Synchronizing time delay systems using variable delay in coupling

Chaos, Solitons & Fractals 44 (2011) 1035–1042

Contents lists available at SciVerse ScienceDirect

Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Synchronizing time delay systems using variable delay in coupling G. Ambika a,⇑, R.E. Amritkar b a b

Indian Institute of Science Education and Research, Pune 411 021, India Physical Research Laboratory, Ahmedabad 380 009, India

a r t i c l e

i n f o

Article history: Received 4 May 2011 Accepted 29 August 2011 Available online 25 September 2011

a b s t r a c t We present a mechanism for synchronizing time delay systems using one way coupling with a variable delay in coupling that is reset at finite intervals. We present the analysis of the error dynamics that helps to isolate regions of stability of the synchronized state in the parameter space of interest for single and multiple delays. We supplement this by numerical simulations in a standard time delay system like Mackey Glass system. This method has the advantage that it can be adjusted to be delay or anticipatory in synchronization with a time which is independent of the system delay. We demonstrate the use of this method in communication using the bi channel scheme. We show that since the synchronizing channel carries information from transmitter only at intervals of reset time, it is not susceptible to an easy reconstruction. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Time delay systems occur naturally in many real situations where the delay arises due to finite signal transmission times, switching times, finite lengths of feed back loops or memory effects among the coupled units of the systems. In such systems transmission delay is known to play a crucial role in the collective dynamics since coupling with time delay can give rise to many interesting emergent phenomena like synchronization [1] and amplitude death [2,3]. Such systems with inherent delays can be considered as prototype models for infinite dimensional systems described by delay differential equations. Experimental studies in such systems flourished recently due to the ease of realizing them using electronic circuits or optical systems [4–6]. The most well studied examples of time delay systems are Mackey Glass and Ikeda systems [7,8]. In both, the dynamics of the system depends crucially on the delay time and as it increases the system displays a range of periodic, chaotic and hyper chaotic behavior. The complexity of the dynamics in these cases makes them ideal for many applications like secure communication and such systems ⇑ Corresponding author. E-mail addresses: [email protected] (G. Ambika), amritkar@ prl.res.in (R.E. Amritkar). 0960-0779/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2011.08.007

can serve as realistic models for coordinated brain activity, pattern recognition etc. [9]. Therefore achieving and studying synchronization in such systems is of high practical importance and relevance. In spite of the increase in the complexity of the dynamics due to the large number of positive lyapunov exponents, it has been established that such systems can be synchronized by a scalar transmitted signal [10–13]. Moreover, in real systems there could be several information exchanges, two or more feedbacks etc. Hence systems with multiple delays serve as more realistic models for interacting complex systems and such multi delay systems are often realized in electro optical semiconductor lasers [14]. Recently, transmission delay induced synchronization transitions have been studied in neuronal networks, where fine tuned transmission delays are found vital for assuring optimally synchronized excitatory fronts [15,16]. Moreover, the magnitude of delay is also found crucial in such systems, since short and long delays can lead to synchronized behaviour while intermediate delays can result in anti phase synchronization and clustering [17]. In physical systems like lasers and related optical devices, delay is identified as an essential ingredient of the dynamics [18]. Therefore adjusting delay or varying delay can be important in controlling the dynamics or switching between different types of dynamical behaviour in such systems.

