More recently, in 1956, Malkin published his famous book [10] dealing with the
method of small parameter. This book also give s a method of successive
approximations in the non-analytic case. Malkin has also given a study of critical
cases in the Lyapunov sense via analytical methods.
From the theoretical results of Andronov’s school, the practical applications are
very large and very important for analysis and synthesis purpose. Now they go
beyond the limit of physics or engineering, and concern also natural sciences,
dynamics of populations, economy, etc. The applications concern all types of
oscillators (electronic type, in particular the theory of multivibrators, mechanical
type with the theory of watches, electro-mechanical type with coupling of elect rical
machines), regulators using relays, steam engines, automatic flight, the dynamics of
flight and of gyroscopes, radio-physics, quantum mechanics, dynamics of systems
heaving a pure delay, etc.
Since 1952, date of the deterioration of the relation between the two groups (one
in Gorki, the other in Mosc ow) of the school, the group of Moscow has developed a
considerable activity in the automatic control field with Chetaev, Lur’e, Aizerman,
Petrov, Meerov, Letov, Tsypkin, etc., and in the optimization theory and practice
with Pontryagin, Boltjanski, Gamkrelidze, Mischenko, etc . .
The Krylov-Bogolyubov school from Kiev, has developed essentially analytical
methods. The foundation of their results is the classical method of perturbations
which has been generalized to nonconservative systems. In 1932, the Krylov-
Bogolyubov method gave a close foundation to the Van der Pol studies about
oscillators. Later, the asymptotic method due to Mitropolski constitutes an
improvement, with the use of only asymptotically convergent series expansions.
It is the same for the averaging method and method for accelerating the conver-
gence [7, 8]. With respect to the Poincare
´
’s small parameter method, these methods
are such that the “full” determination of the first harmonic and of the following
harmonics of a periodic solution does not depend on the determination of the upper
harmonics. The contribution of this school is very large and concerns systems with
one, or several degrees of freedom, the determination of periodical, or quasi and
almost periodical solutions, the determination of transient regimes, the synchroni-
zation phenomenon, and the nonlinear systems having pure delays.
The Hayashi’s school on nonlinear oscillations developed many studies espe-
cially oriented toward electric circuits [11]. Analytical methods as well as qualita-
tive and numerical ones have been intensively used by him and his two disciples
Kawakami and Ueda. Complex behaviours of autonomous oscillators and nonau-
tonomous ones with periodical excitation have been the subject of a lot of publica-
tion. Studies of resonance and synchronization phenomena of harmonic,
subharmonic, higher harmonic and fractional harmonic types, in phase spaces and
in parameter spaces, as well as problems of chaotic behaviours, occupy an impor-
tant place in their publications.
In the last years efforts were also made for collecting the main results concerning
nonlinear vibration in monographs. In this respect we mention the books by
Minorski [12], Stoker [13], McLachlan [14], Kauderer [15], Sansone and Conti
[16], and Roseau [17].
4 1 Introduction