conditions for impulsive synchronization of chaotic and hyperchaotic

International Journal of Bifurcation and Chaos, Vol. 11, No. 2 (2001) 551–560 c World Scientific Publishing Company

CONDITIONS FOR IMPULSIVE SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC SYSTEMS MAKOTO ITOH∗ Department of Information and Communication Engineering, Fukuoka Institute of Technology, Fukuoka 811-0295 Japan TAO YANG and LEON O. CHUA Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA 94720 USA Received June 8, 2000; Revised August 7, 2000 Experimental results show that chaotic and hyperchaotic systems can be synchronized by impulses sampled from one or two state variables. In this paper, we study the conditions under which chaotic and hyperchaotic systems can be synchronized by impulses sampled from a part of their state variables. By calculating the Lyapunov exponents of variational synchronization error systems, we show that this kind of impulsive synchronization can be applied to almost all hyperchaotic systems. We also study the selective synchronization of chaotic systems. In a selective synchronization scheme, the synchronizing signal is chosen in the time periods when the Lyapunov exponents of variational synchronization error systems are negative. Since only driving signals during the time periods when synchronization error can be reduced are applied to reduce the synchronization error, and no signal is applied during the time periods when synchronization error can be increased, selective synchronization scheme can be used to achieve synchronization even in the case when continuous synchronization schemes fail to work.

1. Introduction

The other synchronization scheme is impulsive synchronization. In an impulsive synchronization scheme only samples of state variables (or functions of state variables) called synchronization impulses are used to synchronize two chaotic systems [Panas et al., 1998; Stjanovski et al., 1997; Yang & Chua, 1997a, 1997b]. Impulsive synchronization had been applied to several chaotic spread spectrum secure communication systems, and had exhibited good performance [Stjanovski et al., 1997; Yang & Chua, 1997a, 1997b, 1997c, 1998a, 1998b; Itoh, 1999; Itoh et al., 1999]. Recently, the detailed experiments and performance analysis of impulsive synchronization

A number of methods have been proposed for synchronizing chaotic systems. The most widely used methods are continuous synchronization schemes. In a continuous synchronization scheme, chaotic systems are coupled to each other continuously such that synchronization errors can be controlled within a given range or converge to zero. In a continuous unidirectional-coupled chaotic synchronization scheme, driving signals are transmitted continuously to the driven systems [Pecora, 1990; Chua et al., 1992; Chua et al., 1993; Wu et al., 1996]. ∗

E-mail: [email protected] 551

552 M. Itoh et al.

were carried out for the purpose of applying impulsive synchronization to chaotic communication systems. The following experimental results were reported in [Itoh et al., 1999]: • The accuracy of synchronization depends on both the period and the width of the impulse samples. • The minimum impulse width for synchronization increases as the impulse period increases. • The hyperchaotic systems can be synchronized by transmitting two kinds of samples through a single channel via a time-division scheme. These experimental results showed that chaotic systems can be impulsively synchronized by using impulse samples derived from some of the state variables of the driven system. This is because some chaotic systems can be decomposed into two parts. One part tends to make the synchronization unstable and the other part tends to make the synchronization stable. If we construct impulsive controllers to stabilize the unstable part we can synchronize chaotic systems by using samples from some state variables of the driven system. In this paper, we use variational synchronization error systems to study the stability of different impulsive synchronization schemes. We find the stability of impulsive synchronization is closely connected to the values of the Lyapunov components of the variational systems. We present the conditions in terms of Lyapunov exponents for stable impulsive synchronization in both chaotic and hyperchaotic systems. In a unidirectional–directional synchronization scheme, during some time periods, the driving signal can cause the synchronization error to increase and during some other time periods it can cause synchronization error to decrease. These two kinds of time periods can be distinguished by monitoring the eigenvalues of the variational synchronization error systems. When the driving signal in a time period is detected to be more likely to increase synchronization errors, we do not use it to drive the driven system. By doing this, we can synchronize two chaotic systems much more efficiently than a continuous synchronization scheme. This kind of synchronization scheme is called selective synchronization. The selective synchronization scheme can synchronize chaotic systems which cannot be synchronized by a continuous synchronization scheme under similar conditions.