1036

G. Ambika, R.E. Amritkar / Chaos, Solitons & Fractals 44 (2011) 1035–1042

The general method so far employed for synchronizing time delay systems involve coupling with or with out an added delay or anticipation time and this can lead to complete, lag or anticipatory synchronization. In such cases, analytic estimates of the stability regions is obtained from the frame work of Krasovskii–Lyapunov theory [19,20]. By varying the delay sinusoidally, the existence of transition between these types of synchronized states is reported [21]. In these methods the lag or anticipation in the synchronization depend on the delay in the systems that are coupled. The fact that low dimensional chaotic systems can be easily unmasked by reconstructing the dynamics from time series available from the channel by an intruder, makes the use of hyper chaotic systems derived from time delay systems a better option for enhanced security [22]. There are several methods of implementation reported in this context using time delayed optical systems and electronic circuits [23–28,30,29,31,32]. However, later studies have indicated ways of reconstruction of such systems from time series and extraction of messages masked by hyper chaotic carriers [33–38]. In general, such cracking schemes, are aimed at identifying the time delay parameter from the cipher text in the channel. For enhanced security, several variations like multiple delays, variation and modulation of delays, harmonics and multi channels have been reported [39–42]. In an earlier paper we have introduced a method of synchronizing nonlinear systems in which the delay and anticipation in the coupling vary in the same way as the system dynamics [43]. This makes the value of the coupling function remain constant for a chosen reset interval, at the end of which they are reset to the value at that time. This makes the error dynamics discrete and can be analysed in detail analytically. We have applied this to standard low dimensional systems like Rössler and Lorenz and carried out the stability theory and Lyapunov exponents analysis for such a coupling scheme. Here we further develop the method to apply to hyper chaotic time delay systems and show that such systems can also be considered in the general frame work of the stability theory developed by defining the Lyapunov exponent suitably. Moreover in this case, the delay or anticipation of the synchronized state is independent of the system delay. In this context, we note that several studies on variable and multiple time delays in synchronized time delay systems have been reported[45] with applications to communication [46–49]. But in our scheme, unlike these methods, the variability is in the delay or anticipation time in the coupling function and not in the feedback delay in the system dynamics. As such it provides a simple method for adjusting the delay or anticipation from outside controls for any specific application. This is especially relevant for applications like secure communication where the delay and anticipation time in the coupling can serve as additional keys that are independent of system dynamics. We suggest a bi-channel scheme for its implementation where the synchronization channel carries information about the transmitting dynamics intermittently, separated by intervals of reset time and hence is not easily susceptible to reconstruction by an intruder. Hence at large reset times,

the reported methods of reconstructing the dynamics by first computing the delay time of the system dynamics is not possible.The message channel being separate can be made complex enough. By a suitable choice of the carrier as a combination with multiple delays from the transmitter for effective masking, we show how the message can be retrieved truthfully at the receiver once it synchronizes with the transmitter. The method has the advantage that for small anticipatory times, the reset time can be maximized with large enough system delay. This makes the method highly cost effective with more complex output for enhanced security. Such optimization is possible due to the flexibility among the four different time scales in our method viz. delay in the transmitting system , anticipation in the receiving system, system delay and reset times.We present the stability analysis as applied to time delay systems and supplement this numerically by computing the stability regions for synchronization in the parameter space for a standard time delay system viz. Mackey Glass (MG) system. We also present the study of two MG systems with multiple delays using the present scheme. The applicability for communication is illustrated by a bi-channel scheme for an analogue signal. We establish the enhanced security of this method by explicitly applying the existing methods of cracking applicable to such time delay systems. We further explore the possibility of generalised synchronization with the type of coupling presented here, when the delay in the two systems are not equal. Using the auxiliary system method to detect generalised synchronization, we find this is possible for a wide range of mismatch in the system delays. The threshold value for coupling strength remains almost constant for the whole range. We note that the studies on generalised synchronization reported so far for time delay systems are with no delay or anticipation in the coupling [51–53]. Our results indicate that this can be achieved in time delay systems with mismatch in the delay parameter even when the coupling involves delay or anticipatory times. 2. Synchronization of time delay systems with variable delay in coupling In this section, we describe the method of coupling two time delay systems with a variable delay in coupling and present the linear stability analysis as applied to time delay systems. 2.1. Systems with single delay We consider systems with a single delay in its dynamics whose dynamics follows a time delay equation

x_ ¼ f ðx; xss Þ

ð1Þ

where xss ¼ xðt  ss Þ; ss being the time delay in the system. Two such systems are coupled in the drive response mode with a linear difference coupling where the drive variable is delayed by s1 and the response by s2. These delays vary in time from their initial values with the same time scale as the system within an interval called the reset time sr. At

1037

G. Ambika, R.E. Amritkar / Chaos, Solitons & Fractals 44 (2011) 1035–1042

the end of this interval, each delay is reset to its initial value and the dynamical variable in the coupling set to its value at that time. This makes the coupling function constant over sr and hence the information from the driver is required only at intervals of sr. The dynamics of the two coupled systems thus evolves under four different time scales in addition to the system dynamics, viz the delay time s1, the anticipatory time s2, the reset time sr, the system delay time ss. We prescribe the following dynamics for the coupled systems under this scheme.

