ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Estimation of Topological Entropy for Cocycles with Cellular Automaton As a Base System


Viktoriia Evgenyevna Egorova

Student of Department of Applied Cybernetics,
Faculty of Mathematics and Mechanics
Saint-Petersburg State University.

Volker Reitmann

Doctor of Physical and Mathematical Sciences,
Professor of the Department of Applied Cybernetics,
Faculty of Mathematics and Mechanics,
Saint-Petersburg State University.


We study a discrete non-autonomous control system. It is shown that the cellular automaton has the structure of a dynamical system, therefore it can be considered as a base system. For a discrete non-autonomous control system, a cocycle consisting of a base system and an evolutionary system is constructed. An upper bound for the topological entropy for a cocycle of a discrete non-autonomous control system over a base system generated by a cellular automaton is obtained. As an example, we study the non-autonomous Henon system, whose parameters depends on the cellular automaton, for which the state of the cell is determined by the disjunction of the cell itself and two neighboring cells. The inequality for estimating the growth exponents of the Henon mapping is obtained. In addition, we obtain an estimate for the fractal dimension of the compact invariant set of the Henon system. The behavior of the trajectories of the non-autonomous Henon system with some certain parameters and certain initial data is demonstrated.



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