The present work is a condensed version of one of the same title, published in 1937 in Russian and containing the first extensive treat­ ment of non-linear oscillations. Messrs. Andronow and Chaikin are both members of the Institute of Oscillations founded about a decade and a half ago by the late Soviet physicist L. I. Mandelstam. The work of this Institute has become known to the world at large through Dr. N. Minorsky’s Introduction to Non-Linear Mechanics recently issued by the David Taylor Model Basin. The great interest evoked by this excellent report provided a strong incentive for making available to the scientific and technical public the text of Andronow and Chaikin. This has been made possible with the aid of the Office of Naval Research as part of the project on Non-Linear Differential Equations which it is sponsoring at Princeton University.The theory of harmonic oscillations, or oscillations of sinusoidal type, is well known. It is also quite elementary, since it is based upon linear differential equations whose explicit solutions are readily obtained. Unfortunately nature refuses to remain linear and repeat­ edly presents us with non-linear oscillations, that is to say with oscilla­ tions based upon non-linear differential equations. It is often possible to linearize a problem, i.e. to modify the operating assumptions so as to have a linear situation without causing important deviations. Thus the true equation of the pendulum is never linear, but for very small deviations (a few degrees) it may be satisfactorily replaced by a linear equation. In other words for small deviations the pendulum may be replaced by a harmonic oscillator. This ceases to hold however for large deviations, and in dealing with these one must consider the non-linear equation itself and not merely a linear substitute.There are then valid practical reasons for an extensive study of non-linear oscillations. Since their equations rarely admit an explicit solution, one will be forced to have recourse to various mathematical doctrines to obtain all possible information. The text of this book aims precisely to provide an introduction to this general subject.English Language Edition edited under direction of Solomon Lefschetz