x_ ¼ f ðx; xss Þ y_ ¼ f ðy; yss Þ þ 

1 X ðxt1  yt2 Þvðmsr ;ðmþ1Þsr Þ

ð2Þ

m¼0

where xt1 ¼ xðt  t1 Þ; yt2 ¼ yðt  t 2 Þ; sr is the resetting time and vðt0 ;t00 Þ is an indicator function such that vðt0 ;t00 Þ ¼ 1 for t 0 6 t 6 t00 and zero otherwise. Both the delays t1 and t2 depend on time and we choose this dependence as a simple linear increase given by

t i ¼ si þ t  msr ;

determined by matching solutions across the reset points. We have developed a detailed stability analysis for systems coupled using this method [43]. We can use the same analysis here also with the only difference that in this context k depends on the system time delay ss and has to be evaluated for each case using Eq. (6). This transcendental equation can be solved to show that for j + m > 0, k is positive and as ss increases k decreases, while for j + m < 0, k is negative and as ss increases k increases. This is shown in Fig. 1. If we consider the regime 0 6 s2 6 sr. Then the upper limit of stability in the parameter plane s2/sr  a as derived in [43] has a peak at s2p. For s2 6 s2p, the upper limit of stability is given by

au ¼

eksr þ 1  eksr  1

ð8Þ

2ekðsr s2 Þ

while for larger values of s2(s2p 6 s2 6 sr) it is given by

au ¼

eksr 1  eks2

ð9Þ

Moreover, in the present context, the maximum reset time for a given s2 can be obtained as

i ¼ 1; 2:

This takes care of the variation in the delay times mentioned above. The synchronization manifold for the coupled systems (2) is defined by y(t  s2) = x(t  s1) or y(t) = x(t  s1 + s2) . We show in the next section that under this type of coupling, it is possible to have isochronal, delay or anticipatory synchronization between the systems.

ksrmax ¼  lnð1  eks2 Þ:

ð10Þ

Using Eqs. (6) and (10), srmaxfor given s2 and for different ss can be calculated. The cases where s2 > sr are discussed in the reference [43] and the results are not reproduced here for the sake of brevity.

2.2. Linear stability analysis

2.3. Systems with multiple delays

We now present the linear stability analysis for the synchronized state. For this the transverse system can be defined by D ¼ y  xs1 s2 . Its dynamics in linear approximation can be derived from Eq. (2) as

We further extend this scheme to systems with multiple delays that are more realistic models in many applications. They occur in many situations, an example for such multi-delay systems being the semiconductor laser with multiple opto electronic feed backs, where additional feed back is often required to stabilise the output [50]. Hence such systems are of practical importance especially in the context of communication where the higher complexity can lead to better message security. The dynamics for two such systems coupled using the present scheme is

D_ ¼ fx0 ðx; xss ÞD þ fx0ss ðx; xss ÞDss  

1 X

vðmsr ;ðmþ1Þsr Þ Dm

ð3Þ

m¼0

where Dm = D(t  t2) = D(ms  s2) is a constant in any reset interval. s1 can eliminated by shifting the time scale of drive system and redefining s2. We approximate fx0 in the above equation by an effective time average value j and fx0ss by m. This type of approximation was used in our earlier work [43,44] and is found to give reasonably good estimate of stability. Then Eq. (3) gives,

D_ ¼ jD þ mDss  

1 X

vðmsr ;ðmþ1Þsr Þ Dm

2

ð4Þ

1.5

m¼0

1

D_ ¼ jD þ mDss

ð5Þ

we get

λ

Assuming a solution D = ekt for the equation,

0.5 0

kss

k ¼ j þ me

ð6Þ

Then the solution of Eq. (4) in any reset interval , ms 6 t < (m + 1)s, is

D ¼ aDm þ C m ekt

ð7Þ

where a = /k is the normalized dimensionless coupling constant, and Cm is an integration constant,which can be

-0.5 -1 2

2.5

3

3.5

4

4.5

5

5.5

6

τs Fig. 1. The variation in k as the system delay increases for the two cases, j + m > 0 (upper curve) j + m < 0 (lower curve).

1038 N X

x_ ¼ ax þ

ai f ðx; xssi Þ

i¼0

y_ ¼

N X

ai f ðy; yss Þ þ  i

i¼0

ð11Þ

1 X ðxt1  yt2 Þvðmsr ;ðmþ1Þsr Þ m¼0

A stability analysis similar to that discussed for single delay systems in Section 2.2, will furnish the stability criteria with k replaced by

k¼jþ

N X

ai mi ekssi

ð12Þ

i¼0

Here j is the average value of average value of fx0ss ðx; xssi Þ.

