Abstract
For impulsive differential equations, we establish the existence of invariant stable manifolds under sufficiently small perturbations of a linear equation. We consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy. One of the main advantages of our work is that our results are optimal, in the sense that for vector fields of class C 1 outside the jumping times, we show that the invariant manifolds are also of class C 1 outside these times. The novelty of our proof is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, using the same approach we can also consider linear perturbations.
Similar content being viewed by others
References
Barreira, L., Pesin, Ya.: Lyapunov Exponents and Smooth Ergodic Theory, University Lecture Series 23. Amer. Math. Soc. (2002)
Barreira, L., Pesin, Ya.: Nonuniform Hyperbolicity, Encyclopedia of Mathematics and Its Applications 115. Cambridge University Press (2007)
Barreira, L., Valls, C.: Stability of Nonautonomous Differential Equations, Lect. Notes. in Math. 1926. Springer (2008)
Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics 6. World Scientific (1989)
Oseledets V.: A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–221 (1968)
Pesin Ya.: Families of invariant manifolds corresponding to nonzero characteristic exponents. Math. USSR-Izv. 10, 1261–1305 (1976)
Samoilenko A., Perestyuk N.: Impulsive Differential Equations Nonlinear Science Series A: Monographs and Treatises 14. World Scientific, River Edge, NJ (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barreira, L., Valls, C. Stable Manifolds for Impulsive Equations Under Nonuniform Hyperbolicity. J Dyn Diff Equat 22, 761–785 (2010). https://doi.org/10.1007/s10884-010-9161-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-010-9161-6