2. Impulsive Synchronization of Chaotic Systems In this section, we only consider the impulsive synchronization between two chaotic systems with unidirectional-coupling. In a uni-coupled synchronization scheme, we transmit impulses sampled from one state variable of the driving system (master) to the driven system (slave). To avoid clutter and without loss of generality, we study the case when impulse samples are equidistant. Let P and Q denote the period and width of the impulse samples, respectively. Note that P and Q must satisfy Q ≤ P. Consider the following general form of a continuous dynamical system whose state variables can be grouped into two parts; namely, x(t) and y(t): x˙ = f (x, y) ,

)

(1)

y˙ = g(x, y) ,

where x = (x1 , x2 , . . . , xl )> ∈ ∈ v(t) + w(t)> w(t)

from which we have

(µ − λ)Q + λP < 0 ,

(7)

q



(11)

is satisfied,2 we can expect that



respectively. Let us define |u(t)| as |u(t)| =

|u(nP )| ≈ en(µ−λ)Q+nP λ |u(0)| . Therefore, if the inequality

∂g(x, y) w w˙ = ∂y 

and equivalently1

(10)

x0n

)

= xn ,

slave system A

0 = g(xn , yn0 ), yn+1

(15)

where n ∈ (iP, iP + Q), i = 0, 1, 2, . . . , and x0n+1 = f (x0n , yn0 ), 0 = g(x0n , yn0 ), yn+1

)

slave system B

(16)

where n ∈ (iP + Q, (i + 1)P ). The variational synchronization error systems are given respectively by wn+1 =

∂g(xn , yn ) wn , ∂yn

(17)

The mathematical notation: C ≈ D implies O(C) = O(D). The symbol O(·) indicates Landau’s symbol [Nayfeh, 1973]. Note that Q ≤ P .

554 M. Itoh et al.

Table 1. Discrete-Time Dynamical System Chosa–Golubitsky system

    

xn+1 = Axn + d(x2n − yn2 ), yn+1 = Ayn − 2dxn yn , A=

a(x2n

+ yn2 ) +bxn (x2n −

Fold system xn+1 = yn + axn ,

6yn2 ) + c.

Slopes D and Da for discrete-time dynamical systems. Parameter Values

b = 0.0,

c = −2.25,     d = 0.2.

)

a = −0.1,

Lorenz system xn+1 = (1 + ab)xn − bxn yn ,

)

a = 1.25,

yn+1 = (1 − b)yn + bx2n .

xn+1 = αxn (1 − xn ) −β(zn + γ)(1 − 2yn ), yn+1 = δyn (1 − yn ) + ζzn ,

       

   zn+1 = η((zn + γ)(1 − 2yn ) − 1)     ×(1 − θxn ).

Table 2. Discrete-Time Dynamical System

D

Da

yn

0.273

−0.0788

0.776

0.723

yn

0.0196

−2.30

0.0785

0.0825

xn

0.343

−1.39

0.188

0.198

x n & yn

0.413

−4.30

0.0876

0.0852

)

b = 0.75.

R¨ ossler hyperchaotic system

µ

)

b = −1.7.

yn+1 = x2n + b.

λ

    

a = 1.0,

   

Driving Signals

 α = 3.8,     β = 0.05,      γ = 0.35,    δ = 3.78,

   ζ = 0.2,      η = 0.1,     θ = 1.9.

Slopes D and Da for continuous-time dynamical systems. Parameter Values

Driving Signals

λ

 σ = 10,     8 b= , 3     r = 28.

x

0.907

y

0.907

x

0.0782

µ

D

Da

−1.78

0.338

0.462

−2.67

0.253

0.468

−0.0348

0.692

0.895

Lorenz system

 dx   = σ(y − x),    dt    dy = −xz + rx − y,  dt      dz   = xy − bz. dt

R¨ ossler system

 dx   = −(y + z),    dt    dy = x + ay,  dt      dz  = b + z(x − c).  dt

 a = 0.2,   b = 0.2, c = 9.0.