PN

0 i¼0 ai fx ðx; xssi Þ

and mi is the

x_ ¼ x þ

i

2.4. Generalised synchronization with variable delay in coupling

x_ ¼ f ðx; xss1 Þ y_ ¼ f ðy; yss Þ þ  2

1 X ðxt1  yt2 Þvðmsr ;ðmþ1Þsr Þ m¼0

ð13Þ

1 X z_ ¼ f ðz; zss2 Þ þ  ðxt1  zt2 Þvðmsr ;ðmþ1Þsr Þ m¼0

Here the stability criteria for complete synchronization between y and z serves as that for generalised synchronization between x and y. Following the same local stability analysis given in Section 2.2, we can get the threshold for stability in this case also. The method of synchronization of time delay systems coupled with variable delay and reset, presented in the present section is supported by numerical simulations using a standard time delay system, Mackey Glass system, in the next section. 3. Coupled Mackey Glass systems with variable delay in coupling The Mackey Glass system, first introduced as a model for blood generation in patients with leukaemia, is well studied as a model exhibiting hyper chaos [11]. The synchronization in two such coupled systems has been reported for various types of synchronization[53]. It has been applied extensively in the context of communications, especially with its circuit equivalents [5]. The synchronization regimes of two such systems with multiple delays have been reported recently [54,55]. In this section, we apply the scheme of variable delay described in the last section to two coupled Mackey Glass systems. The dynamics of a single system MG system is

ð14Þ

2xðt  ss Þ 1 þ xðt  ss Þ10

y_ ¼ y þ

Now we consider the effect of parameter mismatches in the synchronization scheme presented. A simple case would be when the system delays in x and y systems are different. We analyze the condition for generalised synchronization in such a case by using the auxiliary system method. For this we consider a z system identical with the y system as driven by x system using the same coupling method. Then generalised synchronization between x and y is established when y and z are in complete synchronization. The dynamics is then given by

bxðt  ss Þ 1 þ xðt  ss Þc

As reported in [11], for a = 1, b = 2 and c = 10, as the delay time ss is varied, the system has a fixed point attractor for ss < 0.471, a limit cycle for 0.417 < ss < 1.33, a period doubling sequence in the range ss = [1.33, 1.68] and chaos for ss > 1.68. Thereafter the number of positive Lyapunov exponents increases linearly with ss, and the system shows hyper chaotic behavior. In our simulations here, we choose a similar set of parameter values with ss between 2.5 and 5.0 corresponding to the hyper chaotic region and couple two such systems using our scheme given in Eq. (2) as

2yðt  ss Þ 10

1 þ yðt  ss Þ

þ

1 X ðxt1  yt2 Þvðmsr ;ðmþ1Þsr Þ m¼0

ð15Þ For a typical choice of parameters, ss = 2.5, sr = 0.5, s1 = .02, s2 = 0.1, and coupling strength,  = 2.0, we present the variation in the coupling function according to our scheme and the corresponding error function in Fig. 2. It is clear that the coupling function is not changing within each reset interval and the synchronization error approaches zero. We illustrate delay and anticipatory synchronization in these systems by plotting the time series of the systems for parameters chosen as ss = 2.5, sr = 0.5 and  = 2.0 (Fig. 3). By numerically integrating the coupled systems in Eq. (15), we calculate the asymptotic correlation coefficient between x1(t) and y1(t) shifted by the effective s2 = js2  s1j using the equation,

hy ðtÞx1 ðt þ s2 Þi C ¼ q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 ffi x1 ðtÞ y1 ðtÞ

ð16Þ

The region of stability of the synchronized state is isolated as the region where C = 0.999 and boundaries of stability fixed when C goes below this value.The stability region

0.25 0.2

Coupling function, Error

x_ ¼

G. Ambika, R.E. Amritkar / Chaos, Solitons & Fractals 44 (2011) 1035–1042

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 10

15

20

25

30

35

40

45

50

Time Fig. 2. The nature of variation of the coupling function having variable delay with reset (full line) and the synchronization error (dashed curve) for two coupled Mackey Glass systems. Here ss = 2.5, sr = 0.5, s1 = .02, s2 = 0.1 and  = 2.0.

