Asymptotic Method for Time Dependent Nonlinear Differential Systems with Slowly Varying Coefficients

Abstract

Almost all perturbation methods are developed to find periodic solutions of nonlinear system where transients are not considered. First Krylov and Bogoliubov introduced a perturbation method which is well known as “asymptotic averaging method” to discuss the transients in the second order autonomous systems with small nonlinearities. Later, this method has been amplified and justified by Bogoliubov and Mitropolskii. Mitropolskii has extended the method for slowly varying coefficients to determine the steady state periodic motions and transient process. In this dissertation, we have modified and extended the KBM method to investigate some fifth order and second order nonlinear systems in both cases with constant and slowly varying coefficients. At first, a fifth order damped nonlinear autonomous differential system is considered and a perturbation solution is developed. Then a procedure is developed for the same system with damped taking three of eigenvalues are real. After then we considered fifth order systems for over damped with small nonlinearity to obtain the transient response. We also developed a formula for fifth order critically damped nonlinear systems to control micro vibration, in micro and nano-technological industries that bring the system to equilibrium as quickly as possible without oscillating. After then we presented an analytical technique based on the extended Krylov-Bogoliubov-Mitropolskii method (by Popov) to determine approximate solutions of nonlinear differential systems whose coefficients change slowly and periodically with time. Furthermore, a non-autonomous case also investigated in which an external force acts in this system. At last, Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to certain damped-oscillatory nonlinear systems with varying coefficients. The implementations of the methods are illustrated by several examples.