Asymptotic methods for second-order over-damped and critically damped nonlinear systems. (Q2748622)

The paper is concerned with some modification of the Krylov-Bogoliubov-Mitropolskii asymptotic method applied to a second-order quasi-linear ordinary differential equation. The two possibilities are investigated: the over-damped situation, that is the nonoscillatory case, and the critically damped case, when the corresponding linear equation contains secular terms in its solution. The author manages to design such a procedure that the coefficients of the asymptotic series do not contain secular-type terms. A few examples are presented, the Duffing equation among them. These examples are also solved numerically, and the results agree with those obtained by asymptotic methods.