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Nonuniform exponential dichotomies and Lyapunov regularity. (English) Zbl 1123.34040
Authors’ summary: The notion of exponential dichotomy plays a central role in the Hadamard-Perron theory of invariant manifolds for dynamical systems. The more general notion of nonuniform exponential dichotomy plays a similar role under much weaker assumptions. On the other hand, for nonautonomous linear equations ${v}^{\text{'}}=A\left(t\right)v$ with global solutions, we show here that this more general notion is in fact as weak as possible: namely, any such equation possesses a nonuniform exponential dichotomy. It turns out that the construction of invariant manifolds under the existence of a nonuniform exponential dichotomy requires the nonuniformity to be sufficiently small when compared to the Lyapunov exponents. Thus, it is crucial to estimate the deviation from the uniform exponential behavior. This deviation can be measured by the so-called regularity coefficient, in the context of the classical Lyapunov-Perron regularity theory. We obtain here lower and upper sharp estimates for the regularity coefficient, expressed solely in terms of the matrices $A\left(t\right)$.

##### MSC:
 34D09 Dichotomy, trichotomy 34D10 Stability perturbations of ODE
##### Keywords:
exponential dichotomies; Lyapunov regularity
##### References:
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