Many dynamical systems are described by differential equations that can
be separated into one part, containing linear terms with constant
coefficients, and a second part, relatively small compared with the
first, containing nonlinear terms. Such a system is said to be weakly
nonlinear. The small terms rendering the system nonlinear are referred
to as perturbations. A weakly nonlinear system is called quasi-linear
and is governed by quasi-linear differential equations. We will be
interested in systems that reduce to harmonic oscillators in the absence
of perturbations. This book is devoted primarily to applied asymptotic
methods in nonlinear oscillations which are associated with the names of
N. M. Krylov, N. N. Bogoli- ubov and Yu. A. Mitropolskii. The advantages
of the present methods are their simplicity, especially for computing
higher approximations, and their applicability to a large class of
quasi-linear problems. In this book, we confine ourselves basi- cally to
the scheme proposed by Krylov, Bogoliubov as stated in the monographs
[6,211. We use these methods, and also develop and improve them for
solving new problems and new classes of nonlinear differential
equations. Although these methods have many applications in Mechanics,
Physics and Technique, we will illustrate them only with examples which
clearly show their strength and which are themselves of great interest.
A certain amount of more advanced material has also been included,
making the book suitable for a senior elective or a beginning graduate
course on nonlinear oscillations.
