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Publicly Available Published by De Gruyter May 4, 2018

Synchronisation and Circuit Realisation of Chaotic Hartley System

  • Metin Varan , Akif Akgül EMAIL logo , Emre Güleryüz and Kasım Serbest

Abstract

Hartley chaotic system is topologically the simplest, but its dynamical behaviours are very rich and its synchronisation has not been seen in literature. This paper aims to introduce a simple chaotic system which can be used as alternative to classical chaotic systems in synchronisation fields. Time series, phase portraits, and bifurcation diagrams reveal the dynamics of the mentioned system. Chaotic Hartley model is also supported with electronic circuit model simulations. Its exponential dynamics are hard to realise on circuit model; this paper is the first in literature that handles such a complex modelling problem. Modelling, synchronisation, and circuit realisation of the Hartley system are implemented respectively in MATLAB-Simulink and ORCAD environments. The effectiveness of the applied synchronisation method is revealed via numerical methods, and the results are discussed. Retrieved results show that this complex chaotic system can be used in secure communication fields.

1 Introduction

Chaos resembles high complexity of nonlinearities. The dynamics of chaos have been studied in many studies [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. The sensitivity for initial conditions and system parameters, complexity, and unpredictability are three essential behaviours of the chaotic systems [1], [2], [3]. Circuit realisation of chaotic signal becomes popular after Chua circuit first revealed the behaviour of chaos in a circuit [3]. After then, Colpitts oscillator has been presented by Kennedy [5]. Semiconductors like junction field effect transistor (JFET) and diode elements are combined by capacitor and inductor elements for generating various chaotic circuit models. Wien-bridge oscillator [6], [7], [8], bi-quad based chaotic oscillator [9], and Twin-T oscillator [10] are some of them. Operational amplifier, resistance, and capacitor circuit elements are combined as a chaotic circuit by Yim et al. [11]. Another simple autonomous chaotic circuit which is made of memristor, inductor, and a capacitor is proposed by Muthuswamy and Chua [12]. An autonomous chaotic circuit that is made of four components is introduced by Barboza and Chua [13]. Hartley oscillator is another realised oscillator that is used in high-frequency telecommunication applications as to its simplicity in structure. Using inductor circuit elements creates noise problems on integrated circuit design. Thus, inductor-based model is transferred into equivalent model in chaotic signals generation. In this context, Tchitnga et al. [14] contributed to obtain the simplest chaotic circuit by using rarely studied chaotic Hartley’s oscillator. They used parasitic capacitors of the JFET for creating an exterior resonant circuit of a Hartley’s oscillator.

By using control and synchronisation techniques, chaotic systems have variety of application fields including finance, chemistry, and biology. The main targets for controlling a chaotic system are stabilisation and regulation of the system [21], [22]. Adaptive [23], active [24], sliding mode [25], and back-stepping control [26] are available methods used for controlling chaotic systems. These methods can be also used in chaotic synchronisation procedures. In control theory, chaos synchronisation is directly related to the observer problem. Generally, the state variable of designed controller’s drive side will make the state variable of designed controller’s response that is tracing the trajectories of the state variable of master system [27].

In this study, the equivalent model developed by Tchitnga et al. is used for synchronisation. Synchronisation of Hartley chaotic system has not been seen in the literature. Also, the electronic circuit design of the system is done in simulation programme. The system is different from many other chaotic systems because it has an exponential term and different statements. Thus, the electronic circuit design process is difficult in these systems.

In the following section, the chaotic Hartley system is described in detail. Section 3 presents electronic circuit realisation of chaotic Hartley system. The synchronisation of Hartley system is discussed in Section 4. The section also reveals the success of synchronisation via numerical simulations. Finally, conclusion remarks are given in the last section.

