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We study a linear cocycle over irrational rotation $\sigma_{\omega}(x) = x +\omega$ of a circle $\mathbb{T}^{1}$. It is supposed the cocycle is generatedby a $C^{1}$-map $A_{\varepsilon}: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ whichdepends on a small parameter $\varepsilon\ll 1$ and has the form of thePoincar\'e map corresponding to a singularly perturbed Schr\"odinger equation.Under assumption the eigenvalues of $A_{\varepsilon}(x)$ to be of the form$\exp(\pm \lambda(x)/\varepsilon)$, where $\lambda(x)$ is a positive function,we examine the property of the cocycle to possess an exponential dichotomy (ED)with respect to the parameter $\varepsilon$. We show that in the limit$\varepsilon\to 0$ the cocycle "typically" exhibits ED only if it isexponentially close to a constant cocycle. In contrary, if the cocycle is notclose to a constant one it does not posesses ED, whereas the Lyapunov exponentis "typically" large.

On singularly perturbed linear cocyles over irrational rotations