On the Asymptotic Theory of Non-Linear Oscillations.

Abstract

Publisher Summary This chapter focuses on the asymptotic theory of non-linear oscillations. New approaches to the asymptotic theory of non-linear oscillations and wave propagation are reported and the asymptotic method of Krilov–Bogolioubov–Mitropolski is discussed. A deductive asymptotic theory is developed, which uses from the outset concepts and methods of asymptotic analysis. The necessary preliminaries are given and the fundamental tool of our method of analysis is introduced: in a suitable (asymptotic) sense, a local average value of the function Y (t , e) is defined. With the aid of this concept, a deductive procedure establishes the fundamental theory of Krilov–Bogolioubov–Mitropolski under the most general conditions. For a class of problems validity of the asymptotic approximation on 0 ≤ t