2 Chaotic Hartley System

The Hartley oscillator includes a tapped coil made of 55 turn upper coil (L1) and 9 turn lower coil (L2) and a JFET. The circuit as shown in Figure 1 is an equivalent circuit of Hartley’s oscillator. The model is used for driving high-frequency small signals [28]. Fundamental nonlinearities arise from the existence of semiconductor materials. iD current and current source between the source and the drain create such an effect on the circuit. The equations set given below is the mathematical model of cut-off, saturation, and triode regions of the JFET function [29], [30], [31]:

Figure 1: Equivalent circuit of Hartley’s oscillator.
Figure 1:

Equivalent circuit of Hartley’s oscillator.

If Kirchhoff’s laws are applied to this circuit design, the circuit equations are obtained as

(1)dvGSdt=1CGS(i1+i2iDId),dvGDdt=1CGD(i2+Id),di1dt=1L1vGS,di2dt=1L2(vGS+vGD+E),

here

(2)Id={0if vGSVGSoff,gm0(vGSVGSoff)2if vGDVGSoff,gm0(vGSvGD)(vGS+vGD2VGSoff)if vGDVGSoff.

When the following transformations are used

(3)a1=I0VTCGSω0,a2=ISVTCGSω0,a3=gm0VTCGSω0,b1=VTL1I0ω0,b2=VTL2I0ω0,α=CGSCGD,e=EVT,

the dynamic equations of Hartley’s circuit can be obtained as [14]

(4)x˙=a1(wz)a2(exp(x)1)a3g(x,y),y˙=α(a1wa3g(x,y)),z˙=b1x,w˙=b2(ex+y),

here

(5)g(x,y)={0if xxm,(xxm)2if yxm,(xy)(x+y2xm)if yxm.

With specific values set at a1=10.7066381, a2=0.0000359, a3=0.0117372, b1=0.0010204, b2=0.00625, α=1.1152239, xm=−56.36, and e=112, the dynamic behaviour of Hartley system (4) exhibits an irregular motion with the initial conditions of (x, y, z, w)=(12.4, −120, −0.2, 5.78). Figures 2 and 3 sketch the routes of chaos for the trajectories of Hartley system (4) via time series and phase portraits, respectively.

Figure 2: Time series of Hartley system for (a) x signals, (b) y signals, (c) z signals, and (d) w signals.
Figure 2:

Time series of Hartley system for (a) x signals, (b) y signals, (c) z signals, and (d) w signals.

Figure 3: Phase portraits of Hartley system for (a) y–w phase plane, (b) z–y phase plane, (c) w–z–y phase plane, (d) w–y–x phase plane, (e) w–z–x phase plane, and (f) z–y–x phase plane.
Figure 3:

Phase portraits of Hartley system for (a) yw phase plane, (b) zy phase plane, (c) wzy phase plane, (d) wyx phase plane, (e) wzx phase plane, and (f) zyx phase plane.

Figure 2 shows the time series for Hartley system. The time series shown consists of the x signals in Figure 2a, y signals in Figure 2b, z signals in Figure 2c, and w signals in Figure 2d.

In Figure 3, phase portraits of the Hartley system are plotted for (a) yw phase plane, (b) zy phase plane, (c) wzy phase plane, (d) wyx phase plane, (e) wzx phase plane, and (f) zyx phase plane. For n-dimensional nonlinear systems positive exponent is a signature of chaotic behaviour [32]. To study complex behaviour of systems, Figure 4 shows the “Lyapunov exponent” diagrams with varying e parameter and the other parameters kept fixed. The spectrum of “Lyapunov exponents” is obtained by varying e between 0 and 150. It is consistent with the chaotic range at positive “Lyapunov exponent”. The system at a1=10.7066381, a2=0.0000359, a3=0.0117372, b1=0.0010204, b2=0.00625, α=1.1152239, xm=−56.36, and e=73.8, has three “Lyapunov exponents”, where LE1=0.001178, LE2=−0.006157, LE3=−0.133, and LE4=−1.732.

Figure 4: The spectrum of “Lyapunov exponents”, obtained by varying e between 0–150 and 70–95.
Figure 4:

The spectrum of “Lyapunov exponents”, obtained by varying e between 0–150 and 70–95.

