Parameter dependence of smooth stable manifolds
J. Korean Math. Soc. 2019 Vol. 56, No. 3, 825-855
Published online May 1, 2019
Luis Barreira, Claudia Valls
Universidade de Lisboa; Universidade de Lisboa
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.
Keywords : growth rates, nonuniform exponential dichotomies, parameter dependence, stable manifolds
MSC numbers : Primary 37D10, 37D25, 34D99
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