Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions

Authors:
Konstantin Mischaikow, Hal Smith and Horst R. Thieme

Journal:
Trans. Amer. Math. Soc. **347** (1995), 1669-1685

MSC:
Primary 34C35; Secondary 34D05, 34K20, 54H20

MathSciNet review:
1290727

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[B]**J. M. Ball,*On the asymptotic behavior of generalized processes, with applications to nonlinear evolution equations.ations*, J. Differential Equations**27**(1978), no. 2, 224–265. MR**0461576****[C1]**C. Conley,*The gradient structure of a flow. I*, Ergodic Theory Dynam. Systems**8***(1988), no. Charles Conley Memorial Issue, 11–26, 9. With a comment by R. Moeckel. MR**967626**, 10.1017/S0143385700009305**[C2]**Charles Conley,*Isolated invariant sets and the Morse index*, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR**511133****[D]**Constantine M. Dafermos,*Semiflows associated with compact and uniform processes*, Math. Systems Theory**8**(1974/75), no. 2, 142–149. MR**0445473****[FS]**John E. Franke and James F. Selgrade,*Abstract 𝜔-limit sets, chain recurrent sets, and basic sets for flows*, Proc. Amer. Math. Soc.**60**(1976), 309–316 (1977). MR**0423423**, 10.1090/S0002-9939-1976-0423423-X**[H1]**Jack K. Hale,*Ordinary differential equations*, 2nd ed., Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. MR**587488****[H2]**Jack K. Hale,*Asymptotic behavior of dissipative systems*, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR**941371****[HV]**Jack K. Hale and Sjoerd M. Verduyn Lunel,*Introduction to functional-differential equations*, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993. MR**1243878****[L]**Joseph P. LaSalle,*Stability theory and invariance principles*, Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974) Academic Press, New York, 1976, pp. 211–222. MR**0594977****[Ma]**L. Markus,*Asymptotically autonomous differential systems*, Contributions to the theory of nonlinear oscillations, vol. 3, Annals of Mathematics Studies, no. 36, Princeton University Press, Princeton, N. J., 1956, pp. 17–29. MR**0081388****[Mi]**R. K. Miller,*Almost periodic differential equations as dynamical systems with applications to the existence of A.P. solutions*, J. Differential Equations**1**(1965), 337–345. MR**0185221****[MS1]**Richard K. Miller and George R. Sell,*A note on Volterra integral equations and topological dynamics*, Bull. Amer. Math. Soc.**74**(1968), 904–908. MR**0233160**, 10.1090/S0002-9904-1968-12075-0**[MS2]**Richard K. Miller and George R. Sell,*Volterra integral equations and topological dynamics*, Memoirs of the American Mathematical Society, No. 102, American Mathematical Society, Providence, R.I., 1970. MR**0288381****[NS]**V. V. Nemytskii and V. V. Stepanov,*Qualitative theory of differential equations*, Princeton Mathematical Series, No. 22, Princeton University Press, Princeton, N.J., 1960. MR**0121520****[R]**Clark Robinson,*Stability theorems and hyperbolicity in dynamical systems*, Proceedings of the Regional Conference on the Application of Topological Methods in Differential Equations (Boulder, Colo., 1976), 1977, pp. 425–437. MR**0494300****[Sa]**Stephen H. Saperstone,*Semidynamical systems in infinite-dimensional spaces*, Applied Mathematical Sciences, vol. 37, Springer-Verlag, New York-Berlin, 1981. MR**638477****[Se1]**George R. Sell,*Nonautonomous differential equations and topological dynamics. I. The basic theory*, Trans. Amer. Math. Soc.**127**(1967), 241–262. MR**0212313**, 10.1090/S0002-9947-1967-0212313-2**[Se2]**George R. Sell,*Nonautonomous differential equations and topological dynamics. II. Limiting equations*, Trans. Amer. Math. Soc.**127**(1967), 263–283. MR**0212314**, 10.1090/S0002-9947-1967-0212314-4**[T1]**Horst R. Thieme,*Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations*, J. Math. Biol.**30**(1992), no. 7, 755–763. MR**1175102**, 10.1007/BF00173267**[T2]**Horst R. Thieme,*Asymptotically autonomous differential equations in the plane*, Rocky Mountain J. Math.**24**(1994), no. 1, 351–380. 20th Midwest ODE Meeting (Iowa City, IA, 1991). MR**1270045**, 10.1216/rmjm/1181072470

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
34C35,
34D05,
34K20,
54H20

Retrieve articles in all journals with MSC: 34C35, 34D05, 34K20, 54H20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1290727-7

Keywords:
Chain recurrence,
asymptotically autonomous semiflow,
Lyapunov function

Article copyright:
© Copyright 1995
American Mathematical Society