer. Math. Soc. 127 (1967), 241262. MR 0212313 (35:3187a)
 [Se2]
 , Nonautonomous differential equations and topological dynamics. II. Limiting equations, Trans. Amer. Math. Soc. 127 (1967), 263283. MR 0212314 (35:3187b)
 [T1]
 H. R. Thieme, Convergence results and a PoincaréBendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol. 30 (1992), 755763. MR 1175102 (93e:34042)
 [T2]
 , Asymptotically autonomous differential equations in the plane, Rocky Mountain J. Math. 24 (1994), 351380. MR 1270045 (96a:34095)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199512907277
PII:
S 00029947(1995)12907277
Keywords:
Chain recurrence,
asymptotically autonomous semiflow,
Lyapunov function
Article copyright:
© Copyright 1995
American Mathematical Society
