A High Order Generalized Method of Averaging

We develop a high order generalized perturbation technique that extends the Krylov–Bogoliubov–Mitropolsky method of averaging to vector systems written in normal form with multiple angular components. An algorithm is presented that iteratively gives the terms in the asymptotic approximation. A nonresonance condition is assumed that guarantees the smoothness of the terms. The main result establishes that the absolute error between the unaveraged normal system and its Nth order approximation is of the order of the Nth power of the perturbation parameter for a time interval of length the order of the reciprocal of the perturbation parameter. The high order algorithm is applied to a coupled van der Pol oscillator system. Some numerical results are given to show that the main result reflects actual computational experience.