Taming of chaos and synchronisation in RCL-shunted Josephson junctions by external forcing

Taming of chaos and synchronisation in RCL-shunted Josephson junctions by external forcing

S.K. Dana, P.K. Roy, G.C. Sethia, A. Sen and D.C. Sengupta

Abstract: The use of external forcing to influence the dynamics of RCL-shunted Josephsonjunction is reported either to control chaos or to synchronise two identical coupled junctions. Thefrequency of an external periodic signal is used as the control parameter for taming or suppressingchaos in a junction. Numerical investigations show that the junction oscillation is amplitudemodulated at forcing frequencies much lower than the natural frequency of the junction. Whenthe forcing frequency is increased to a higher range close to the natural frequency of the junction,phase locking (n :m) of the sinusoidal forcing is observed. Two important routes to chaos, perioddoubling and torus breakdown are observed as the forcing frequency range is changed.Synchronisation of two identical junctions is implemented using an external negative pulsing.

1 Introduction

The superconducting Josephson junction (JJ) findspractical use as a millimetre or sub-millimetre waveoscillator [1, 2], in digital systems [3] and in SQUIDmagnetometer applications [4]. Most of the earlier studieson JJ dynamics have used an RC-shunted junction model[1–3], which shows chaotic behaviour in the presence of anrf magnetic field. However, a resistive-capacitive-inductiveshunted junction (RCLSJ) model provides a more accuraterepresentation of experimentally observed [5–7] junctionbehaviour. The junction shows chaotic behaviour [8, 9] in aselected parameter space even in the absence of any externalforcing. A suitable mechanism of suppression or taming ofchaos in the RCLSJ is thus essential for useful applicationsof the superconducting device. Controlling chaos by theOGY method [10] and by using weak periodic forcing [11]are well known. These methods try to stabilise an unstableperiodic orbit (UPO) embedded in the chaotic attractor ofany system and are not very simple to implement. Simplertechniques of controlling chaos or converting chaos toperiodicity would be useful for many practical applications.In this spirit, the use of an external rf signal for controllingchaotic patterns in an RC-shunted junction has alreadybeen investigated by others [12]. In this method, theamplitude and phase-delay of the rf signal are chosen ascontrol parameters. In addition, the taming of chaos in a

E-mail: [email protected]

S.K. Dana is with the Instrument Division, Indian Institute of ChemicalBiology, Jadavpur, Kolkata 700032, India

P.K. Roy is with the Department of Physics, Presidency College, Kolkata700073, India

G.C. Sethia and A. Sen are with the Institute for Plasma Research, Bhat,Gandhinagar 382 428, Gujarat, India

D.C. Sengupta is with the Department of Mathematics and Computer Sciences,Elizabeth City State University, Elizabeth City, North Carolina 27909, USA

r The Institution of Engineering and Technology 2006

IEE Proceedings online no. 20045141

doi:10.1049/ip-cds:20045141

Paper first received 23rd September 2004 and in revised form 13th December2005

IEE Proc.-Circuits Devices Syst., Vol. 153, No. 5, October 2006

single RC-shunted junction using weak perturbation hasbeen reported [13] and also using a Melnikov function nearhomoclinic orbits in a driven junction [14]. In the latter,the suppression of chaos is made using an additionalperiodic forcing. The authors addressed, in this paper,a simple approach of taming chaos in RCLSJ using asingle sinusoidal forcing. The forcing frequency is used asthe control parameter for the taming of chaos. The RCLSJmodel shows chaotic behaviour in a selected parameterspace [8]. When the forcing frequency is much lower thanthe natural frequency of the junction, the junction voltage isamplitude modulated. The junction oscillation becomesperiodic (period-n, n is an integer) with different phaselocking ratios for forcing frequency close to the naturalfrequency of the junction and its sub-harmonics. The choiceof forcing frequency to control chaos is made from a prioriknowledge of the bifurcation of junction dynamics withforcing frequency. The bifurcation of the junction voltageshows chaotic regimes with intermediate periodic windows.The choice of forcing amplitude does need a bit of trial anderror but this is not a difficult task. The stability of thecontrolling technique is also investigated in the presence ofnoise.

Synchronisation between coupled chaotic systems [15] isan interesting area of study for understanding the collectivebehaviour of nonlinear systems [16]. Synchronisation ofthe superconducting junction arrays is also important forthe purpose of generating reasonably large output power.In this paper, a method for synchronising two identicaljunctions by using a negative pulse forcing is also proposed.The robustness of this synchronisation technique to whitenoise is also established.

