Width of parametric resonance regions for equations on a torus (Q1282515)

Considering the differential equation on a torus \(T^2\): \[ dx/dt= \delta K_{01}(x,t)+\varepsilon K_{10}(x,t) \] with two small parameters \(\delta\) and \(\varepsilon\), and \(K_{01}, K_{10}\) are analytic functions and periodic with respect to \(t\) and \(x\), the author gives an estimation of the parameter set when \(2\pi\)-periodic solutions exist. As a consequence, new proofs of a theorem on the width of the forbidden regions for the Hill equation with a small potential and theorem on the width of the parametric resonance regions for the first-order differential equation on a torus are given.