Intermittent Lag and Complete Synchronization in Two Coupled

Murali-Lakshmanan-Chua’s circuits

In this presentation we demonstrate experimentally intermittent and complete synchronizationin two coupled Murali-Lakshmanan-Chua (MLC ) circuit. Here the circuits are coupled in unidi-rectional drive response mode. Both the drive and response are kept in double scroll chaotic state.The transition to intermittent lag synchronization precedes that of complete synchronization onincreasing the coupling parameter. Numerical simulations also are provided.

Synchronization is a process where in two or morechaotic systems (either equivalent or non-equivalent) ad-just a given property of their motion to a common be-haviour, due to coupling or forcing. This ranges fromcomplete agreement of trajectories to locking of phases.Several regimes of chaotic synchronization like CompleteSynchronization (CS), Intermittent Lag synchronization(ILSA), Lag Synchronization (LS), Phase Synchroniza-tion (PS), etc. can be recognized, depending upon thenature of the coupling. The coincidence of phase and am-plitude of two identical chaotic systems in the presenceof strong interaction is termed Complete Synchroniza-tion. Intermittent Lag Synchronization has been charac-terized in terms of the existence of a set of lag times τn(n= 0,1,2,3,....) such that the system verifies x1(t − τn)≈ x2(t)for a given n. This occurs due to local instabil-ities and leads to occasional bursts of nonsynchronousbehaviour.

In this study, we use two Murali-Lakshmanan-Chua(MLC) circuits [1] coupled using resistive coupling in adrive - response mode.MLC circuit is a simple two dimen-sional series LCR circuit with Chua’s diode as the non-linear element. The circuit diagram is shown in Fig. 1.

C

LR

iL

f(t)RS

NR

iC

−

+

C

L

f(t)

RS

NR

v’v

RC

i’Ci’L+ +

Buffer

R

FIG. 1: Schematic circuit realization of two identical MLCcircuits with unidirectionally coupling resistor RC .

The equations of the circuit are

drive:

Cdv

dt= iL − g(v), (1a)

LdiL

dt= −RiL − RsiL − v + (f sin(Ωt). (1b)

response:

Cdv′

dt= i′

L− g(v′) + ǫ(v − v′), (1c)

Ldi′

L

dt= −Ri′

L− Rsi

′

L− v′ + (f sin(Ωt). (1d)

When the strength of coupling ǫ =0.362 we find thesystem exhibits an ILS synchronization state. Increas-ing the coupling strength ǫ to 0.048 results in the on-set of complete synchronization. These are shown bythe x1(t)−x2(t) phase portraits obtained experimentallyand numerically. In order to verify the extent of syn-

1.5

0

-1.51.50-1.5

x 2

x1

1.5

0

-1.51.50-1.5

x 2

x1

(a) (b)

FIG. 2: Numerical phase portrait in the (x1 − x2) plane.

(a) (b)

FIG. 3: Experimental phase portrait in the (vc1 − vc2) plane.

chronisation between the drive and response circuits thesimilarity function introduced in [2] and later on usedby many authors is employed. The intermittent synchro-

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

x(t)

- x

(t+

tau)

time

’ils_mlc.dat’ev 500 u 1 0.6

0.4

0.2

0 0 0.5 1 1.5 2 2.5

Sim

ilari

ty fu

nctio

n

tau

(a) (b)

FIG. 4: Similarity function versus lag time plot.

nisation of x(t)− x(t + τ) vs timet is shown in Fig. 4(a).

2

The similarity function for both CS and ILS are shownin Fig. 4(b). For ILS this has a small non-zero minimumat τ =0.11 sec.

[1] K. Murali,M. Lakshmanan, and L. O. Chua, IEEE Trans.Circuits Syst.I, 41, 462-463,1994a.

[2] , M. G. Rosenblum, S. Pikovsky and J. Kurths, Phys. RevLett.78, 4193(1997).

[3] P. K. Roy, S. Chakraborty and S. K. Dhana, Chaos,13(1),342(2003).