The paper is organised as follows. In the nextSection, the RCLSJ model is described. The control ofchaos by external forcing with phase locking is described inSection 3. Two important bifurcation routes to chaos,namely, periodic doubling and torus breakdown underperiodic forcing are also discussed in this Section.Synchronisation of two identical coupled junctions and itsrobustness to white noise is described in Section 4. Asummary is given in Section 5.

453

2 Driven RCLSJ model

The circuit schematic of the RCLSJ model [8] is shown inFig. 1a. L is the shunt inductance with a series resistance Rs

and Is is the current flowing through it. Ic is the junctioncritical current, y is the phase difference of the super-conductor pair while C is the junction capacitance and Iext isthe external bias current. In dimensionless form, thegoverning equations of a single RCLSJ are

dxdt¼ y ð1aÞ

dydt¼ 1

bC½iext � gðyÞy � sinðxÞ � z� ð1bÞ

dzdt¼ 1

bL½y � z� ð1cÞ

Iext

Is

R(V) Rs

TK ON Rn

OFF

Rsg

0 1

0

2

y

z

JJ

V

Iext = Ic Iext

C

L

a

b

c

Fig. 1 Josephson junctiona RCLSJ model of a single junctionb Idealised current-voltage characteristics of Josepshon junction. Thejunction is ON for critical bias current Iext¼ Ic and in return path thejunction voltage tends to zero (OFF state) for bias current Iexto Ic

c Phase portrait of junction dynamics: junction voltage (y) againstshunt inductive current (Is) plot [bL¼ 2.6,bC¼ 0.707, i0¼ 1.2]

454

and

gðyÞ ¼

Rs

Rnif jyj4 Vg

IcRs

Rs

Rsgif jyj � Vg

IcRs

��

ð1dÞ

where t¼o0t, o0¼ 2peIcRs/h, bC¼ 2peIcRs2C/h, bL¼

2peIcL/h. The y, v¼V/IcRs and is¼ Is/Ic are the normalisedstate variables represented by x, y, z respectively. h isPlanck’s constant and e is the electronic charge. Rn is thejunction normal state resistance, Rsg is the subgap resistanceand Vg is the junction gap voltage. The external currentiext¼ i0+i1 sin(ost) consists of a dc bias i0 and a sinusoidalforcing A(t)¼ i1 sin(ost). The external dc bias i0¼ I0/Ic andforcing amplitude i1¼ I1/Ic are normalised currents. Theos¼o/o0 is the dimensionless angular frequency of thesinusoidal forcing. The step function g(y) approximatesthe experimental I–V characteristic of the junction asshown in Fig. 1b, which switches from Rs/Rsg¼ 0.061 toRs/Rn¼ 0.366 when 7y7Z2.9 at temperature T¼ 4.22K forselected parameters used in [5–8]. The hysteresis effect isclear from the I–V characteristic in Fig. 1b as indicated by acounter-clockwise arrow. The phase portrait of the chaoticattractor of the junction is shown in Fig. 1c for bL¼ 2.6,bC¼ 0.707, i0¼ 1.2. The natural frequency oN of theJosephson junction (autonomous) in dimensionless form, isgiven by

o2N ¼

1

bLbC� 1

b2Lð2Þ

3 Effect of sinusoidal forcing on RCLSJ

The effect of external periodic perturbation on self-sustainedoscillator is an important area of research in electronics [17],lasers [18] and in neural oscillators [19]. High amplitudeperiodic forcing can regularise chaotic self-oscillation whilesmall amplitude forcing can induce [20] phase synchronisa-tion or phase locking provided the forcing frequency is closeto the mean frequency of the chaotic oscillator. For smallforcing amplitude, if the forcing frequency is much lowerthan the natural frequency of any oscillator, amplitudemodulation is usually seen as in a Hodgkin-Huxleyoscillator [19] and in lasers [21]. Encouraged by theseobservations, we have attempted to study the effect ofperiodic forcing on the complex dynamics of a Josephsonjunction using the forcing frequency as a control parameter.The RCLSJ model in (1) shows chaotic oscillations in theparameter space [bC¼ 0.707, bL ¼ 2.6 and i0¼ 1.2] in theabsence of periodic forcing (i1¼ 0). When a sinusoidalcurrent (i1a0) is applied to the junction, in addition to thedc bias, the junction voltage is amplitude modulated forweak forcing and frequencies much lower than the junctionnatural frequency. As the forcing frequency becomes closerto the junction natural frequency and its sub-harmonics, thejunction output changes to periodic oscillations which arephase locked to the forcing signal.