1039

G. Ambika, R.E. Amritkar / Chaos, Solitons & Fractals 44 (2011) 1035–1042

x1(t), y1(t)

(a) 1 0.6

x1(t), y1(t)

0.2

(b)

1 0.6 0.2 1940

1950

1960

1970

1980

1990

2000

Time Fig. 3. The time series of the coupled MG systems with single delay for ss = 2.5, sr = 0.2, (a) Delay synchronization with s1 = 0.02 and s2 = 0.42 so that y(t) lags behind x(t) by 0.4 units and (b) Anticipatory synchronization in which y(t) anticipates x(t) by 0.4 units.

thus obtained in the parameter plane s2   is plotted in Fig. 4. The region below the upper curve and above the lower curve corresponds to total synchronization with correlation coefficient 1.0 . The dotted and dashed curves for the upper boundary correspond to the curves obtained by fitting Eqs. (8) and (9) obtained from the stability analysis, while the points correspond to the values obtained by direct numerical simulations using the correlation coefficient. The agreement is found to be very good. We extend our numerical computations to the case of coupled MG systems with multiple system delays. The dynamics of two such systems with four delays is given by

x_ ¼ x þ y_ ¼ y þ

x_ ¼ x þ y_ ¼ y þ z_ ¼ z þ

2xðt  ss1 Þ 1 þ xðt  ss1 Þ10 2yðt  ss2 Þ 10

1 þ yðt  ss2 Þ 2zðt  ss2 Þ 1 þ zðt  ss2 Þ

þ

1 X ðxt1  yt2 Þvðmsr ;ðmþ1Þsr Þ m¼0

1 X þ ðxt1  zt2 Þvðmsr ;ðmþ1Þsr Þ 10 m¼0

ð18Þ

4 X

bi xðt  ssi Þ

i¼1

1 þ xðt  ssi Þ10

4 X

bi yðt  ssi Þ

i¼1

1 þ yðt  ssi Þ10

þ

1 X

ðxt1  yt2 Þvðmsr ;ðmþ1Þsr Þ

m¼0

ð17Þ Using ðbi ; ssi Þ values as (2.0, 2.5), (0.2, 2.2), (0.05, 2.3) and (0.1, 2.4), the time series of the system showing delay

18 16 14 12

ε

and anticipation in the synchronized states is plotted in Fig. 5. We consider an auxiliary system z and make couplings as given below to study generalised synchronization.

10 8 6 4

We set all the parameters as in the above cases but with a mismatch of 0.5 in system delays by taking ss1 ¼ 2:5 and ss2 ¼ 3:0. The system in Eq. (18) numerically. The time series for x(t) and y(t  s2) thus obtained are plotted in Fig. 6a, while that for y(t) and z(t) are shown in Fig. 6b. The complete synchronization between y and z systems confirms generalised synchronization between x and y systems. This is further established in Fig. 6c and d where the synchronization manifold is plotted in each case. To study the stability of generalised synchronization for a range of mismatch in delay parameters, we study the synchronization threshold for the coupling strength as indicated by that for complete synchronization for y and z systems by keeping ss1 ¼ 2:5 and varying ss2 . We find the threshold up to which generalised synchronization is stable, remains almost constant over a wide range for mismatch. Fig. 7 shows the results of this study for two different sets of parameters. In the next section we illustrate how the present method can be applied in secure communication by taking the case of MG systems with multiple delays.

2 0

0

0.1

0.2

τ2

0.3

0.4

0.5

Fig. 4. The limits of stability of the synchronized state for two coupled Mackey Glass systems in the parameter plane s2  . The dotted and dashed curves for the upper boundary correspond to the curves obtained by fitting Eqs. (8) and (9) obtained from the stability analysis, while the points correspond to the values obtained by direct numerical simulations using the correlation index. Here ss = 2.5, sr = 0.2 and s1 = .02.

4. Application to secure communication In this section we propose a scheme for secure communication using hyper chaotic systems coupled under the method of synchronization discussed in this paper. We also show how the method can lead to enhanced security. We follow the chaotic masking method as applied to a bi-channel scheme. The transmitter and receiver dynamics in our

x1(t), y1(t)

G. Ambika, R.E. Amritkar / Chaos, Solitons & Fractals 44 (2011) 1035–1042

1.2

x1(t), y1(t)

1040

1.2

(a)

0.8 0.4 0

(b)

0.8 0.4 0 1940

1950

1960

1970

1980

1990

2000

Time Fig. 5. The time series of two coupled MG systems with 4 delays showing (a) delay synchronization with sr = 0.2, s2 = .02 and s1 = .62. and (b) anticipatory synchronization for sr = 0.2, s1 = .02 and s2 = .42.

