Chaos, Solitons and Fractals 31 (2007) 105–111

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Chaos synchronization in RCL-shunted Josephson junctionvia active control

Ahmet Ucar a,*, Karl E. Lonngren b, Er-Wei Bai b

a Department of Electrical and Electronics Engineering, Firat University, Elazıg 23119, Turkeyb Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA 52242, USA

Accepted 9 September 2005

Abstract

This paper investigates the synchronization of coupled RCL-shunted Josephson junction that is of interest in high-frequency applications. A nonlinear controller is developed in order to achieve the desired behavior. The synchroniza-tion is obtained using the slave–master technique and the controller ensures that the states of the controlled chaoticslave system exponentially synchronize with the state of the master system. Numerical simulations are illustrate andverify the proposed method.� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

Different models have been introduced to represent Josephson junctions (JJ) [1]. Amongst them are the shunted lin-ear resistive–capacitive junction (RCSJ), the shunted nonlinear resistive–capacitive junction (SNRCJ) and the shuntednonlinear resistive–capacitive–inductance junction (RCLSJ). The study of the chaotic behavior of Josephson junctionsis of fundamental and even practical interest [2–4]. The first two models of the Josephson junction contain two statevariables [5] and show chaotic behavior with the external sinusoidal signal while the shunted inductance junctionincludes the third state variable and causes the RCLSJ model to generate a chaotic oscillation with external dc biasonly. The RCLSJ model is found to be more in high-frequency applications [4]. The chaotic behavior of the RCLSJmodel has been extensively studied [4].

In this paper, the synchronization of two coupled JJ models using active control techniques that were introduced in[6] and successfully applied in several other systems [7–9] is employed.

In Section 2, the dynamics of the RCLSJ model is briefly presented. In Section 3, the synchronization of coupledRCLSJ models is discussed. Numerical simulations are provided in Section 4 to illustrate and verify the veracity ofthe controller that is obtained using this method. Finally a concluding remark is given in Section 5.

0960-0779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2005.09.035

* Corresponding author. Tel.: +90 424 2370000x5214; fax: +90 424 2415526.E-mail addresses: aucar1@firat.edu.tr (A. Ucar), lonngren@engineering.ulowa.edu (K.E. Lonngren), erwie@engineering.ulowa.edu

(E.-W. Bai).

106 A. Ucar et al. / Chaos, Solitons and Fractals 31 (2007) 105–111

2. The RCLSJ model of Josephson junction

The RCLSJ model proposed in [10] and studied in [11–13] is found to be a more accurate and appropriate model inhigh-frequency applications and in chaotic digital communication systems [4]. A standard form of the nonlinear dynam-ical equation RCLSJ model proposed in [10] may be expressed as

Fig. 1.z(0)) =solutio

_x1 ¼ y1

_y1 ¼ 1=bC i� gðy1Þy1 � sinðx1Þ � z1½ �_z1 ¼ 1=bLðy1 � z1Þ

ð1Þ

where bC and, bL represent the capacitive and inductance constants respectively and are considered to be the modelparameters. i represents the external current consisting of a dc component only. The nonlinear damping term g(y1)is approximated with a current–voltage relation between the two junction resistances and is represented by the follow-ing step function

gðy1Þ ¼0:366 if jy1j > 2:9

0:061 if jy1j 6 2:9

�ð2Þ

The state variables in Eq. (1) x1, y1, z1 correspond to phase difference, junction voltage, and current through shuntedinductance, respectively. For the actual physical model and details of the junction, the reader should refer to [10–12].The system without the external current i, has the equilibrium points xe = (xe1,ye1,ze1) = (±np, 0,0) wheren = 0,1,2,3, . . . The system with an external dc current i has been analyzed in [10,11] and the chaotic behavior has beenobserved. In the numerical simulation, the parameters were chosen to have the numerical value of bL = 2.6, bC = 0.707