The natural frequency of the junction is oNE0.629 asestimated from (2) for the selected parameters. Thebifurcation diagram of ymax with forcing frequency os isshown in Fig. 2, which shows periodic windows nearfrequencies os¼ 0.25, 0.4, 0.5 and 0.6 (forcing amplitudei1¼ 0.1) intermediate to chaotic windows. Chaos appears inthese windows via two important routes, period doublingand torus breakdown. For control of chaos, the forcingfrequencies are chosen near the periodic windows. Thestability of the controlling techniques has also been testedusing additive white noise in numerical simulations.


3.1 Amplitude modulationThe time evolutions of the junction voltage (y) are shown insolid lines in Figs. 3a and b for two different amplitudesi1¼ 0.10, 0.25 respectively at forcing frequency os¼ 0.05.The sinusoidal current in dotted lines is superimposed ontop of the junction voltage. The envelopes of the junctionvoltage amplitudes are found to follow their respectivesinusoidal amplitudes. The total external currentiext¼ i0+A(t) is shown in the middle in solid lines whereA(t)¼ i1 sin(ost). The horizontal line represents the criticalcurrent value icE1.0002. For i1¼ 0.10 in Fig. 3a, iextalways remain above the critical line where max(iext)¼ 1.3and min(iext)¼ 1.1. But for moderately larger i1¼ 0.25, thejunction voltage decays exponentially to zero, as shown inFig. 3b, during negative swing of the external sinusoidalcurrent. The total external current iext then goes belowthe critical value (icE1.0002) when min(iext)¼ 0.95 and thejunction voltage decays to zero or approaches a stableequilibrium state. The junction voltage starts growing againwhen the iext moves above the critical value during thepositive swing of the external current and iext becomes amaximum as max(iext)¼ 1.45. It clearly shows periodic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.72

2.5

3

ymax

Angular frequency, ω

Fig. 2 Bifurcation diagram of junction voltage maxima ymax withos for i1¼ 0.10: [bL¼ 2.6,bC¼ 0.707, i0¼ 1.2]

0 100 200 300 400 500 6002

0

2

4

y

time

0 100 200 300 400 500 6002

0

2

4

y

time

a

b

Fig. 3 Amplitude modulation of junction voltage [bL¼ 2.6,bC¼ 0.707, i0¼ 1.2 and os¼ 0.05]a i1¼ 0.10b Bursting oscillation, i1¼ 0.25. Forcing sinusoidal amplitude in dottedlines is superimposed on top of junction voltages in solid lines, criticalcurrent ic¼ 1.0002 in solid horizontal lines. Total external currentiext¼ i0+i1sin(os � t) is shown in the middle


bursting [21] for this moderately larger forcingamplitude (i1¼ 0.25) and low frequency (os¼ 0.05)although amplitude modulation still persists. The originof such bursting is currently being investigated and theresults will be reported elsewhere [22]. It has also beenchecked that the effect of amplitude modulationdisappears beyond the forcing frequency osE0.20 fori1¼ 0.10 although details are not produced here. Fordifferent sinusoidal current amplitudes, the limiting rangeof forcing frequency for amplitude modulation differs. Notethat the forcing amplitudes are moderately low(i1¼ 0.10, 0.25) in comparison to amplitudes (B3.0) ofjunction voltage.

3.2 Taming of chaosAs mentioned above, the forcing frequency is chosen as thecontrol parameter for controlling or taming of chaos in thesense of converting chaos to periodic oscillations in anRCLSJ. The junction chaos is converted to periodbehaviour (period-n, n is an integer) by an appropriateselection of the forcing frequency at a fixed forcingamplitude as shown in Fig. 4. For selected parameters[bC¼ 0.707, bL ¼ 2.6 and i0¼ 1.2], the junction oscillationbecomes period-1 (1 : 1) for forcing frequency osE0.698while period-2 (1 : 2;osE0.405) and period-3 (1 : 3;osE0.245) are found near its sub-harmonics as shown inupper row phase portraits of Figs. 4a–c. Phase lockingratios and forcing frequencies are given within parenthesis.The bar graphs in the lower row show different phaselocking of the external signal to the junction voltage. Thebar graph represent the maximum of junction voltage ymax

