Ju l 2 00 9 On the small – amplitude approximation to the differential equation ẍ + ( 1 + ẋ 2 ) x = 0

We obtain the radius of convergence of the small–amplitude approximation to the period of the nonlinear oscillator ẍ + (1 + ẋ2)x = 0 with the initial conditions x(0) = A and ẋ(0) = 0 and show that the inverted perturbation series appears to converge smoothly from below. The interest in the nonlinear oscillator ẍ(t) + [1 + ẋ(t)]x(t) = 0 x(0) = A, ẋ(0) = 0 (1) apparently arouse from the fact that the first–order harmonic balance method yielded the approximate frequency [1] ω(A) = 2 √ 4− A2 (2) that suggests that the frequency of the oscillator is not defined for A > 2. By straightforward analysis of the dynamical trajectories in the x−y plane, where 1 e–mail: [email protected] Preprint submitted to Elsevier 20 July 2009 y = ẋ, Beatty and Mickens [2] concluded that such a restriction is merely an artifact of the harmonic balance method. Later, Mickens [3] derived an explicit expression for the period T (A) = 4A 1 ∫ 0 du √ eA2(1−u2) − 1 (3) where u = x/A. By means of this expression Mickens [3] proved that dT/dA < 0 and obtained upper and lower bounds to the period. Kalmár–Nagy and Erneux [4] derived the behaviour of the period for small and large values of A