-2 -1 0 1 2 3-0.5

0

0.5

1

1.5

-2 -1 0 1 2 3-0.5

0

0.5

1

1.5

-2 -1 0 1 2 3-0.5

0

0.5

1

1.5

2

-1 0 1 2 30

0.5

1

1.5

2

(a) (b)

z 1

z 1

y1 y1

y1 y1

z 1 z 1

(c) (d)

Phase portrait of y1 and z1 of the system (1) with the parameters bC = 2.6 and bL = 0.707 and initial conditions (x(0),y(0),(0,0,0): (a) two periods solution for i = 1.01; (b) a limit cycle for i = 1.12; (c) chaotic behavior for i = 1.20; (d) two periods

n for i = 1.23.

A. Ucar et al. / Chaos, Solitons and Fractals 31 (2007) 105–111 107

and 0 < jij 6 2 with the current voltage characteristic defined in Eq. (2). The initial conditions were(x(0),y(0), z(0)) = (0,0,0). For the 0 < jij 6 1, the system trajectory decays to the origin. Increasing i results in multipleperiodic solutions and eventually a chaotic solution. Fig. 1(a) and (b) show a period two and a limit cycle for i = 1.10and i = 1.12 respectively. A further increase of i leads to chaotic behavior and period two solutions. Fig. 1(c) and (d)respectively show chaotic and period two solutions for i = 1.2 and i = 1.5. It has been shown in [4] that the systemexhibits chaotic behavior for low values of inductance (bL = 2.6) as observed in Fig. 1(c). For large values of inductancebL = 26 with i = 1.2 the junction voltage y1 behaves as a relaxation oscillator.

Our interest is to design an active controller to synchronize two RCLSJ systems and discuss its validity in the fol-lowing section.

3. Chaos synchronization for the RCLSJ model

Consider the RCLSJ chaotic system defined in (1) with the subscript �1� as the master system with which the slavesystem needs to be synchronized. Since the external dc input i has been found to be very important in determining thechaotic attractors observed from (1), we were led to ask whether it would be possible to synchronize two different cha-otic systems together.

Consider the second RCLSJ chaotic system with the subscript �2� that contains a different value of the system param-eter ~i

_x2 ¼ y2 þ laðtÞ_y2 ¼ 1=bC

~i� gðy2Þy2 � sinðx2Þ � z2

� �þ lbðtÞ

_z2 ¼ 1=bLðy2 � z2Þ þ lcðtÞð3Þ

where la(t), lb(t), and lc(t) are active control signals that are yet to be determined. Here, the aim of the control signals isto force the slave system 2 to follow the master system 1. Thus, one-way synchronization of the two RCLSJ chaoticsystems from system 1 to system 2 will be achieved. Without the controls la(t), lb(t), and lc(t), the second system de-fined by (2) will have a chaotic behavior depending on the chosen value of ~i.

In order to obtain active control signals, we define the error state between the dependent variables of system 2 andsystem 1 as

ex ¼ x2 � x1

ey ¼ y2 � y1

ez ¼ z2 � z1

ð4Þ

Subtracting the first system (1) from the second system (2), which includes the control signals, we obtain

_ex ¼ ey þ laðtÞ

_ey ¼1

bC

½~i� i� � 1

bC

½gðy2Þy2 � gðy1Þy1 þ sinðx2Þ � sinðx1Þ� �1

bC

ez þ lbðtÞ

_ez ¼1

bL

ðey � ezÞ þ lcðtÞ

ð5Þ

Since the nonlinear terms in both systems are known, the control signals can be defined as

laðtÞ ¼ u1ðtÞ

lbðtÞ ¼ �1

bC

½~i� i� þ 1

bC

½gðy2Þy2 � gðy1Þy1 þ sinðx2Þ � sinðx1Þ� þ u2ðtÞ

lcðtÞ ¼ u3ðtÞ

ð6Þ

Substitute Eq. (6) into (5) leads to

_ex

_ey

_ez

264

375 ¼

0 1 0

0 0 �1=bC

0 1=bL �1=bL

264

375

ex

ey

ez

264

375þ

1 0 0

0 1 0

0 0 1

264

375

u1

u2

u3

264

375 ð7Þ

Eq. (7) describes the error dynamics and can be considered in terms of a control problem where the system to becontrolled is now a linear system with the control input u(t) = [u1,u2,u3]T [14]. Since the error dynamics is a full state

