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Visiiyk NTIJU KP1 Servia Radiolekhnika Radioaparat.obuduuannia, "2018, Iss. 75, pp. 61—65
Control of the Modified Chaotic Chua's Circuit Using Threshold Method
Rusyn V. B.1, Pribylova L.2, Dimitriu D.-G.3,
i-Yuriy Fedkovych Chernivtsi National University, Ukraine 2Masaryk University, Czech Republic 3Alexandru loan Cuza University of Iasi, Romania
E-mail: rusyn_ v&ukr.ncl■
The modified Cliua's circuit that, realize chaotic behaviour is presented. This circuit having a simple nonlinear element designed to be accurately piecewise-linear modelled. The circuit was modelled by using Mult.iSim software environment. System's behaviour is investigated through numerical simulations, by using well-known tools of nonlinear theory, such as chaotic at.tract.or and time distributions of the chaotic coordinates. Using threshold method was practical realization of the control of chaotic att.ract.or. This modified Chua's circuit, that, generate a chaotic and controlled at.tract.or with a fixed period can be used in modern systems transmitting and receiving information. Number of periodic (controlled) at.tract.or can be used as a keys for masking of information carrier.
Key words: chaos: Cliua: control: threshold method
DOI: 10.20535/RADAP. 2018.75.61-65
Introduction
Chaos theory have in different for
application, snch as biology [1]. economy [2 4]. plasmas [5]. magnetism [G]. memristor [7 19]. electronics schemes [20.21]. etc. There are many different circuit realizations of the chaotic Chna's generator.
For chaos control have been proposed many different approaches or techniques, snch as linear feedback control. OGY. inverse optimal control, etc [22 30]. The theoretical basis of most known methods for control chaos is stabilizing the unstable periodic orbits via parameter perturbation.
For modelling, analysis and demonstrate results was selected software MnltiSim.
1 Modelling and Analysis of Non-Linear Element
Nonlinear elements these are elements in which the relation between voltage and current is a nonlinear function. An example is a diode, in which the current is an exponential function of the voltage. Circuits with nonlinear elements are harder to analyze and design, often requiring circuit simulation computer programs snch as SPICE.
The circuit realization for modelling and analysis of the non-linear element is displayed in Fig. 1. with component: one operational amplifier TL082: resistors R1 = R2 = 220 Q, R3 = 1,2 kQ, R4 = 6 kQ, R5 = 800 Q; two diodes 1N4148; voltage - ±9 V.
Fig. 1. Circuit realization for modelling and analysis of nonlinear characteristic
Fig.
AC 0 ¡DË] AC 0 [k] - SBgle Normal Auto |Mûrie |
V/I characteristic of nonlinear element
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Rusvn V. tí., Pribylova L. , Dimitriu i). -G.
The nonlinear characteristic was modelled by the following parameters: E = 9 V, f = 1 kHz, R = 6 kQ. Fig. 2 shows result of modelling of nonlinear element using MultiSim. The simulation parameters: U 1 = 5 V/div, U2 = 5 V/div.
2 Modelling and Analysis of the Modified Chaotic Chua's Generator
Fig. 3 shows simulated scheme of the modified chaotic Chua's generator by using MnltiSim. Circuit was realized on the one operational amplifier TL082, powered by a 9 V. two diodes 1N4148. resistors R1 R2 = 220 Q, R3 = 1.2 kQ, R4 = 800 Q, potentiometer R5 = 2 kQ (1.7 kQ), two capacitors CI = 10 nF, C2 = 100 nF. inductor LI 18 mH.
Fig. 3. The simulated circuit of the modified chaotic Chua's generator
Fig. 4 shows the result of circuit simulation. Generated chaotic signal in the plane XY presented on the virtual oscilloscope. Coordinate X in the circuit correspond voltage UC2, coordinate Y - voltage Uci- The simulation parameters: Ui = 1 V/div, U2 = 2
Fig. 4. Chaotic attractor
Fig. 5. Time dependences of tlie coordínate X and Y
In Fig. 5 shows time dependences of tlie coordinates X and Y. The simulation parameters for Fig. : U = 2 V/div, U2 = 5 V/div, time scale 2 ms/div.
3 Threshold method for control of chaotic oscillations
Consider a general N-dimerisiorial dynamical system, described by the evolution equation x = F(x, t) where x = (xi,x2, ...,x^) are the state variables, and variable 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*) and the variable will then be reset to x*. The dynamics
continues till the next occurrence of exceeding the
x*
F( x)
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 one-dimensional maps and numerically for multidimensional systems that the threshold mechanism yields stable orbits of all orders by simply varying the threshold level. But so far there had been no direct experimental verification of this control scheme [31]. Now to experimentally demonstrate the range and efficacy of the method, we implement it on the modified chaotic Chna's circuit. We consider a realization of
the double scroll chaotic Chua's attractor given by the following set of (rescaled) three coupled ODEs:
^ = a[y — x — g{x)},
dy
— = x — y + z, at
-T = —to, dt '
(1)
(2)
(3)
where a = 10, fl = 14.87, g(x) — piecewise linear function. Chaotic oscillations were if system parameters a = 2,b = 6.7, and dynamic variables x = 1.2, y = 0.8, z = 1.4.
