Asymptotic Methods for some third order Nonlinear Differential Equations

Abstract

In this thesis, we investigate the oscillations of third order nonlinear systems by the asymptotic method. The asymptotic method of Krylov-Bogoliubov-Mitropolskii (KBM) is a popular technigue for obtaining analytic solution of a second order nonlinear oscillatory system. First a third order nonlinear differential system modeling nonoscillatory process and characterized by critical damping is considered and a new perturbation technigue is developed, based on the work of Krylov-Bogoliubov-Mitropolskii, to £ind the solution of the system. Then a method is presented unifying both third order damped and overdamped systems. This method is a generalization of Bogoliubov·s asymptotic method and covers all the cases when the roots of the corresponding linear equation are real, real and complex, and real and purely imaginary.Later a third order forced nonlinear differential system modeling oscillatory process is considered and a new perturbation technique is developed to find the solution of the system. The methods are illustrated by several examples.