The bifurcation diagram has varying e parameter and the other parameters kept fixed. As seen in Figure 5, after e parameter reaches over the value of 73.8, the system confirms the chaotic behaviours. The initial states of the 4D chaotic Hartley system are x (0)=12.4, y (0)=−120, z (0)=−0.2, and w (0)=5.78. For a step size in q of 0.001 and the running time of 32,500 s, the bifurcation diagram in Figure 5 was obtained.

Figure 5: Bifurcation diagram for parameter e.
Figure 5:

Bifurcation diagram for parameter e.

3 The Electronic Circuit Realisation of Chaotic Hartley System

There are some important works related to circuit realisations in the literature [33], [34], [35]. In this section, the circuit design of chaotic Hartley system (1) is implemented in simulation programme. The Hartley system has an exponential term. There are a few works in the literature related to the exponential term, and it is very difficult for circuit realisation. The Hartley circuit is designed for the aforementioned parameters and initial values of the system. The circuit of chaotic Hartley system is shown in Figure 6.

Figure 6: The circuit schematic of the chaotic Hartley system.
Figure 6:

The circuit schematic of the chaotic Hartley system.

R1=224 kΩ, R2=4444 kΩ, R3=R14=5333 kΩ, R4=555 kΩ, R5=296 kΩ, R6=944 kΩ, R7=402 kΩ, R8=1000 kΩ, R9=544 kΩ, R10=1734 kΩ, R11=3278 kΩ, R12=8571 kΩ, R13=10,000 kΩ, R15=R17=R18=100 kΩ, R16=500 kΩ, C1=C2=C3=C4=1 nF, Vn=−V, and Vp=15 V were taken. Also, the exponential block circuit is given in Figure 7. R1=R2=R5=100 kΩ, R3=3 kΩ, R4=89.5 kΩ, and V1=1 V were chosen in exponential block circuit.

Figure 7: The exponential block circuit of the chaotic Hartley system.
Figure 7:

The exponential block circuit of the chaotic Hartley system.

The simulation outputs of chaotic Hartley system in ORCAD-PSpice are seen in Figures 8 and 9. Figure 8 shows the time series for x, y, z and w outputs in 1000 ms. The phase portraits are given in Figure 9 for xy, xz, yz, and yw state combinations.

Figure 8: The time series of the circuit schematic of the chaotic Hartley system.
Figure 8:

The time series of the circuit schematic of the chaotic Hartley system.

Figure 9: The phase portraits of the circuit schematic of the chaotic Hartley system.
Figure 9:

The phase portraits of the circuit schematic of the chaotic Hartley system.

4 Synchronisation of Hartley System

For the synchronisation, two coupled chaotic Hartley systems are considered with different initial conditions. The drive and response systems are denoted by subscript 1 and subscript 2, respectively. Firstly, drive system is expressed as follows:

(6)x˙1=a1(w1z1)a2(exp(x1)1)a3g(x1,y1),y˙1=α(a1w1a3g(x1,y1)),z˙1=b1x1,w˙1=b2(ex1+y1),

and similarly, the response system is established as below:

(7)x˙2=a1(w2z2)a2(exp(x2)1)a3g(x2,y2)+u1,y˙2=α(a1w2a3g(x2,y2))+u2,z˙2=b1x2+u3,w˙2=b2(ex2+y2)+u4,

here u1, u2, u3, and u4 in system (7) are the control functions to be determined. The state errors e1, e2, e3, and e4 are defined as

(8)e1=x2x1,e2=y2y1,e3=z2z1,e4=w2w1.

This leads to

(9)e˙1=a1(e4e3)a2(exp(x2)exp(x1))a3(g(x2,y2)g(x1,y1))+u1,e˙2=α(a1e4a3(g(x2,y2)g(x1,y1)))+u2,e˙3=b1e1+u3,e˙4=b2(e1+e2)+u4.