The circuit realization of the above is displayed in Fig. 6, with component values: capacitors CI 100 nF, C2 10 nF, DA1-DA4 operational amplifier TL082, powered by a 9 V, GB1 threshold reference voltage, inductor LI = 18 mH, resistors R1 = R2 = 1.71 kQ, R3 = R4 = 220 Q, R5 = 800 Q, R6 = 1.2 kQ, R7 = 1 kQ potentiometer R8 = 100 kQ, diodes VD1-VD3 -1N4148.
Fig. 7. Uncontrolled chaotic attractor in the V1 — V2 plane
Fig. 8. 2-period controlled attractor obtained when x*=2.7 V in the V1 — V2 plane
Fig. 6. Modified chaotic Chua's circuit with threshold level controlling circuit (shown in the dotted box). Vt is the threshold controlled signal
We implement an even more minimal thresholding. Instead of demanding that the x variable be reset to x* if it exceeds x*, we only demand this in Eq. ( ). This has very easy implementation, clS it avoids modifying the value of x in the nonlinear element g(x), which is harder to do. So then all we do is to implement dy/dt = x* — y + z instead of Eq. ( ), when x > x*, and there is no controlling action if x < x*. In the circuit, the voltage Vr corresponds to x*.
Fig. 7 Fig. 9 shows experimental results of the control of chaotic oscillations.
Fig. 9. 3-period controlled attractor obtained when x*=2.71 V in the V1 — V2 plane
Conclusions
For the first time was used threshold method for control of chaotic oscillations for modified Chua's
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Rusvn V. B., Pribylova L. , Dimitriu D. -G.
chaotic generator. This modified Chua's circuit that generate a chaotic and 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.
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Управлшня м од и ф i ко в а н о ю хаотичною схемою Чуа пороговим методом
Русин В., Прибилова Л., Длштргу Д.-Г.
В робот! представлена модифшовапа хаотична схема Чуа, яка реал!зуе хаотичпу поведшку. Ця схема мае простий пелипйпий елемепт. спроектовапий так. щоб мати кусково-лшшпу характеристику. Ця модифшовапа схема Чуа, яка геперуе хаотпчпий та коптрольовапий аттрактор з ф1ксовапим перюдом. може використову-ватися в сучаспих системах передаваппя та приймаппя шформацп. Число перюдичпих (коитрольова1шх) атра-ктор1в може використовуватися як ключ! для маскува-ппя шформацишого пооя. За допомогою ирограмиого середовища Mult.iSim проведено схемотехшчпий апал!з i представлено результати моделюваппя пелшшпого еле-мепта та модиф1ковапо1 хаотично! схеми Чуа. Досл1дже-па поведшка системи за допомогою чиселыюго моделюваппя. використовуючи в1дом! шетрумептп пелшшпо! теорп. так! як хаотичпий атрактор i часов! розподгли хаотич1шх координат. Приведено опис порогового методу для здшепеппя управлишя хаотичпими коливаппями та представлено результати практичного застосуваппя
дапого методу до модифшовапо! хаотично! схеми Чуа. Практичпими результатами е видшеш 2- та З-перюдш коптрольоваш орб!ти 1з хаотичного атрактора.
Ключовг слова: хаос: Чуа: управлишя: пороговий метод
Управление модифицированной хаотичной схемой Чуа пороговым методом
Русый В., Прибылова Л., Длштриу Д.-Г.
В работе представлена модифицированная хаотичная схема Чуа, которая реализует хаотическое поведение. Эта схема имеет простой нелинейный элемент, спроектированный так. чтобы иметь кусочпо-липейпую характеристику. Эта модифицированная схема Чуа, которая генерирует хаотический и контролируемый аттрактор с фиксированным периодом, может использоваться в современных системах передачи и получения информации. Количество периодических (контролируемых) аттракторов может использоваться как ключи для маскировки информационного носителя. С помощью программной среды МиШвпп проведен схемотехнический анализ и представлены результаты моделирования нелинейного элемента и модифицированной хаотической схемы Чуа. Исследовано поведение системы с помощью численного моделирования, используя известные инструменты нелинейной теории, такие как хаотичный аттрактор и временные распределения хаотических координат. Приведено описание порогового метода для осуществления управления хаотическими колебаниями и представлены результаты практического применения данного метода к модифицированной хаотической схеме Чуа. Практическими результатами являются выделенные 2- и З-периодпые контролируемые орбиты с хаотического аттрактора.
Ключевые слова: хаос: Чуа: управление: пороговый метод