on top of which the forcing signals A(t)¼ i1 sin(ost) aresuperimposed. Higher periodic orbits, period-4 (3 : 4;osE0.55), period-5 (2 : 5;osE0.30) and period-6 (4 : 6;osE0.50) with other (m :n) phase locking ratios are shownin Fig. 5. It should be noted that the values of the forcingfrequency for the various cases of phase locking correspondto specific periodic windows in the bifurcation diagram inFig. 2. The time evolution of junction voltage andsinusoidal signal are shown in Figs. 5a–c for period-4,period-5 and period-6 respectively. Stroboscopic plotting isused to determine the phase locking ratios for higherperiodic orbits. In stroboscopic plots, number of piercingpoints on the Poincar!e surface of section by the trajectory ofthe junction oscillation are indicated for kT (k is an integer)time intervals. T is the time period of the sinusoidal forcing.The stroboscopic plots of zp(t+kT) against zp(t) are shownin Figs. 5d–f. zp denotes Poincar!e points of state variable zon the selected surface of section y¼ 1.8. Four piercingpoints are seen in Fig. 5d indicating a period-4 junctionoscillation locked to 3-period (k¼ 3, time interval¼3TE34.26) of the forcing signal. Similarly, other phaselocking 2 : 5 and 4 : 6 of the forcing signal to junctionoscillations of period-5 and period-6 are shown in Fig. 5eand Fig. 6f respectively.

3.3 Route to chaosThe autonomous junction is chaotic for the selectedparameters. Under the influence of the external periodicforcing, the junction shows chaotic behaviour but withmany intermediate periodic windows as shown in thebifurcation diagram of Fig. 2. The transition of periodicityto chaos with changing frequency, in these windows, mainlyfollows period doubling bifurcation. However, it showstorus breakdown near the largest periodic window at theextreme right. The junction oscillation is period-3 nearos¼ 0.245 (i1¼ 0.10) in Fig. 2, which shows a narrowperiod-3 window. This is again confirmed by three points in

455

0 1 2

0

2

4

z

y

z

y

z

y

500 550 600 650

2

3y m

ax

time

500 550 600 650

2

3

time

500 550 600 650

2

3

time

y max

y max

−2

0

2

4

−2

0

2

4

−2−1 0 1 2−1 0 1 2−1

a b c

d e f

Fig. 4 Control of junction chaos [bL¼ 2.6,bC¼ 0.707, i0¼ 1.2]Phase portraits of:a Period-1 for i1¼ 0.10 and os¼ 0.698 with (1 : 1) phase-locking in db Period-2 for i1¼ 0.05, os¼ 0.405 with (1 : 2) phase-locking in ec Period-3 for i1¼ 0.10, os¼ 0.245 with (1 : 3) phase-locking in fBar graphs in d–f represent junction voltage maxima ymax, which conforms 1 : 1, 1 : 2 and 1 : 3 phase locking respectively

the Poincar!e map of Fig. 6a. The surface of section ischosen at y¼ 1.8. The forcing frequency is then slowlydecreased. The period-6 oscillation appears for os¼ 0.2365via period doubling as indicated by six bands in thePoincar!e map in Fig. 6b. It may be noted that the twobands in the bottom right appears to be a single band, but acloser inspection shows them as two separate bands. Then itbecomes chaotic for os¼ 0.233 as shown in Fig. 6c. Perioddoubling during transition to chaos is also seen for forcing

2

0

2

4

y

2

0

2

4

y

500 575 6502

0

2

4

y

time

1 1.5 21

1.5

2

Zp(

t+3T

)

5 10 15

5

10

15

Zp(

t+2T

)

1.4 1.5 1.6

1.4

1.5

1.6

Zp(t)

Zp(

t+4T

)

a d

e

f

b

c

Fig. 5 Phase locking of junction oscillation [bC¼ 0.707, bL ¼ 2.6and i0¼ 1.2, i1¼ 0.1] and stroboscopic plots (d–f)a Period-4 at os¼ 0.55b Period-5 at os¼ 0.30c Period-6 at os¼ 0.50d Period-4 with 3 : 4 phase lockinge Period-5 with 2 : 5 phase lockingf Period-6 with 4 : 6 phase locking

456

frequencies near other periodic windows, but details are notpresented here. However, the situation is different forforcing frequencies near os¼ 0.63 when quasi-periodicity isobserved near the largest periodic window at the extreme

1.45 1.53 1.61.45

1.53

1.6

Zp(

n+1)

1.568 1.57 1.5711.568

1.57

1.571

1.46 1.51 1.561.45

1.51

1.56

Zp(

n+1)

