On Some Asymptotic Solutions of Fourth Order Critically Damped Nonlinear Systems

Abstract

Krylov and Bogoliubov introduced a perturbation method named "asymptotic averaging method". The method was developed only to obtain the periodic solution of second order nonlinear differential systems. Now the method is used to obtain the solutions of oscillatory, damped oscillatory, over-damped, critically damped and more critically damped systems with of second, third, fourth etc. order nonlinear differential equations by imposing some restrictions to make the solutions uniformly valid. The method of Krylov and Bogoliubov has been improved and justified by Bogoliubov and Mitropolskii. In this dissertation, we have modified and extended the Krylov-Bogoliubov-Mitropolskii (KBM) method to investigate the fourth order critically damped and more critically damped nonlinear systems. We have imposed some restrictions on the eigenvalues to determine the unknown functions which are related to the variational equations. To get the solutions of the variational equations, we have replaced the variables by their corresponding linear values. For justification of the solution obtained by the extended KBM method, we have compared the results to those obtained by the fourth order Runge-Kutta method.