Normalization of perturbed non-linear Hamiltonian oscillators

We present a new way of handling perturbed non-linear oscillators. The solution of the equations of motion of a non-linear oscillator, generally, involves elliptic functions. The main key of our method consists in keeping these elliptic functions in the unperturbed part, and expanding the perturbation as a Fourier series of the amplitude, in which coefficients are powers of the Jacobian nome. Once the Fourier expansion is obtained, Lie based perturbation methods may be applied to it. As an illustration, the hard Duffing oscillator perturbed by sextic term is considered. Comparisons between our method and the exact solution (that is given here) are carried out. (C) 1997 Elsevier Science Ltd.