2

(a)

1

y(t), z(t)

x(t), y(t-τ2)

2 1.5

(d)

0.5 0

(b)

1.5 1 0.5 0

0

10

20

30

40

50

0

10

20

1.4 1.2 1 0.8 0.6 0.4 0.2

(c)

0.2 0.4 0.6 0.8

30

40

50

Time

z(t)

y(t-τ2)

Time

1

1.2 1.4

1.4 1.2 1 0.8 0.6 0.4 0.2

(d)

0.2 0.4 0.6 0.8

x(t)

1

1.2 1.4

y(t)

Fig. 6. Generalised synchronization in coupled MG systems. (a) time series x(t) and y(t  s2), (b) time series y(t) and z(t), (c) the synchronization manifold for x and y systems and (d) that for y and z systems. Here ss1 ¼ 2:5; ss2 ¼ 3:0; s1 ¼ 0:02; s2 ¼ 0:1; sr ¼ 0:2 and the coupling strength =3.0.

scheme are given by Eq. (17). Here we take ðbi ; ssi Þ as (2.0, 5.0), (0.2, 4.0), (0.05, 3.0), (0.1, 2.0). The synchronizing signal S(t) is the intermittent values of x(t- s1) sent at intervals of the reset time sr. The message to be transmitted is assumed to be a sine signal given by

mðtÞ ¼ 0:01sinðxtÞ

ð19Þ

This message is encrypted using the method given in reference [54] for bi-channel scheme. The carrier is generated by combining the output of the transmitter at eleven different delay times using a nonlinear function as given below.

CWðtÞ ¼

10 X

ai xðt  ssi Þ

i¼0

1 þ xðt  ssi Þ10

ð20Þ

Here the values of ðai ; ssi Þ used are (4.0, 2.0), (5.0, 1.9), (6.0, 1.8), (7.0, 1.7), (8.0, 1.6), (9.0, 1.5), (10.0, 1.4), (11.0, 1.3),(12.0, 1.2), (13.0, 1.1), (14.0, 1.0). This carrier is used to mask the message and the cipher text C(t) thus obtained is

CðtÞ ¼ mðtÞ þ

10 X

ai xðt  ssi Þ

i¼0

1 þ xðt  ssi Þ10

ð21Þ

This cipher text is communicated to the receiver along one channel, while the synchronizing signal S(t) is transmitted through another channel. Once the systems at both ends are synchronized, the signal is recovered at the receiving end as plain text.

PTðtÞ ¼ CðtÞ 

10 X

ai yðt  ssi Þ

i¼0

1 þ yðt  ssi Þ10

ð22Þ

We display graphs of the cipher text C(t) and recovered plain text PT(t) to illustrate the procedure in Fig. 8. It is seen that after the transients, the message can be recovered truthfully. To illustrate the enhanced security of the method, we note that S(t), derived using the present scheme, has information about the dynamics of the transmitter only at intervals of rest time sr and as such can not be used to derive the dynamics using conventional methods of cracking reported so far. We show this explicitly by considering the method used in [33,34], where the time

1041

G. Ambika, R.E. Amritkar / Chaos, Solitons & Fractals 44 (2011) 1035–1042

500

20

400

N( τ)

εc

15

10

300

200

5 100

0 0

1

2

3

4

5

0

6

0

δ τs

5

10

15

20

25

τ

Fig. 7. The threshold for coupling strength up to which stable generalised synchronization is observed in coupled MG systems. This is obtained numerically using correlation coefficient as an index for complete synchronization in y and z systems. The upper curve corresponds to values ss1 ¼ 2:5s2 ¼ 0:04 and sr = 0.2 while the lower curve is with ss1 ¼ 0:5s2 ¼ 0:2 and sr = 0.5. The mismatch in delay parameters dss obtained by varying ss2 is plotted in the x-axis while the threshold for coupling strength c is along the y axis.

Fig. 9. The method reconstruction in [34] applied to the synchronizing signal to extract the system delay ss. It is clear that there is no single absolute minimum.

