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**Sergei Andreevich Brygin**

Saint-Petersburg State University

Faculty of Mathematics and Mechanics

Dept. of Math.Analysis

student

**Alexandr Alekseevich Florinskiy**

Saint-Petersburg State University

Faculty of Mathematics and Mechanics

Dept. of Math.Analysis

Assoc. Prof.

Dynamical system generated by a nonlinear operator acting on the real
line or an ordered metric space and having increasing trajectories is considered.
Such an operator is said to be mapping (operator), majorized below by identity map.
The effect of changing the generating operator by the operator with lesser
or greater values is studied.

The main results of the work are the following:

(a) For the composition of generating mappings
the sufficient condition to reserve the property of the stabilization
of all system trajectories is obtained;

(b) It is proved that if the set of all unbounded trajectories of the
operator acting on the real line is not empty, than any greater operator
has the same property (with respect to the pointwise order relation).

(c) It is proved that there are systems on real line with increasing trajectories,
such that we may change their generating operators by arbitrary close ones
(by subtracting small constants from their values),
and obtain the systems having hidden attractors (in the sense of N.Kuznetsov).

The example illustrating (c) is given.

- discrete dynamical systems
- hidden attractor
- mappings majorized by the identity map
- unbounded trajectories

- N. V. Kuznetsov. Analitiko-chislennie metodi issledovania skritih kolebanii, Dissertation Doctor in Physics . and Mathematics, Sakt-Peterburgckii gos. universitet, Sankt-Peterburg, 2016
- С. В. Чистяков, Элементы динамической теории классических кооперативных игр. СПб, РОПИ Изд-ва С. -Петерб. ун-та, 2001. 30 с
- A. L. Kryukova, On idempotent elements of semigroup of increasing monotonous mappings, Izv. Saratov Univ. (N. S. ), Ser. Math. Mech. Inform., 2011, Volume 11, Issue 4, Pages 27-33