108 A. Ucar et al. / Chaos, Solitons and Fractals 31 (2007) 105–111

controllable entity, the feedback gains can be designed to stabilize the state of error system, [ex,ey,ez]T so that the error

signals converge to zero as time t goes to infinity. This implies that two RCLSJ chaotic systems with different values ofthe parameter i are synchronized. There are many possible choices for the control u(t). We choose

Fig. 2i = 1.1

u1

u2

u3

264

375 ¼ �

k11 k12 k12

k21 k22 k23

k31 k32 k33

264

375

ex

ey

ez

264

375 ð8Þ

where the constants kij�s are controller gains. For the proper choice of the constants kij�s the closed loop system char-acteristic matrix must have all of the eigenvalues with negative real parts. In this case the control signal defined in (8)yields the error dynamic as the function of kij�s as

_ex

_ey

_ez

264

375 ¼

�k11 1� k12 �k13

�k21 �k22 �1=bC � k23

�k31 1=bL � k32 �1=bL � k33

264

375

ex

ey

ez

264

375 ð9Þ

The rate of convergence is now determined by the numerical values of the parameters of the feedback gains kij�s. For theparticular choice of feedback gains

k11 k12 k12

k21 k22 k23

k31 k32 k33

264

375 ¼

5 1 0

0 5 �1:4144

0 0:3846 4:6154

264

375 ð10Þ

The error system given in (9) is stable and the closed loop system has eigenvalues that are found to be �5, �5 and �5.This choice of control gains will lead to a stable error system and to synchronization of the two RCLSJ chaotic systems.

4. Simulation results

Here the numerical results are given to verify the proposed method. In order to demonstrate the procedure, we keepthe external input for the master system at i = 1.135 and chose different values of the external input ~i for the slave sys-tem. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve the master and slave systemsdefined in (1) and (3) respectively with a time step size of 10�3. Zero initial conditions and the parameters bL = 2.6,bC = 0.707 are chosen and fixed for the master and slave systems. For the value of external dc input i = 1.135 the mastersystem exhibits the chaotic behavior as depicted in Fig. 2.

First, consider the case of the slave system has the parameter ~i ¼ 1:2, i.e., the chaotic behavior depicted in Fig. 1(c).The controller gains are chosen as defined in (10) in order to achieve the synchronization within 1 s after controller acti-

. Phase portrait of y1 and z1 of the master system (1) with zero initial conditions and the parameters bC = 2.6, bL = 0.707,35.

Fig. 3. The time response of the error signals of coupled RCLSJ chaotic systems with active controller applied at a time s = 200. Theexternal input for master system (1) was i = 1.135 and the slave system (3) parameter was ~i ¼ 1:2.

A. Ucar et al. / Chaos, Solitons and Fractals 31 (2007) 105–111 109

vation time s on. When the control signal is activated at s = 200 s, the simulation results illustrated in Fig. 3 for theerror system of (9) clearly demonstrate that synchronization has occurred. Fig. 3(a) shows the error signal ex = x2 � x1;Fig. 3(b) shows the error signal ey = y2 � y1 and Fig. 3(c) shows the error signals ez = z2 � z1, respectively. Fig. 3 showsthat the error signals converge to zero and this illustrates that the synchronization of the chaotic behavior of the slavesystem depicted in Fig. 2 with the chaotic behavior of the master system depicted in Fig. 1(c) is achieved.