 

Conditions for Impulsive Synchronization of Chaotic and Hyperchaotic Systems 555 Table 2. Discrete-Time Dynamical System Chua’s oscillator dx = α(y − x − f (x)), dt dy = x − y + z, dt dz = −β(y + γz), dt f (x) = bx + 0.5(a − b) ×(|x + 1| − |x − 1|). Hyperchaotic system dx = α(g(z − x) − y), dt dy = x + (δ − ε)y, dt dz = −β(g(z − x) + w), dt dw = γz − εw, dt g(u) = bu + 0.5(a − b) ×(|u + 1| − |u − 1|).

and "

vn+1 wn+1

#



              

Parameter Values

α = 10, β = 15, γ = 0.1,

             

   a = −1.27,     b = −0.68.

                  

 α = 2,     β = 20,      γ = 1.5,   δ = 1,   ε = 0.05,      a = −0.2,     b = 3.

                 

∂f (xn , yn ) ∂f (xn , yn )

 ∂xn ∂yn  =  ∂g(xn , yn ) ∂g(xn , yn )

∂xn

       



" #  v n  ,   wn

∂yn (18)

where vn = xn − x0n and wn = yn − yn0 . All conclusions are the same as those for the continuous dynamical systems. We then examine the inequality (12) by computer simulations when different kinds of chaotic systems are used. First, we suppose that the master and slave systems are synchronized by the continuous synchronization scheme. Then, without loss of generality, assume µ < 0 and λ > 0, and transform inequality (12) into Q > DP ,

(19)

where D = λ/(λ − µ). The slope D can be found by the following two methods: (1) Calculate the Lyapunov exponents λ and µ and substitute them into D = λ/(λ − µ).

(Continued ) Driving Signals

λ

µ

D

Da

x

0.298

−0.550

0.351

0.439

y

0.298

−0.100

0.749

1.02

g(z − x) & δy

0.188

−0.025

0.882

0.935

(2) Calculate the minimum impulse width Q under various frame lengths P , and obtain the slope Da by using least the mean square approximation. Tables 1 and 2 show the simulation results for discrete-time and continuous-time dynamical systems, respectively. In the case of discretetime dynamical systems, the slope D gives a good ∼ approximation for Da , that is, Da = D. In the case of continuous-time dynamical systems, D seems to be a lower bound of Da ; namely, Da ≥ D.

3. Impulsive Synchronization of Hyperchaotic Systems In this section, we examine the stability of impulsive synchronization between hyperchaotic systems via a single communication channel. We study only the unidirectional-coupling case in this section and assume that we can group the state variables of the hyperchaotic system into two parts: xn and yn . Since the conclusions of continuous hyperchaotic systems and discrete hyperchaotic systems are the

556 M. Itoh et al.

Table 3.

Configuration of impulsive synchronization for discrete-time hyperchaotic systems.

R¨ ossler Hyperchaotic System

Parameter Values

Driving Signals di and Slopes D, Da

master       = αxn (1 − xn ) − β(yn + γ)(1 − 2yn ),      = f (x , y , z )

xn+1 = f1 (xn , yn , zn ) yn+1

2

n

n

n

= δyn (1 − yn ) + ζzn , zn+1 = g1 (xn , yn , zn ) = η((zn + γ)(1 − 2yn ) − 1)(1 − θxn ). slave x0n+1 = f1 (x0n , yn0 , zn0 ) +e1 (d1 + εx0n − f1 (x0n , yn0 , zn0 )), 0 = f2 (x0n , yn0 , zn0 ) yn+1

+e2 (d2 + εyn0 − f1 (x0n , yn0 , zn0 )), 0 = g1 (x0n , yn0 , zn0 ) zn+1

= η((zn0 + γ)(1 − 2yn0 ) − 1)(1 − θx0n ). ?

         

 α = 3.8,     β = 0.05,      γ = 0.35,    δ = 3.78,    ζ = 0.2,      η = 0.1,     θ = 1.9.

          

d1 = f1 (xn , yn , zn ) − εxn = αxn (1 − xn ) −β(zn + γ)(1 − 2yn ) −εxn , d2 = f1 (xn , yn , zn ) − εyn = δyn (1 − yn ) + ζzn − εyn , (ε = 0.4)

D = 0.315,

         

Da = 0.362.

