Some examples of singular perturbations (Q1972717)

Consider the (singularly) perturbed differential equations (1) \( \varepsilon dx/dt = e^{it} x+1\) and (2) \( \varepsilon dx/dt = (a+ e^{it}) x+1\) where \(\varepsilon\) and \(x\) are complex, \( 0 < |a|< 1.\) Let \(\sum^\infty_{n=0} \varepsilon^n x_n (t)\) with \(2\pi\)-periodic \(x_n\) be a formal solution to these equations. It can be shown that the formal periodic solution does not represent an asymptotic expansion of a periodic solution to (1) and (2): (1) has no periodic solution for \(\varepsilon \neq 0,\) in case of equation (2) the formal periodic solution does not represent a true periodic solution. The peculiarity of these examples is that the mentioned property occurs not only for small \(\varepsilon\).