1.563 1.567 1.5711.563

1.567

1.571

1.4 1.5 1.61.4

1.5

1.6

Zp(n)

Zp(

n+1)

1.55 1.558 1.5651.55

1.558

1.565

Zp(n)

a d

eb

c f

Fig. 6 Poincar!e map of forced junction [bC¼ 0.707, bL¼ 2.6 andi0¼ 1.2, i1¼ 0.1]Period-doubling near forcing frequenciesa os¼ 0.245b os¼ 0.2365c os¼ 0.233Torus breakdown near frequenciesd os¼ 0.63e os¼ 0.623f os¼ 0.6


Fig. 7 Effect of noise on control of chaos: white noise of noise strength D¼ 0.00005a Phase portraits of period-1 for i1¼ 0.10 and os¼ 0.698b Period-2 for i1¼ 0.05, os¼ 0.405c Period-3 for i1¼ 0.10, os¼ 0.245

right in Fig. 2. The torus spirally breaks into chaos as theforcing frequency is decreased from os¼ 0.63 to 0.6 asillustrated in Figs. 6d–f respectively.

3.4 Effect of noiseThe effect of noise on the control strategy is discussed here.We tried with additive white noise (mean zero and deltacorrelated) with maximum strength D¼ 0.00005 for threeperiodic cases period-1, period-2 and period-3. The noise isdefined as x(t) with strength D as discussed in detail in thenext section. It may be seen from Fig. 7 (cf. Figs. 4a–c) thatthe trajectories of the period orbits are smeared by noise forall three cases, however, they maintain their periodicity.Furthermore, we estimated the fluctuation of the trajectoriesfrom the stable periodic orbit. We took a Poinc!are surfaceof section on the trajectories for z¼ 0.8. The Poinc!are planewill return n-points (n is integer) for period-n orbit. Inthe presence of noise, the fluctuation in the number ofpiercing points on the Poinc!are plane decides the levelof deviation from the stable periodic trajectory. Wemeasured the standard deviation of these points as tomeasure the robustness of the periodic state to noise. Theresults of standard deviation with noise strength D are givenin Table 1 for comparison. The phase portrait of periodicorbits and their standard deviations indicate that thecontrolling technique is reasonably robust in the presenceof a moderate amount of noise.

Table 1: Fluctuation of periodic trajectory in presence ofnoise

Periodicity D, noisestrength

Standarddeviation

Period-1 1� 10� 5 0.008

2� 10� 5 0.0109

5� 10� 5 0.0109

Period-2 1� 10� 5 0.0435

2� 10� 5 0.0467

5� 10� 5 0.071

Period-3 1� 10� 5 0.023

2� 10� 5 0.035

5� 10� 5 0.047


4 Synchronisation of coupled junctions

Studies on synchronisation of chaotic systems are importantin science [20]. Synchronisation of superconducting multi-junction arrays is also necessary for reasonably high-poweroutput. A perfect synchronisation [15] in terms of both theamplitude and phase of two chaotic oscillators connected indrive-response mode is possible when the oscillators areidentical. The driver signal is transmitted to the responseoscillator for such synchronisation. However, synchronisa-tion is affected by the presence of noise since the transmittedsignal is distorted by channel noise. Recently Inaba andNitanai [23] proposed a pulsing technique for synchronisinga pair of Rayleigh circuits with idealised diodes where someof the timings of one circuit are only sent to the othercircuit. Even if the timing is offset by noise, it is no problemto recover it at the receiver end. The pulsing technique isfound to be more effective for diode-like switching circuits.The superconducting device indeed behaves as an idealswitch. The device develops a voltage (ON state) across thejunction when a dc bias current is applied across thejunction above a critical value. Otherwise, the junctionremains at zero voltage (OFF state) at a bias current belowthe critical value.