The driving signals d1 and d2 are transmitted to the slave system via a time-division scheme.

Table 4.

Configuration of impulsive synchronization for continuous-time hyperchaotic systems.

R¨ ossler Hyperchaotic System

Parameter Values

Driving Signals di and Slopes D, Da

master  dx   = α(g(z − x) − y),   dt     dy   = x + (δ − ε)y,   dt  dz = −β(g(z − x) + w),    dt    dw    = γz − εw,   dt   g(u) = bu + 0.5(a − b)(|u + 1| − |u − 1|). slave dx0 dt dy 0 dt dz 0 dt dw0 dt ?

= α(e1 d1 − y 0 + (1 − e1 )g(z 0 − x0 )), = x0 + e2 d2 − εy 0 + (1 − e2 )δy 0 ,

     β = 20,      γ = 1.5,    δ = 1,    ε = 0.05,      a = −0.2,     b = 3. α = 2,

d1 = g(z − x), d2 = δy, D = 0.448, Da = 0.441.

           

  = −β(e1 d1 + w0 + (1 − e1 )g(z 0 − x0 )),         0 0  = γ(z − εw ).

The driving signals d1 and d2 are transmitted to the slave system via a time-division scheme.

Conditions for Impulsive Synchronization of Chaotic and Hyperchaotic Systems 557

same, in this section we consider only the following discrete hyperchaotic master system: xn+1 = F (xn , yn ),

)

master

yn+1 = G(xn , yn ),

(20)

where

yn = (yn1 , yn2 , . . . , ynm )> , F = (f1 , f2 , . . . , fl ) , G = (g1 , g2 , . . . , gm )> .

Wn+1 =

The slave system can be given by )

slave

(21)

where E = diag(e1 , e2 , . . . , el ) , H = (h1 , h2 , . . . , hl )> = D − F (x0n , yn0 ) + εx0n , D = (d1 , d2 , . . . , dl )> = F (xn , yn ) − εxn , ε > 0, (

ej =

1, if iP + (j − 1)Q < n ≤ iP + jQ 0, else

(i, j : integers; j = 1, 2, . . . , l; diag: diagonal matrix). Here, the driving signals dj = fj (xn , yn ) − εxnj are transmitted sequentially via a time-division scheme. We then study the variational equation with respect to the variable Vnj = xnj − x0nj . For the interval Ij1 = {n |iP + (j − 1)Q < n ≤ iP + jQ}, the variational system is given by V(n+1)j = εVnj ,

(22)

where 1 ≤ j ≤ l. For the remaining interval Ij2 = {n|n ≤ iP +(j−1)Q, iP +jQ < n ≤ (i+1)P }, the variational equation is written as 

V(n+1)j

= e(µ−λ)Q+λP |V0j | ,

∂fj (xn , yn ) ∂fj (xn , yn ) = ∂xn ∂yn

"

Vn Wn

#

, (23)

0 . where 1 ≤ j ≤ l and Wnj = ynj − ynj

(24)

Therefore, the hyperchaotic systems are synchronized if all Lyapunov exponents of the following system are negative

>

0 yn+1 = G(x0n , yn0 ) ,

|VP j | ≈ eµQ+λ(P −Q) |V0j |

where µ = ln ε. By choosing ε sufficiently small, we get (25) Vnj → 0 for n → ∞ .

xn = (xn1 , xn2 , . . . , xnl )> ,

x0n+1 = F (x0n , yn0 ) + EH ,

Let λ be the largest Lyapunov exponent of (23). Then we can approximate |VP j | by

∂G(xn , yn ) Wn . ∂yn

(26)

If not all Lyapunov exponents of (26) are negative, then we incorporate some of gj (xn , yn ) into F (xn , yn ) until all Lyapunov exponents become negative. Therefore, the time-division based impulsive synchronization can synchronize almost all kinds of hyperchaotic systems. Table 3 shows the impulsive synchronization between two discrete hyperchaotic systems. For continuous hyperchaotic systems, we can draw similar conclusions. One example of impulsive synchronization between continuous hyperchaotic systems is shown in Table 4. The slope D gives a good approximation for Da . In our numerical study, the Lyapunov exponent λ of the variational equation (23) is approximated by the Lyapunov exponent of the hyperchaotic systems because the impulse width Q is usually sufficiently small compared to the frame length P .

