Method of Complex Amplitudes: Harmonically Excited Oscillator With Strong Cubic Nonlinearity

Application of method of complex amplitudes for investigation of dynamics of nonlinear oscillatory systems is considered. The method is analyzed in details for harmonically forced Duffing oscillator without damping for the case of exact 1:1 resonance. It is demonstrated that the method of complex amplitudes may be formalized as a generalization of standard multiple–scales procedure. Two alternatives for elimination of secular terms in high-order approximations are proposed and compared for various values of small parameter. It is demonstrated that regardless the exact procedure chosen the necessary condition to avoid secularities is the appropriate choice of functions determining the correction of the slow time scale in any approximation. Criterion for use of the method of complex amplitudes is formulated. All analytic results are compared with data of numerical simulation and considerable agreement is observed.