The proposed synchronisation scheme with negativepulsing technique is shown in Fig. 8a for two coupledRCL-shunted junctions. Two identical RCL-shunted junc-tions are coupled with a resistance R in the unidirectionaldrive-response mode. The RCL-shunted junctions arechaotic for appropriate choice of junction parameters andDC bias. Two similar state variables of the junctions aremonitored and their difference is taken as the error signal tocontrol an analogue switch (P). A negative current source isalways active to act as an external trigger to theanalogue switch. When the error signal is positive theanalogue switch in ON and the negative pulse isforced to both the junctions. Alternatively, for zero errorsignal, the negative pulse is withdrawn from the junctionsby switching OFF the analogue switch. The junctionsare automatically restored to the original chaotic regimeby the withdrawal of negative pulse. The junction voltagesare then found synchronized since they are now switchedON to a chaotic state at identical initial conditions.The choice of switching time is very important, whichis decided by the measure of the difference of twosynchronising junction voltages. This synchronisationapproach has been found successful for both unidirectionaland bi-directional coupling. However, the results of onlyunidirectional coupling are presented here. The governing

457

equations of the unidrectionally coupled junctions, indimensionless form, are

dxdt¼y

dydt¼ 1

bC½i0 þ IpðtÞ � gðyÞ � y � sinðxÞ � z�

dzdt¼ 1

bL½y � x�

dx0

dt¼y0

dy0

dt¼ 1

bC½i0 þ IpðtÞ � gðy0Þ � y0 � sinðx0Þ

� z0 þ aðy � y0Þ� þffiffiffiffiffiffi

2Dp

xðtÞdz0

dt¼ 1

bL½y0 � x0�

ð3Þ

where

IpðtÞ ¼i04Ic if t � t1i0 � IssoIc if t4t1i04Ic if t14t2

��

and X¼ [x,y, z]T and X0 ¼ [x0,y0, z0]T are the state vectors ofthe drive and response junctions. T denotes the transpose ofa matrix. The x(t) represents zero mean and delta correlatedwhite noise with strength D. The normalised junctioncapacitance bC, parallel inductance bL and junctionresonance frequency o0 are already defined in (1). Thecoupling coefficient a is defined by Rs/R. The pulse currentIp(t) is shown in Fig. 8b. The junctions are chaotic in theON state at selected values of biasing current i0 and other

Ip(τ)

ON ON

IC

ISS

JJ1 JJ2

R

ττ1 τ2 τ3

OFF OFF

EXT. TRIG.

Error Signal

P

a

b

Fig. 8 Synchronisation schemea Two RCL-shunted Josephson junctions coupled unidirectionallywith a series resistance R. The junction voltages y and y 0 are comparedto switch ON the pulse generator (P) while the external trigger is usedfor manual triggering at the start of the negative pulseb The synchronising negative pulse: ON state specifies self-oscillatingjunction, junction voltage becomes zero during OFF state when thepulse source (P) is negative for the duration t1 to t2 and the totaljunction current goes below the critical current Ic

458

parameters. The negative pulses are triggered ON to thejunctions at t¼ t1 when the junctions start moving towardsthe OFF state for a selected pulse height Iss such that i0� Iss

is less than Ic. The junction voltage reaches the OFF stateasymptotically owing to hysteresis in the I–V characteristicshown in Fig. 1b. When the negative pulses are withdrawnat t¼ t2, the junctions become ON again.

The uncoupled junctions are chaotic for parametersbL¼ 2.6, bC¼ 0.606, i0¼ 1.18. The coupling parameter isthen set at a¼ 0.25. The negative pulse of amplitude � Iss

starts at any arbitrary time t¼ t1 but it is withdrawn att¼ t2 when the difference between the junction voltagesDy¼ y� y0 goes below a threshold level e1 (Dy-e1B10� 15

within computational precision). The junctions becomesynchronous for t4 t2 but they may lose synchronisationafter certain elapse of time. The state variables y and y0 ofthe two oscillators are monitored to test the loss ofsynchronisation. When the difference signal or synchronisa-tion error y� y0 ¼Dy goes above another threshold levelDy4 e2, the negative pulse � Iss is again forced into thejunctions at t¼ t3. The junctions again become synchro-nised and this process can be repeated to achieveintermittent synchronisation of the coupled junctions. Inour numerical simulations, the pulse amplitude is chosen asIss¼ � 0.35 and the pulse is enforced at an arbitrary timet1¼ 50 and withdrawn at t2¼ 275 when the Dy approachesa threshold level chosen as e1¼ � 2.629� 10� 14. The timeevolution of y in solid line and synchronisation error Dy indotted lines and the pulse in dashed line are shown inFig. 9a. The dotted Dy line is drawn along the zerohorizontal line but shifted down the scale to separate it fromthe junction voltage plot. The pulse ‘dashed line’ is alsoscaled down to make it separate from the synchronisationerror signal (Dy) and the junction voltage (y). It is clear fromthe fluctuating Dy line in Fig. 9a that the loss ofsynchronisation starts near t2¼ 625. The synchronisationmanifold as a plot of y against y0 in Fig. 9b shows a thick

