Almost-periodic solutions to weakly nonlinear equations in critical cases
Consider the differential system \((*)\) \(dz/dt=Az+f(t)+\varepsilon Z(z,t,\varepsilon)\) under the assumptions (i) \(\varepsilon \) is a small parameter; (ii) \(A\) is a constant \(n \times n\)-matrix whose spectrum intersects the imaginary axis; (iii) \(f\) is quasi-periodic and can be represented as a Fourier series; (iv) \(Z\) is sufficiently smooth and quasi-periodic in \(t\). The author derives conditions on \(f\) and \(F\) guaranteeing the existence of an \(r\)-parametric family of quasi-periodic solutions to \((*)\) in case \(\varepsilon =0\) and the existence of a unique quasi-periodic solution to \((*)\) with \(\varepsilon\) sufficiently small, respectively.
scientific article; zbMATH DE number 1557304
scientific article