Now, the goal is to keep the error system asymptotically stable at the origin for the synchronisation. The Lyapunov function for system (9) is selected as follows:

(10)V(e1,e2,e3,e4)=12(e12+e22+e32+e42).

Lyapunov function (10) is positive definite, and it is equal to 0 at the equilibrium of system (9). The time derivative of Lyapunov function V is

(11)V˙=e˙1e1+e˙2e2+e˙3e3+e˙4e4=[a1(e4e3)a2(exp(x2)exp(x1))a3(g(x2,y2)g(x1,y1))+u1]e1+[α(a1e4a3(g(x2,y2)g(x1,y1)))+u2]e2+[b1e1+u3]e3+[b2(e1+e2)+u4]e4.

There are many alternatives for the controllers u1, u2, u3, and u4. If they are considered as

(12)u1=a1(e4e3)+a2(exp(x2)exp(x1))+a3(g(x2,y2)g(x1,y1))k1e1,u2=α(a1e4a3(g(x2,y2)g(x1,y1)))k2e2,u3=b1e1k3e3,u4=b2(e1+e2)k4e4,

then V˙ becomes

(13)V˙=k1e12k2e22k3e32k4e42,

where it is negative definite when the control parameter values k1, k2, k3, k4>0. Finally, the zero solution of system (9) is asymptotically stable with the choice of (12) by using Lyapunov direct method. This shows that the synchronisation of two identical chaotic Hartley oscillators is achieved.

Figure 10 depicts that synchronisation in time series with the controllers are fired at t=200 ms for (a) x signals, (b) y signals, (c) z signals, and (d) w signals. After the activation of synchronisation system, success of fully synchronised system is seen for all signals.

Figure 10: Synchronisation in time series with the controllers are fired at t=200 ms for (a) x signals, (b) y signals, (c) z signals, and (d) w signals.
Figure 10:

Synchronisation in time series with the controllers are fired at t=200 ms for (a) x signals, (b) y signals, (c) z signals, and (d) w signals.

Figure 11 sketches multiple signals of synchronisation in time series with the controllers are fired at t=200 ms for (a) e1 signals, (b) e2 signals, (c) e3 signals, and (d) e4 signals. After activation of controllers, the success of fully synchronised system is seen for all signals with zero synchronisation error.

Figure 11: Synchronisation errors in time series with the active controllers are fired at t=200 ms for e1 signals, e2 signals, e3 signals, and e4 signals.
Figure 11:

Synchronisation errors in time series with the active controllers are fired at t=200 ms for e1 signals, e2 signals, e3 signals, and e4 signals.

Figure 12 shows the phase plains for the synchronisation errors and the state variables x, y, z, and w signals for the drive and response system. As seen in the aforementioned figures it can be appreciated that drive and response are pretty well synchronised.

Figure 12: Phase plains for the synchronisation errors and the state variables (a) x, (b) y, (c) z, and (d) w signals for the drive and response system.
Figure 12:

Phase plains for the synchronisation errors and the state variables (a) x, (b) y, (c) z, and (d) w signals for the drive and response system.

5 Conclusion

To the knowledge of the authors, this is the first paper that investigates circuit realisation and synchronisation of the Hartley system with the Lyapunov based nonlinear control method. Numerical simulations are demonstrated to show the effectiveness of proposed synchronisation method, and the results are discussed. The synchronisation success of chaotic Hartley systems is demonstrated with its zero equilibrium point. And the synchronisation success of the system is demonstrated via time series of synchronised and error signals. It is also shown that the synchronisation is observed in an effective amount of time. Hartley chaotic system has an exponential term and different statements. Also, it is very sensitive as seen in its parameters and initial conditions. Thus, physical realisation is difficult. In this work, the electronic circuit application is also realised in a simulation programme.

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Received: 2018-01-15
Accepted: 2018-04-10
Published Online: 2018-05-04
Published in Print: 2018-06-27

©2018 Walter de Gruyter GmbH, Berlin/Boston

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