80 70 60

delay in the dynamics of the systems is to be extracted first to reconstruct the transmitter system. For this the number of pairs of extrema separated in time by different s in the time series obtained from the channel is taken and the absolute minimum gives a clue to the time delay in the system ss. Once this is obtained the functions in the transmitter dynamics can be derived [34]. For applying this to our method, we reconstruct the time series from the synchronizing channel by removing the reset intervals and use this collapsed time series to take the count of extrema for different separations. This is given in Fig. 9. It is clear that as the minima happen at a large number of points due to finite reset times, an estimate of the time delay ss cannot be obtained. We also use the same technique for checking possibility of reconstruction from the cipher text. Here for the same combination of eleven nonlinear functions used above for the cipher text in Eq. (21), we find that the information about the system delay is sufficiently hidden as is clear from Fig. 10. Moreover, the eleven different sets of parameters (bi ; ssi Þ used in constructing the cipher

N( τ)

50 40 30 20 10 0

0

5

10

τ

15

20

25

Fig. 10. The method of reconstruction in [34] applied to the cipher text. The complexity of the cipher text gives several minima and hence it is difficult to get an indication of ss.

text gives added security, by providing extra keys. In addition to this, in our method the two time delays s1 and s2 serve as additional basic keys.

80

(a)

CT(t)

60 40 20

PT(t)

0 0.03

(b)

0.01 -0.01 -0.03 60

80

100

120

140

160

180

200

Time Fig. 8. (a) The cipher text CT(t) given in Eq. (21) generated by masking the signal m(t) using carrier wave given in Eq. (20), (b) the plain text PT(t) or recovered signal using Eq. (22).

1042

G. Ambika, R.E. Amritkar / Chaos, Solitons & Fractals 44 (2011) 1035–1042

5. Conclusion We present a coupling scheme for synchronizing two one way coupled time delay systems where the delay in coupling varies with system dynamics within intervals of the reset time. In this method, the delay or anticipatory time of the synchronization can be independent of the system delay and hence can be adjusted independently. The local stability analysis developed for such couplings gives a good estimate of the region of stable synchronization in the parameter plane. The method is applied to a standard time delay system like Mackey Glass system with single and multiple delays. We further establish the existence of generalised synchronization using the auxiliary system approach in this method of coupling with varying delay and anticipation times. We suggest the possibility of applying this method in communication with enhanced security. For this, we propose a bi-channel method for implementation of the same. Our method has the advantage that synchronizing channel carries only intermittent information from the driver. We show that this enhances the security of transmission as extracting system delay from this channel is not possible by the methods reported so far. The cipher text channel can be made complex resulting in extra keys. We illustrate this by using a carrier wave constructed with eleven different delays from a four delay Mackey Glass system as driver . Such bi-channel schemes have been reported earlier also. But our scheme promises extra security in the synchronizing channel due to intermittent transmission and the delay and anticipation in synchronization in our method function as additional keys that are independent of system delay. The stability of generalised synchronization over a wide range of parameter mismatch can also be made use of for such applications. References [1] Oguchi T, Nijmeijer H, Yamamoto T. Chaos 2008;18:037108. [2] Ramana Reddy DV, Sen A, Johnston GL. Phys. Rev. Lett. 1998;80:5109. [3] Prasad A. Phys. Rev. E 2005;72:056204. [4] Kim MY, Sramak C, Uchida A, Roy R. Phys. Rev. 2004;E74:016211. [5] Sano S, Ushida A, Yoshimori S, Roy R. Phys. Rev. E 2007;75:016207. [6] Kittel A, Parisi J, Pyragas K. Physica D 1998;112:459. [7] Mackey MC, Glass L. Science 1997;197:287. [8] Ikeda K, Daido H, Akimoto O. Phys. Rev. Lett. 1980;45:709; IKeda K, Matsumoto K. Physica D 1987;29:223. [9] Rodriguez E, George N, Lachaux J-P, Martinerie J, Renault B, Varela FJ. Nature (London) 1999;397:430. [10] Peng JH, Ding EJ, Ding M, Yang W. Phys. Rev. Lett. 1996;76:904.

