Notes on generalized exponential dichotomies (Q1367108)

Consider the linear differential system \[ dx/dt= A(t)x\tag{\(*\)} \] with \(x\in\mathbb{R}^n\) and \(A\in C(J, L(\mathbb{R}^n, \mathbb{R}^n))\) where \(J\) is some interval in \(\mathbb{R}\). Let \(X(t)\) be the fundamental matrix of \((*)\) satisfying \(X(0)= I\). \((*)\) is said to have a generalized exponential dichotomy (GED) on \(J\) if there are real constants \(\alpha\), \(\beta\), \(k\) with \(\alpha>\beta\), \(k>0\), and a constant projection \(Q\) in \(\mathbb{R}^n\) such that \[ |X(t) QX^{-1}(s)|\leq ke^{\beta(t- s)},\quad |X(s)(I- Q)X^{-1}(s)|\leq ke^{\alpha(t- s)} \] for \(t,s\in J\), \(t\geq s\). The author gives two roughness results and a criterion for GED. With respect to the linear differential-difference system \[ dx/dt= A(t+ h)x \] and to the time-scaled system \[ dx/dt= A(\mu t)x, \] the author derives conditions to determine by means of spectral gaps the GED for these systems and their \(L^1\)-limit systems. Two examples are considered.