ON COMPRESSIBILITY AREA OF UNSTABLE DIFFERENCE-DYNAMIC SYSTEMS AND DETERMINATED CHAOS

Аннотация

In this paper saddle-point bifurcation is studied. It is shown that as a result of bifurcation or collision
of stable and unstable points, they leave the area. In other words, they go into chaos, i.e. in a state of disorder. Here
the normalization method is used to identify the bifurcation point. Unstable in the sense of Lyapunov differencedynamic
systems are considered.
In the first part of the paper, the transformation of linear systems in instability case is given. A linear system
with a diagonal matrix is considered. It is shown that in a neighborhood of zero this system is not reduced to a
special form with the help of non-degenerate transformations. It is proved that for non-degenerate transformations
the trajectories of a system of a special form from a neighborhood of the origin cannot be displayed in the trajectory
of solutions of a given linear system. Thus, the topology of the neighborhood of the zero point of a system of a
special form does not transform into the topology of a given system in a neighborhood of zero. The causes of the
contradiction obtained by applying this method are shown.
In the second part, analytic difference-dynamical systems and analytic homeomorphisms are considered. Also
compressive difference-dynamical systems are investigated. The concept of compressing difference-dynamical
systems is given. It is shown to which system the  -compressing difference-dynamical systems are isomorphic.