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analytic characteristic multipliers coefficient completely stable completes the proof condition for existence consider corresponding denote derivative with respect derivatives of F derivatives of x(t directly unstable ef(t equation 4.1 equation 7.9 evaluated existence theorem exists a unique F and G fixed points found in Theorem Fourier series higher order terms hypotheses of Theorem implicit function theorem increasing frequency characteristic integral least common multiple let g(x Let x(t linear differential equation modified implicit function nonconstant periodic solutions Nonlinear obtain odd function orthogonality condition periodic function periodic solution found periodic solution x(t perturbation theory piecewise continuous positive amplitude PROOF OF THEOREM right member satisfied say that g(x second order simple zero solution is directly solution of period solutions of 4.1 stability properties subharmonic of order sufficiently small Theorem 12 theorem holds unique periodic solution vanish identically variation equation write x(t Wronskian Xg(t zero of F