4. Selective Synchronization When a driving signal is applied to the slave system, two kinds of effects can be observed. On the one hand, the driving signal in some time periods can have strong desynchronizing effect. On the other hand, in some other time periods, the driving signal can have strong synchronizing effect. In continuous synchronization schemes, since the driving signal is continuously applied to the slave system, these two kinds of effects cannot be distinguished and therefore in many cases, the continuous synchronization schemes have low synchronizing efficiency. In this section, we present a synchronization scheme called selective synchronization to selectively use only those time periods of driving signals which can show strong synchronizing effect

558 M. Itoh et al. Table 5.

Selective synchronization scheme for discrete-time dynamical systems.

Discrete-Time Dynamical System

Parameter Values

Driving Signal

Synchronization

Chosa-Golubitsky system master xn+1 = f (xn , yn ) = Axn + d(x2n − yn2 ), yn+1 = g(xn , yn ) = Ayn − 2dxn yn , A(xn , yn )

a(x2n

= + yn2 ) + bxn (x2n −

6yn2 ) + c.

                    

a = 1.0, b = 0.0,

    

c = −2.25,     d = 0.2.

x0n = xn

No

(Continuous synchronization)

(100% of driving signal was used)

slave

     = A0 xn + d(xn 2 − yn2 ),    0 0 yn+1 = g(xn , yn )     = A0 yn0 − 2dxn yn0 ,    0 0 0 A = A(xn , yn ). x0n+1 = f (x0n , yn0 )

∂g(x0n , yn0 ) ≤ 1.1, if ∂yn0

Yes

then x0n = xn .

(38.3% of driving signal was used)

yn0 = yn

No

(Continuous synchronization)

(100% of driving signal was used)

∂f (x0n , yn0 ) ≤ 0.5, if ∂x0

Yes

Lorenz system master xn+1 = f (xn , yn ) = (1 + ab)xn − bxn yn , yn+1 = g(xn , yn ) = (1 − b)yn +

bx2n .

slave

0 = yn+1

=

   

a = 1.25,

)

b = 0.75.

    

x0n+1 = f (x0n , yn0 ) =

    

(1 + ab)x0n − bx0n yn0 ,  g(x0n , yn0 )    0 0 2 (1 − b)yn + bxn .

Table 6.

n

then

yn0

= yn .

(15.0% of driving signal was used)

Selective synchronization scheme for continuous-time dynamical systems.

Continuous-Time Dynamical System

Parameter Values

Driving Signal

Synchronization

Lorenz system master

 dx   = σ(y − x),    dt   dy = −xz + rx − y,  dt     dz   = xy − bz. dt

 σ= 10,   8  b= , 3    r = 28.

z0 = z

No

(Continuous synchronization)

(100% of driving signal was used)

slave  dx0   = σ(y 0 − x0 ),    dt   0 dy 0 0 0 0 = −x z + rx − y ,  dt     dz   = x0 y 0 − bz 0 . dt

if z 0 ≥ 23, then z 0 = z.†

Yes (48.8% of driving signal was used)

Conditions for Impulsive Synchronization of Chaotic and Hyperchaotic Systems 559 Table 6. Continuous-Time Dynamical System

(Continued )

Parameter Values

Driving Signal

Synchronization

Chua’s oscillator (with an eventually passive nonlinearity) master  dx  = α(y − x − f (x)),     dt   dy = x − y + z,  dt     dz   = −β(y + γz), dt

α = 10, β = 15, γ = 0.1, a = −1.27,

            

   b = −0.68,      c = 0.5,     d = 2.3,

z0 = z

No

(Continuous synchronization)

(100% of driving signal was used)

f (x) = cx + 0.5(a − b)(|x + 1| − |x − 1|) + 0.5(b − c)(|x + d| − |x − d|). slave

 dx0  0 0 0 = α(y − x − f (x )),    dt    0 dy 0 0 0 = x −y +z ,  dt    0  dz  0 0  = −β(y + γz ), dt

† ?

if |x0 | ≥ d, then z 0 = z.?

