Asymptotical behaviour of complicated oscillations
in
models of autogenerators with nonlinear delaying
feedbacks
Sergey A. Kaschenko
Yaroslavl State University, Yaroslavl, Russia
Dynamics of the equations
and
can be studied by asymptotical methods. Here the parameters a,
b and T are positive, and the nonlinear function F(x) is finite in
Eq.(1), i.e. F(x)=0 for 
(p;SPMgt;0), and is the following in
Eq.(2): F(x)=const for . The main assumption allowing to
apply the special asymptotical method of a large parameter developed by
the author is that 
.
We give comparative analysis of dynamics of Eqs.(1) and (2). Provided
, stable slowly oscillating relaxational periodic oscillations is
typical for each of these equations. We have found their asymptotical
behaviours, and the difference of such solutions of Eqs.(1) and (2) has
just quantitative nature.
If 
the difference is rather essential. We show that provided
each of Eqs.(1) and (2) has an attractor which
dynamics can be described by structure of the solutions of one-dimensional
mappings (each equation has its own mapping). If 
(m is an integer) Eqs.(1) and (2) are reduced to
(2m+1)-dimensional mappings, that could be analytically defined.