Periodic Solutions of X ̋+ Cx ́+ G(x) |
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analytic B₂ C₁p(t C₂ Cg(x characteristic multipliers coefficient completely stable completes the proof condition for existence consider corresponding derivatives of x(t differential equation directly unstable evaluated existence of x(t exists a unique fixed points found in Theorem Fourier series h₁(t h₂ higher order terms hypotheses of Theorem implicit function theorem increasing frequency characteristic integral KLg(A least common multiple least period let g(x Let x(t linear modified implicit function nonconstant periodic solutions o(eČ obtain odd function orthogonality condition partial derivatives periodic function periodic solution found periodic solution x(t perturbation series phase plane piecewise continuous PROOF OF THEOREM r₁ reduce to x(t right member satisfied say that g(x Section simple zero small ɛ solution of period solutions of 4.1 subharmonic of order sufficiently small Theorem 12 theory translations of x(t unique periodic solution vanish identically variation equation write x(t Wronskian x₁(t zero of F εε