THRESHOLD METHOD FOR CONTROL OF CHAOTIC OSCILLATIONS

: In this paper, threshold method for control of chaotic (nonlinear) oscillations is presented. Main principles as how to use this method are describe more detail. One of the practical realizations, i


Volodymyr Rusyn
Chaos is the most interdisciplinary thematic areas; it includes very interesting, complex, nonlinear phenomena that have been intensively studied and regard many different areas ranging from social systems to sciences, mathematics and engineering [1][2][3][4].
In the area of engineering, chaos has great potential in many technological disciplines, such as in information and computer sciences, power systems protection, liquid mixing, flow dynamics, economics, magnetism, biomedical systems analysis etc. [5][6][7][8].
Chaotic signals can be generated by electronic circuits [9][10][11][12], memristors [13,14].These signals depend on the system's initial conditions and this dependence is very sensitive.Thus, they demonstrate the feature of being unpredictable.At the same time, chaotic signals are wide-band signals.
Although they seem to be random, they are fully deterministic, highly sensitive to the system parameters, as well.
THE PURPOSE OF THE ARTICLE.This article aims to present a method that allows to control of chaotic oscillations.

THE MAIN MATERIAL.
Consider a general N-dimensional dynamical system, described by the evolution equation x=F(x, t) where x=(x 1 , x 2 , …, x N ) are the state variables, and variable xi is chosen to be monitored and threshold controlled.The prescription for threshold control in this system is as follows: control will be triggered whenever the value of the monitored variable exceeds a critical threshold x * (i.e., when x i > x * ) and the variable x i will then be reset to x * [15].The dynamics continues till the next occurrence of x i exceeding the threshold, when control resets its value to x * again.
No run-time knowledge of F(x) is involved, and no computation is needed to obtain the necessary control.The method only involves monitoring a single variable and no parameters are perturbed in the original system.The theoretical basis of the method does not involve stabilizing unstable periodic orbits, but rather involves clipping desired time sequences (symbol sequences in maps) and enforcing a periodicity on the sequence through the thresholding action which acts as a resetting of initial conditions.The effect of this scheme is to limit the dynamic range slightly, i.e., «snip» off small portions of the available phase space, and this small controlling action is effective in yielding a range of stable behaviors.In fact, chaos is advantageous here, as it possesses a rich range of temporal patterns which can be clipped to different behaviors.This immense variety is not available from thresholding regular systems.
It can be shown analytically for onedimensional maps and numerically for multidimensional systems that the threshold mechanism yields stable orbits of all orders by simply varying the threshold level [15].But so far there had been no direct experimental verification of this control scheme.To the best For example, this method can be used for chaotic Chua's circuit.Fig. 1 shows a circuit that realize this method practically.

CONCLUSIONS AND SUGGESTIONS.
This method can be used for circuit that generate chaotic oscillations.Controlled attractor with a fixed period can be used in modern systems transmitting and receiving information.Number of periodic (controlled) attractor can be used as a keys for masking of information carrier.
Ph.D. (Engineering), Assistant Professor of the Department of Radio Engineering and Information Security Yuriy Fedkovych Chernivtsi National University UKRAINE FORMULATION OF THE PROBLEM.
International multidisciplinary scientific journal «ΛΌГOΣ.The art of scientific mind» ■ №6 ■ October, 2019 ENGINEERING AND IT All rights reserved Creative Commons Attribution 4.0 International License of our knowledge, this work is the first such attempt.Now to experimentally demonstrate the range and efficacy of the method, we implement it on three different chaotic electrical circuits, including a hyperchaotic one.The results from our experiments are presented in detail in the sections below.