Parameter dependence of smooth stable manifolds
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초록
We establish the existence of $C^1$ stable invariant manifolds for differential equations $u'=A(t)u+f(t,u,\lambda)$ obtained from sufficiently small $C^1$ perturbations of a \emph{nonuniform} exponential dichotomy. Since any linear equation with nonzero Lyapunov exponents has a nonuniform exponential dichotomy, this is a very general assumption. We also establish the $C^1$ dependence of the stable manifolds on the parameter $\lambda$. We emphasize that our results are optimal, in the sense that the invariant manifolds are as regular as the vector field. We use the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, we can also consider linear perturbations, and thus our results can be readily applied to the robustness problem of nonuniform exponential dichotomies.
Abstract
We establish the existence of $C^1$ stable invariant manifolds for differential equations $u'=A(t)u+f(t,u,\lambda)$ obtained from sufficiently small $C^1$ perturbations of a \emph{nonuniform} exponential dichotomy. Since any linear equation with nonzero Lyapunov exponents has a nonuniform exponential dichotomy, this is a very general assumption. We also establish the $C^1$ dependence of the stable manifolds on the parameter $\lambda$. We emphasize that our results are optimal, in the sense that the invariant manifolds are as regular as the vector field. We use the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, we can also consider linear perturbations, and thus our results can be readily applied to the robustness problem of nonuniform exponential dichotomies.
참고문헌 (11)
- L. Barreira, "Nonuniform Hyperbolicity" (2007), Cambridge University Press
- L. Barreira, "Characterization of stable manifolds for nonuniform exponen-tial dichotomies", Discrete Contin. Dyn. Syst, 21(4), pp.1025~1046 (2008)
- L. Barreira, "Optimal regularity of stable manifolds of nonuniformly hyperbolic dynamics", Topol. Methods Nonlinear Anal, 38(2), pp.333~362 (2011)
- A. Fathi, "A proof of Pesin's stable manifold theorem", Geometric dynamics, Lecture Notes in Math, 1007, pp.177~215 (1983), Springer
- R. Mane, "Lyapounov exponents and stable manifolds for compact transformations", Geometric dynamics Lecture Notes in Math, 1007, pp.522~577 (1983), Springer
- V. I. Oseledec, "A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems", Trudy Moskov. Mat. Obsc, 19, pp.179~210 (1968)
- Ya. Pesin, "Families of invariant manifolds corresponding to nonzero characteristic ex-ponents", Math. USSR-Izv, 40, pp.1261~1305 (1976)
- Ya. Pesin, "Characteristic Ljapunov exponents, and smooth ergodic theory", Uspehi Mat. Nauk, 32(4), pp.55~112 (1977)
- C. Pugh, "Ergodic attractors", Trans. Amer. Math. Soc, 312(1), pp.1~54 (1989)
- D. Ruelle, "Ergodic theory of differentiable dynamical systems", Inst. Hautes Etudes Sci. Publ. Math, (50), pp.27~58 (1979)
- D. Ruelle, "Characteristic exponents and invariant manifolds in Hilbert space", Ann. of Math, 115(2), pp.243~290 (1982)