Abstract
Two tracking properties for trajectories on attracting sets are studied. We prove that trajectories on the full phase space can be followed arbitrarily closely by skipping from one solution on the global attractor to another. A sufficient condition for asymptotic completeness of invariant exponential attractors is found, obtaining similar results as in the theory of inertial manifolds. Furthermore, such sets are shown to be retracts of the phase space, which implies that they are simply connected.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Bing, R. H. (1951). Concerning hereditarily indecomposable continua. Pacific J. Math. 1, 43–51.
Constantin, P., Foias, C., Nicolaenko, B., and Temam, R. (1988). Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer, Berlin.
Chow, S.-N., Lu, K., and Sell, G. R. (1992). Smoothness of inertial manifolds. J. Math. Anal. Appl. 169, 283–312.
Eden, A., Foias, C., Nicolaenko, B., and Temam, R. (1990). Ensembles inertiels pour des équations d'évolution dissipatives. C. R. Acad. Sci. Paris I 310, 559–562.
Eden, A., Foias, C., Nicolaenko, B., and Temam, R. (1994). Exponential Attractors for Dissipative Evolution Equations, RAM, Wiley, Chichester.
Fenichel, N. (1971). Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 23, 226–309.
Foias, C., Sell, G. R., and Temam, R. (1988). Inertial manifolds for dissipative evolution equations. J. Diff. Eq. 73, 311–353.
Foias, C., Sell, G. R., and Titi, E. (1989). Exponential tracking and approximation of inertial manifolds for dissipative nonlinear equations. J. Dyn. Diff. Eq. 1, 199–244.
Günther, B. (1995). Construction of differentiable flows with prescribed attractor. Topol. Appl. 62, 87–91.
Günther, B., and Segal, J. (1993). Every attractor of a flow on a manifold has the shape of a finite polyhedron. Proc. Am. Math. Soc. 119, 321–329.
Hale, J. K. (1988). Asymptotic behavior of dissipative systems. In Math. Surv. Monogr., Vol. 25, Am. Math. Soc., Providence, RI.
Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., Vol. 840, Springer-Verlag, New York.
Kuratowski, K. (1968). Topology, Vol. II, Academic Press, London.
Mañé, R. (1981). On the dimension of compact invariant sets of certain nonlinear maps. Springer Lect. Notes Math. 898, 230–242.
Marlin, J. A., and Struble, R. A. (1969). Asymptotic equivalence of nonlinear systems. J. Diff. Eq. 6, 578–596.
Robinson, J. C. (1996). The asymptotic completeness of inertial manifolds. Nonlinearity 9, 1325–1340.
Robinson, J. C. (1997). Some closure results for inertial manifolds. J. Dyn. Diff. Eq. 9, 373–400.
Robinson, J. C. (1998). Global attractors: Topology and finite-dimensional dynamics. J. Dyn. Diff. Eq. (in press).
Rosa, R., and Temam, R. (1996). Inertial manifolds and normal hyperbolicity. Acta Appl. Math. 45, 1–50.
Temam, R. (1988). Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, AMS 68.
Temam, R. (1990). Inertial manifolds. Math. Intell. 12, 68–73.
Vishik, M. I. (1992). Asymptotic Behaviour of Solutions of Evolutionary Equations, Lezioni Lincee, Cambridge University Press, Cambridge.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Langa, J.A., Robinson, J.C. Determining Asymptotic Behavior from the Dynamics on Attracting Sets. Journal of Dynamics and Differential Equations 11, 319–331 (1999). https://doi.org/10.1023/A:1021933514285
Issue Date:
DOI: https://doi.org/10.1023/A:1021933514285