| Contents |
| We consider a class of the dynamical systems generated by
non-invertible generally, map $F$ of the form $F y = \Phi(y) A y$
in some compact $\mathrm{X} \subseteq \mathbb{R}^n$. Here
$\mathrm{R}^n$ is $n$- dimensional real vector-space, $\Phi(y)$ is
a scalar function and $A$ is a matrix of $n$ order. The dynamical
systems are proposed as mathematical models with a limiting factor
(nonlinear matrix models). Let $n$ be a number of macro system's
components, $y$ be a vector of components' characteristics, $A$ be
a matrix of components' interrelations and $\Phi(y)$ be a limiting
function (limiting factor). Then the systems describe the dynamics
of model and real macro systems in the presence of limiting
factors. There are some problems with the nonlinear matrix
modelling one of which concerns reducing the system's dimension or
the matrix order. Since the number of the system's components may
be very large or the matrix structure may be very sparse then
constructing appropriate algorithms for determining asymptotic
behavior of the system is essential. In this talk we discuss the
algorithm which we propose for determining the asymptotic behavior
of macro systems governed by the class of the dynamical systems
considered. It is based on the qualitative theory which we develop
for this class of the systems. As the fist step of the algorithm,
an asymptotically stabilized macro system structure is obtained by
matrix $A$, i.e. using linear dynamical system. As the second
step, the dynamics of macro system with the stabilized structure
is determined by the one-dimensional nonlinear dynamical system
with many parameters to which $n$- dimensional dynamical system
reduces. Some problems of the qualitative theory are under
consideration as well. In particular, we define the number of parameters by
which the dynamics of $n$- dimensional systems' family depending
on $A$ is described. |
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