One of the advantage of the nonlinear controller proposed in [6–8] is decentralizing error system is feasible. Considerthe error system defined in (5) with the following controller

laðtÞ ¼ �ey � k11ex

lbðtÞ ¼ �1

bC

½~i� i� þ 1

bC

½gðy2Þy2 � gðy1Þy1 þ sinðx2Þ � sinðx1Þ� �1

bC

ez � k22ey

lcðtÞ ¼ �1

bL

ey � k33 �1

bL

ez

� � ð11Þ

leads to the error dynamics given in (12)

_ex

_ey

_ez

264

375 ¼

�k11 0 0

0 �k22 0

0 0 �ðk33 � 1=bLÞ

264

375

ex

ey

ez

264

375 ð12Þ

Secondly, to the synchronization of the same chaotic behavior of the slave system with the limit cycle behavior of themaster system for i = 1.12 within 5 s using the controller gains of (12) k11 = k22 = 1 and k11 = 3.6/2.6 can be achieved.

Fig. 4. The time response of the error signals of coupled RCLSJ systems with active controller applied at a time s = 200. The externalinput for master system (1) was i = 1.12 and the slave system (3) parameter was ~i ¼ 1:14.

110 A. Ucar et al. / Chaos, Solitons and Fractals 31 (2007) 105–111

The numerical results of the error system are depicted in Fig. 4 for s = 200. The results of this calculation clearly indi-cate that the slave system follows the master system after the control signal is activated.

5. Conclusion

In this paper, we have obtained a nonlinear active controller that can be used to synchronize two RCLSJ chaoticsystems together such that the frequency of oscillation of the slave system will follow the master system. The controlleris designed such that the error dynamics can be decentralized and the desired speed of the convergence time of synchro-nization can be modified by the linear feedback gain.

References

[1] Likharev KK. Dynamics of Josephson junction and circuits. New York: Gordon and Breach; 1986.[2] Cirillo M, Pedersen NF. On bifurcation and transition to chaos in a Josephson junction. Phys Lett A 1982;90:150.[3] Graham R, Schlautmann M, Keymer J. Dynamical localization in Josephson junction. Phys Rev Lett 1991;67:255.[4] Dana SK, Sengupta DC, Edoh KD. Chaotic dynamics in Josephson junction. IEEE Trans Circuits Syst I 2001;48:990–6.[5] Stewart WC. Current–voltage characteristics of Josephson junctions. Appl Phys Lett 1968;12:277.[6] Bai EW, Lonngren KE. Sequential synchronization of two Lorenz systems using active control. Chaos, Solitons & Fractals 2000;

11:1041–4.

A. Ucar et al. / Chaos, Solitons and Fractals 31 (2007) 105–111 111

[7] Ucar A, Lonngren KE, Bai EW. Synchronization of the unified chaotic systems via active control. Chaos Solitons & Fractals2006;27:1285–91.

[8] Ucar A, Lonngren KE, Bai EW. Synchronization of chaotic behavior in nonlinear Bloch equations. Phys Lett A 2004;314:96–101.[9] Bai EW, Lonngren KE, Ucar A. Secure communication via multiple parameter modulation in a delayed chaotic system. Chaos,

Solitons & Fractals 2005;23:1071–6.[10] Whan CB, Lobb CL, Forester MG. Effect of inductance in externally shunted Josephson tunnel junction. J Appl Phys Rev

1995;77:382–9.[11] Whan CB, Lobb CL. Complex dynamic behaviors in RLC-shunted Josephson tunnel junction. Phys Rev E 1996;53:405–13.[12] Cawthorne AB, Whan CB, Lobb CL. Complex dynamics of resistively and inductively shunted Josephson junction. J Appl Phys

1998;84:1126–32.[13] Yang XS, Li Q. A computer-assisted proof of chaos in Josephson junctions. Chaos, Solitons & Fractals 2006;27:25–30.[14] Goodwin GC, Graebe SF, Salgado ME. Control system design. New Jersey: Prentice Hall; 2001.