Periodic Solutions of X ̋+ Cx ́+ G(x) |
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amplitude B₁ B₂ C₁ C₁p(t C₂ Cg(x characteristic multipliers coefficient completely stable completes the proof condition for existence consider corresponding derivatives of F derivatives of x(t differential equation directly unstable equation 7.9 evaluated exists a unique fixed points found in Theorem Fourier series h₁(t higher order terms hypotheses of Theorem implicit function theorem increasing frequency characteristic integral k₂(t least common multiple least period let g(x Let x(t linear modified implicit function Nonlinear 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 reducing to x(t right member satisfied say that g(x Section simple zero small ɛ solution of 3.1 solution of period subharmonic of order sufficiently small Theorem 12 theory unique periodic solution vanish identically variation equation write x(t Wronskian x₁ x₁(t x₂ zero of F εε