[11] [12] [13] [14]

[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55]

Pyragas K. Phys. Rev. E 1998;58:3067. Masoller C. Physica A 2001;295:301. Zhou SB, Li H, Wu ZF. Phys. Rev. E 2007;75:037203. Shahverdiev EM, Shore KA. Int. J. Mod. Phys. 2007;21:5207; Shahverdiev EM, Shore KA. IET Optoelect. 2009;3:326; Shahverdiev EM, Shore KA. Chaos Solitons Fract. 2008;38:1298. Wang G, Perc M, Duan Z, Chen G. Phys. Rev. E 2009;80:026206. Wang G, Chen G, Perc M. PLoS One 2011;6:e15851. Wang G, Duan Z, Perc M, Chen G. Euro. Phys. J. 2008;83:50008. Arenas A, Diaz-Guilera A, Kurths J, Moreno Y, Zhou CS. Phys. Rep. 2008;469:93. Senthilkumar DV, Kurths J, Lakshmanan M. Phys. Rev. E 2009;79:066208. Sun Y, Ruan J. Chaos 2009;19:043113. Senthilkumar DV, Lakshmanan M. J. Phys. 2005;23:300. Pyragas K. Int. J. Bifur. Chaos 1998;8:1839. Vanwiggeren GD, Roy R. Int. J. Bifur. Chaos 1999;9:2129. Li C, Liao X, Wong K. Physica D 2004;194:187. Cruz-Hern’andez C. Nonlinear Dyn. Syst. Theory 2004;4:1. Ohtsubo J. IEEE J. Quantum Electron. 2002;38:1141. Banerjee S. Chaos Solitons Fract. 2009;42:745. Kim Chil-Min, Kye Won-Ho. Sunghwan Rim and Soo–Young Lee. Phys. Lett. A 2004;333:235. Larger L, Lacourt PA, Poinsot S, Udaltsov V. Laser Phys. 2005;15:1209. Larger L, Goedgebuer JP, Udaltsov V. C.R. Phys. 2004;5:669. Argyris A, Syvridis D, Larger L, Annovazzi-Lodi V, Colet P, Fischer I, et al. Nature 2005;438:343. Lavrov R, Peil M, Jacquot M, Larger L, Udaltsov V, Dudley J. Phys. Rev. E 2009;80:026207. Bezruchko BP, Karavaev AS, Ponomarenko VI, Prokhorov MD. Phys. Rev. E 2001;64:056216. Ponomarenko VI, Prokhorov MD. Phys. Rev. E 2002;66:026215. Zhou C, Lai CH. Phys. Rev. E 1999;60:320. Udaltsov VS, Goedgebuer Jean-Pierre, Larger L, Cuenot Jean-Baptiste, Levy P, Rhodes WT. Phys. Lett. A 2003;308:54. Udaltsov VS, Larger L, Goedgebuer JP, Locquet A, Citrin DS. J. Opt. Technol. 2005;72:373. Yan Z, Yaowen L, Yinghai W. Chinese J. Phys. 2004;42:323. Wu L, Zhu S. Phys. Lett. A 2003;308:157. Yaowen L, Guangming Ge, Hong Z, Yinghai W, Liang G. Phys. Rev. E 2000;62:7898. Shahverdiev EM, Shore KA. Optics Commun. 2009;282:3568. Rontani D, Locquet A, Sciamanna M, Citrin DS. Optics Lett. 2007;32:2960. Ambika G, Amritkar RE. Phys. Rev. E 2009;79:056206. Resmi V, Ambika G, Amritkar RE. Phys. Rev. E 2010;81:046216. Gjurchinovski A, Urumov V. Phys. Rev. E 2010;81:016209. Ghosh D, Banerjee S, Roy Chowdhury A. Eur. Phys. Lett. 2007;80:30006. Senthilkumar DV, Lakshmanan M. Chaos 2007;17:013112. Zheng Z, Yu P, Laio X. Int. J. Bifur. Chaos 2008;18:160. Souza FO, Palhares RM, Mendes E, Torres L. Int. J. Bifur. Chaos 2008;18:187. Shahverdiev EM, Shore KA. Chaos Solitons Fract. 2008;38:1298. Zhan M, Wang X, Gong X, Wei GW, Lai C-H. Phys. Rev. E 2003;68:036208. Shahverdiev EM, Nuriev RA, Hashimov RH, Shore KA. Chaos Solitons Fract. 2006;29:854. Chen M, Kurths J. Phys. Rev. E 2007;76:036212. Hoang TM. Int. J. Elect. Electron. Eng. 2007;2:240. Hoang TM, Minh DT, Nakagawa M. J. Phys. Soc. Jpn. 2005;74:2374.