Fig. 9 Synchronisation of coupled junctiona Time evolution of y in solid line and Dy¼ y� y0 in dotted line,negative pulse in dashed lineb Synchronisation manifold of y and y0

[bC¼ 0.707, bL ¼ 2.6 and i0¼ 1.2, a¼ 0.25 and Ip¼ � 0.35]


451 line indicating the loss of perfect synchronisation. Toovercome this loss of synchronisation, the pulse is repeatedat t3¼ 600 and withdrawn again at t4¼ 800, when Dy

Fig. 10 Perfect synchronisation of coupled junctionsa Time evolution of y in solid line and Dy¼ y� y0 in dotted line,intermittent negative pulsing in dashed lineb Synchronisation manifold of y and y’[bC¼ 0.707, bL¼ 2.6 and i0¼ 1.2, a¼ 0.25 and Ip¼ � 0.35]

4

2

0

– 2

– 40 125 250 375 500 625 750 875 1000

y

y-y'

IP

time

0.08

0.06

0.04

–14 –12 –10 –8 –6

In(D)

a

b

Stan

dard

Dev

iatio

n (%

)

Fig. 11 Robustness to noisea Time evolution y in solid line and difference signal y� y 0 in dottedlineb Standard deviation (%) in synchronisation error plotted with naturallogarithm of noise strength D[bC¼ 0.707, bL¼ 2.6 and i0¼ 1.2, a¼ 0 and Ip¼ � 0.35]


approaches the assigned threshold as e1¼ 2.629� 10� 14.Perfect synchrony of the two junctions is then observedsince the synchronisation error Dy is always seen hori-zontal in Fig. 10a and corresponding synchronisationmanifold (y against y 0 plot) in Fig. 10b is represented bythe 451 line.

Robustness of our proposed synchronisation techniquein presence of noise is checked using zero mean anddelta correlated white noise in the driven junction. Thenoise term x(t) is introduced in the fourth equation of (3)with noise strength D. In the absence of noise (D¼ 0),a 4th-order Runge-Kutta algorithm is used for numericalsimulations with constant step size 0.01. In the presenceof noise, the Euler method is used for integration withstep size 0.001 to deal with the stochastic differentialequation. The junction parameters and timing of the pulset1¼ 50, t2¼ 600 and t3¼ 800 are chosen as earlier when wefind the perfect synchronisation in absence of noise. Inpresence of additive noise of strength D¼ 10� 4 the coupledsystem maintained reasonably small synchronisation errorDy with standard deviation 70.063 as shown in Fig. 11a.We also plotted the deviation of synchronisation frommeanzero synchronisation error by its standard deviation withnoise strength as ln(D) in Fig. 11b, which gives an ideaabout the influence of additive noise on synchronisation.The line plot is the fitted curve to the actual standarddeviation in circles.

5 Summary

The RCLSJ shows complex dynamical behaviour withexternal dc bias. The dynamics of the junction are found tobe sensitive to the amplitude and frequency of an externalperiodic forcing. The junction voltage has been found to beamplitude modulated in response to changes in ampli-tude of an external sinusoidal force. However, theamplitude modulation of the junction voltage is possiblein the low frequency range much below the naturalfrequency of the unforced junction. Control of chaos, inthe sense of converting chaos to periodicity (period-n), inRCSLJ can also be implemented using external periodicforcing using the driving frequency as a control parameter.Period-doubling and torus breakdown bifurcation route tochaos have been observed with changes in the drivingfrequency. Synchronisation of two coupled junctions hasbeen successfully observed by using negative pulsing.Robustness of the synchronisation technique is tested inthe presence of white noise.

6 Acknowledgments

The authors wish to thank the Director, Indian Institute ofChemical Biology, Calcutta, India for his permission topublish this paper. This work is partly supported by NASAGlen Research Centre, Ohio, USA under grant # NAG3-2150 and partly by the BRNS/DAE under grant # 2000/34/13/BRNS.

