Abstract
For semiflows generated by ordinary differential equations v’=A(t)v admitting a nonuniform exponential dichotomy, we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v’=A(t)v+f(t,v). As an application, we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow. In particular, we obtain smooth invariant manifolds for a class of vector fields that need not be C1+α for any α ∈ (0,1). To the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting. We emphasize that we do not need to assume the existence of an exponential dichotomy, but only the existence of a nonuniform exponential dichotomy, with sufficiently small nonuniformity when compared to the Lyapunov exponents of the original linear equation. Furthermore, for example in the case of stable manifolds, we only need to assume that there exist negative Lyapunov exponents, while we also allow zero exponents. Our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones.
Similar content being viewed by others
References
Barreira, L., Pesin, Ya.: Lyapunov Exponents and Smooth Ergodic Theory. University Lecture Series 23. Providence, RI: Am. Math. Soc., 2002
Barreira, L., Pesin, Ya.: Smooth ergodic theory and nonuniformly hyperbolic dynamics. In: Handbook of Dynamical Systems 1B, B. Hasselblatt, A. Katok (eds.), Elsevier, to appear
Barreira, L., Valls, C.: Nonuniform exponential dichotomies and Lyapunov regularity. Preprint
Barreira, L., Valls, C.: Stable manifolds for nonautonomous equations without exponential dichotomy. J. Differ. Eqs. To appear
Chicone, C., Latushkin, Yu.: Evolution Semigroups in Dynamical Systems and Differential Equations. Mathematical Surveys and Monographs 70, Providence, RI: Am. Math. Soc., 1999
Coppel, W.: Dichotomies in Stability. Theory. Lect. Notes in Math. 629, Berlin-Heidelberg-New York: Springer, 1978
Hale, J.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs 25, Providence, RI: Am. Math. Soc., 1988
Hale, J., Lunel, S.: Introduction to Functional Differential Equations. Applied Mathematical Sciences 99, Berlin-Heidelberg-New York: Springer, 1993
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lect. Notes in Math. 840, Berlin-Heidelberg-New York: Springer, 1981
Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, with a supplement by A. Katok and L. Mendoza. Encyclopedia of Mathematics and its Applications 54, Cambridge: Cambridge University Press, 1995
Mañé, R.: Lyapunov exponents and stable manifolds for compact transformations. In: Geometric Dynamics (Rio de Janeiro, 1981), ed. J. Palis, Lect. Notes in Math. 1007, Berlin-Heidelberg- New-York: Springer, 1983, pp. 522–577
Pesin, Ya.: Families of invariant manifolds corresponding to nonzero characteristic exponents. Math. USSR-Izv. 10, 1261–1305 (1976)
Pugh, C.: The C1+α hypothesis in Pesin theory. Inst. Hautes Études Sci. Publ. Math. 59, 143–161 (1984)
Ruelle, D.: Characteristic exponents and invariant manifolds in Hilbert space. Ann. Math. (2) 115, 243–290 (1982)
Sell, G., You, Y.: Dynamics of Evolutionary Equations. Applied Mathematical Sciences 143, Berlin-Heidelberg-New York: Springer, 2002
Thieullen, P.: Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapunov. Entropie. Dimension. Ann. Inst. H. Poincaré. Anal. Non Linéaire 4, 49–97 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J.L. Lebowitz
Supported by the Center for Mathematical Analysis, Geometry, and Dynamical Systems, and through Fundação para a Ciência e a Tecnologia by Program POCTI/FEDER, Program POSI, and the grant SFRH/BPD/14404/2003.
Rights and permissions
About this article
Cite this article
Barreira, L., Valls, C. Smoothness of Invariant Manifolds for Nonautonomous Equations. Commun. Math. Phys. 259, 639–677 (2005). https://doi.org/10.1007/s00220-005-1380-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-005-1380-z