Yes (12.1% of driving signal was used)

If z 0 ≥ 27, then all eigenvalues of the variational system have negative real part. If x0 ≥ d, then all eigenvalues of the variational system have negative real part.

to the slave system and shut off the driving signals in some other time periods when they have strong desynchronizing effects. First, we calculate the eigenvalues ηj of ∂G(xn , 4

yn )/∂yn and define ρj = ln |ηj |. Then, we find the region where the driving signal has large ρj . That is, we find the region that has strong desynchronizing effects. If ρj in a region is greater than the Lyapunov exponent of the system, then we do not apply driving signal to the driven system in this region. Otherwise, we apply the driving signal to the driven system because the synchronization error can be reduced rapidly if ρj is less than the Lyapunov exponent. We show some typical examples in Tables 5 and 6. From the results in Tables 5 and 6 we can see that the selective synchronization scheme can be used to achieve synchronization even in the case when continuous synchronization schemes fail to work.

5. Conclusions In this paper, we studied the stability of impulsive synchronization of chaotic and hyperchaotic sys-

tems by using Lyapunov exponents of the variational synchronization error systems. Our results are consistent with what we have observed in experiments. We also presented an efficient synchronization scheme called selective synchronization.

Acknowledgments This work is supported in part by the Computer Science Laboratory at the Fukuoka Institute of Technology. T. Yang and L. O. Chua are supported by the Office of Naval Research under grant No. 00014-96-1-0753.

References Chua, L. O., Kocarev, L., Eckert, K. & Itoh, M. [1992] “Experimental chaos synchronization in Chua’s circuit,” Int. J. Bifurcation and Chaos 2(3), 705–708. Chua, L. O., Itoh, M., Kocarev, L. & Eckert, K. [1993] “Chaos synchronization in Chua’s circuit,” J. Circuits Syst. Comput. 3(1), 93–108. Itoh, M. [1999] “Spread spectrum communication via chaos,” Int. J. Bifurcation and Chaos 9(1), 153–213. Itoh, M., Yang, T. & Chua, L. O. [1999] “Experimen-

560 M. Itoh et al.

tal study of impulsive synchronization of chaotic and hyperchaotic circuits,” Int. J. Bifurcation and Chaos 9(1), 1393–1424. Nayfeh, A. L. [1973] Perturbation Methods (John Wiley, NY). Panas, A. I., Yang, T. & Chua, L. O. [1998] “Experimental results of impulsive synchronization between two Chua’s circuits,” Int. J. Bifurcation and Chaos 8(3), 639–644. Pecora, L. & Carroll, T. [1990] “Synchronization in chaotic systems,” Phys. Rev. Lett. 64(8), 821–824. Stjanovski, T., Kocarev, L. & Parlitz, U. [1997] “Digital coding via chaotic systems,” IEEE Trans. Circuit Syst. 44(6), 562–565. Wu, C. W., Yang, T. & Chua, L. O. [1996] “On adaptive synchronization and control of nonlinear dynamical systems,” Int. J. Bifurcation and Chaos 6(3), 455–471. Yang, T. & Chua, L. O. [1997a] “Impulsive control

and synchronization of nonlinear dynamical systems and application to secure communication,” Int. J. Bifurcation and Chaos 7(3), 645–664. Yang, T. & Chua, L. O. [1997b] “Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication,” Trans. Circuits Syst. I 44(10), 976–988. Yang, T. & Chua, L. O. [1997c] “Chaotic digital code-division multiple access (CDMA) communication systems,” Int. J. Bifurcation and Chaos 7(12), 2789–2805. Yang, T. & Chua, L. O. [1998a] “Applications of chaotic digital code-division multiple access (CDMA) to cable communication systems,” Int. J. Bifurcation and Chaos 8(8), 1657–1669. Yang, T. & Chua, L. O. [1998b] “Error performance of chaotic digital code-division multiple access (CDMA) systems,” Int. J. Bifurcation and Chaos 8(10), 2047–2059.