7 References

1 Likharev, K.K.: ‘Dynamics of Josephson junction and circuits’(Gordon and Breach, NY, 1986)

2 Kornev, V.K., and Arzumanov, A.V.: ‘Josephson-junction oscillationspectral linewidth for some phase-locked multijunction systems’, J.Phys. IV Colloq., 1998, 8, (3), pp. 279–282

3 Kanasugi, A., Morisue, M., Noguchi, H., Yamadaya, M., andFurukawa, H.: ‘Oscillation modes in a Josephson circuit and itsapplications to digital systems’, IEICE Trans. Electron., 1996, E79-C,(9), pp. 1206–1212

4 Spargo, J.W. (Ed.): ‘2002 Applied Superconductivity Conf. (SpecialIssue)’, IEEE Trans. Appl. Supercond., 2003, 13, Parts I–III

459

5 Whan, C.B., Lobb, C.J., and Forester, M.G.: ‘Effect of inductance inexternally shunted Josephson tunnel junctions’, J. Appl. Phys. Rev.,1995, 77, (1), pp. 382–389

6 Whan, C.B., and Lobb, C.J.: ‘Complex dynamical behaviours inRCL-shunted Josephson tunnel junction’, Phys. Rev. E, 1996, 53, (1),pp. 405–413

7 Cawthorne, A.B., Whan, C.B., and Lobb, C.J.: ‘Complex dynamics ofresistively and inductively shunted Josephson junction’, J. Appl. Phys.,1998, 84, (2), pp. 1126–1132

8 Dana, S.K., Sengupta, D.C., and Edoh, K.: ‘Chaotic dynamics inJosephson junction’, IEEE Trans. Circuit Syst. I, Fundam. TheoryAppl., 2001, 48, (8), pp. 990–996

9 Neumann, E., and Pikovsky, A.: ‘Slow-fast dynamics in Josepshonjunction’, Eur. Phys. J., 2003, B34, pp. 293–303

10 Ott, E., Grebogi, C., and Yorke, J.A.: ‘Controlling chaos’, Phys. Rev.Lett., 1990, 64, (11), pp. 1196–1199

11 Braiman, Y., and Goldhirsch, I.: ‘Taming of chaotic dynamics byweak periodic perturbation’, Phys. Rev. Lett., 1991, 66, pp. 2545–2548

12 Olsen, O.H., and Samuelsen, M.R.: ‘Control of chaos patterns in aJosephson junction model’, Phys. Lett. A, 2000, 266, pp. 123–133

13 Hsu, R., Su, H.-T., Chern, J.-L., and Chen, C.-C.: ‘Conditions tocontrol chaotic dynamics by weak periodic pereturbation’, Phys. Rev.Lett., 1997, 78, pp. 2936–2939

14 Chacon, R., Palmero, F., and Balibrea, F.: ‘Taming chaos in drivenJosephson Junction’, Int. J. Bifurcation Chaos Appl. Sci. Eng., 2001,11, (7), pp. 1897–1909

460

15 Pecora, L.M., Carroll, T.L., Johnson, G.A., and Mar, D.J.:‘Fundamentals of synchronization in chaotic systems, concepts andapplications’, Chaos, 1997, 7, (4), pp. 520–543

16 Haken, H.: ‘Synergetics: from pattern formation to pattern analysisand pattern recognition’, Int. J. Bifurcation Chaos Appl. Sci. Eng.,1994, 4, pp. 1069–1083

17 Pikovsky, A.S.: ‘Synchronization of the stochastic self-excitedoscillation phase using a periodic external signal’, Radiotekh. Electron.,1985, 35, (10), pp. 1970–1974

18 Allaria, E., Arecchi, F.T., di Garbo, A., and Meucci, R.: ‘Synchro-nization of homoclinic chaos’, Phys. Rev. Lett., 2001, 86, pp. 791–794

19 Aihara, K., Matsumoto, G., and Ikegaya, Y.: ‘Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator’,J. Theor. Biol., 1984, 109, pp. 249–269

20 Pikovsky, A., Rosenblum, M., and Kurths, J.: ‘Synchronization, auniversal concept in nonlinear science’ (Cambridge University Press,Cambridge, England, 2001)

21 Meucci, R., di Garbo, A., Allaria, E., and Arecchi, F.T.: ‘Autono-mous bursting in a homoclinic system’, Phys. Rev. Lett., 2002, 88,pp. 144101–144104

22 Dana, S.K., Sengupta, D.C., and Hu, C.-K.: ‘Spiking and bursting inJosephson junction’, IEEE Trans. Circuits Syst. II, Analog Digit.Signal Process., 53, (10), in press

23 Inaba, N., and Nitanai, T.: ‘Synchronization of chaos in a pair offorced Rayleigh circuits with diodes’, IEEE Trans. Circuits Syst. I,Fundam. Theory Appl., 1999, 46, (5), pp. 645–647


Documents

Taming of chaos and synchronisation in RCL-shunted Josephson junctions by external forcing