Weakly Coupled Harmonic Oscillators

Sufficient conditions are given here for the existence and stability of a possibly infinite sequence of isolated periodic solutions of the nonautonomous system of differential equations \[ \ddot y_k + \omega _k^2 y_k = \mu f_k \left(y_1 , \cdots ,y_n ,\dot y_1 , \cdots ,\dot y_n ,t\right) \], where $0 < \mu \ll 1,k = 1, \cdots ,n$. An $O(\mu )$ estimate of the amplitude of each of these periodic solutions is also obtained.