Letters

International Journal of Bifurcation and Chaos, Vol. 11, No. 6 (2001) 1723–1735c© World Scientific Publishing Company

UNCERTAINTY IN CHAOS SYNCHRONIZATION

GUO-QUN ZHONG, KIM-FUNG MAN and KING-TIM KODepartment of Electronic Engineering, City University of Hong Kong,

Kowloon, Hong Kong, China

Received August 3, 2000; Revised September 1, 2000

In this paper a variety of uncertainty phenomena in chaos synchronization, which are causedby the sensitive dependence on initial conditions and coupling strength, are numerically inves-tigated. Two identical Chua’s circuits are considered for both mutually- and unidirectionally-coupled systems. It is found that initial states of the system play an important role in chaossynchronization. Depending on initial conditions, distinct behaviors, such as in-phase synchro-nization, anti-phase synchronization, oscillation-quenching, and bubbling of attractors, mayoccur. Based on the findings, we clarify that the systems, which satisfy the standard synchro-nization criterion, do not necessarily operate in a synchronization regime.

1. Introduction

Synchronization is one of the interesting discov-eries in the study of chaos [Pecora & Carroll,1990; Afraimovich et al., 1986]. To generate a setof chaotic systems so that they can be synchro-nized among themselves, a variety of approaches forcoupling the systems have been proposed [Pecora& Carroll, 1990; Chua et al., 1993; Kapitaniaket al., 1994; Chua et al., 1996; Wu & Chua, 1993;Wu et al., 1996; Pecora & Carroll, 1991]. For allthese coupled systems there exists a critical value,or synchronization threshold, that corresponds toa blowout bifurcation [Ott & Sommerer, 1994]. Inthe general case, the blowout bifurcation describesthe loss of stability of the smooth invariant man-ifold which exists within the phase space of thedynamical system. Two interesting phenomena,namely, on–off intermittency and attractors withriddled basins of attraction are expected to takeplace in the vicinity of the bifurcation point. Thephenomenon of riddled basins has become an impor-tant theme in the study of chaotic dynamics, andreceived many attentions especially in chaos syn-chronization [Kapitaniak et al., 1998; Nakajima &Ueda, 1997; Maistrenko et al., 1998; Kim, 1997;Astakhov et al., 1997].

In contrast to the studies before, we investi-gate the uncertainty of chaos synchronization in thesense that the systems, which satisfy the standardsynchronization criterion, do not necessarily oper-ate in a synchronization regime. Depending on theinitial states of the systems, distinct behaviors ofsynchronization, such as in-phase synchronization,anti-phase synchronization, oscillation-quenching,and attractor bubbling, may take place. Our at-tentions are focused on the dynamical behaviors oftwo mutually and unidirectionally coupled identicalchaotic systems, respectively. The results observedare based on two coupled identical Chua’s circuits.

This paper is organized as follows. The un-certainty of chaos synchronization is discussed inSec. 2. The dynamical behaviors related to the un-certainty phenomenon in coupled Chua’s circuit aredemonstrated in Sec. 3. Some concluding remarksare given in Sec. 4.

2. Uncertainty in Synchronizationof Chaos

The sensitive dependence on initial conditions isa fundamental property of chaos motion. It hasbeen a topic of common interest and has receivedmuch attention in this field. It was reported [Ott &

1723

1724 G.-Q. Zhong et al.

Sommerer, 1994; Ott et al., 1993; Ashwin et al.,1994] that an attractor’s basin is riddled by an-other attractor’s basin due to the sensitivity. Thisfinding naturally gives rise to such questions: dothe systems, which satisfy the standard synchro-nization criterion, operate definitely in a synchro-nization regime? Can the synchronized state of twocoupled chaotic systems certainly be determined?

To answer the above questions, we shall ex-amine below the dynamical behaviors in coupledchaotic systems.

2.1. System model for synchronization

Consider two (or more) n-dimensional autonomouschaotic systems

ui = fi(ui) , ui ∈ Rn , 1 ≤ i ≤ N (N ≥ 2) ,(1)

where ui = ui(t) are state vectors.Obviously, an intuitive measure of the quality

of synchronization for systems ui and uj is

ρ = ‖ui − uj‖ , i 6= j ,

which is the length of a vector between a point inthe subspace ui and its corresponding point in thesubspace uj.

The synchronization of the two systems impliesthat their submanifolds will converge to each otherin such a way that

limt→∞‖ui − uj‖ = 0 , i 6= j . (2)

To achieve the synchronization of two chaoticsystems, the linear coupling technique is commonlyused [Chua et al., 1993; Kapitaniak et al., 1994;Pecora & Carroll, 1991]. For a system with two sub-systems u and v mutually-coupled, the state equa-tions of the system can be described as below:

u = f1(u) +K(v − u)

v = f2(v) +K(u− v)(3)

where u, v ∈ Rn and K = diag[k1, . . . , kn]T .As for the unidirectionally-coupled system, we

haveu = f1(u)

v = f2(v) +K(u− v)(4)

where u, v ∈ Rn and K = diag[k1, . . . , kn]T .Hence the synchronization problem can be for-

mulated as follows: find K such that ρ = ‖v(t) −

u(t)‖ → 0 as t→∞; in other words, the manifoldsu(t) and v(t) are tracking with each other in thecase of the mutually-coupled system (3), and themanifold v(t) will synchronize with the manifoldu(t) in the unidirectionally-coupled system (4).

2.2. Dynamical behavior in coupledsystems

It will be shown that two coupled chaotic systemswould exhibit a variety of distinct performances ofsynchronization, depending on the coupling and ini-tial conditions. We refer to the dynamical behavioras the uncertainty of synchronization in the sensethat the systems, which satisfy the standard syn-chronization criterion, do not necessarily operate ina synchronization regime. Here we shall explore theinfluence of initial conditions on chaos synchroniza-tion while the initial conditions are classified intothe following three main categories: (a) identicalinitial conditions, (b) equal initial conditions withopposite signs, and (c) otherwise.

2.2.1. Identical initial conditions

Theorem A. Suppose that in system (3), f1 = f2 =f which satisfies f(0) = 0, and that the gain matrixK is chosen such that

‖f ′(0)‖ < 2‖K‖ . (5)

Then, with u(t = 0) = v(t = 0) we have u(t) = v(t)for all t ≥ 0, independent of the value of K ≥ 0.

Proof. We prove the result by using the Poincare–Lyapunov Theorem [Verhulst, 1996]. From (3), wehave

u− v = f(u)− f(v) + 2K(v − u) .

Taylor expand f at 0:

f(u) = f(0) + f ′(0)u + H.O.T.

f(v) = f(0) + f ′(0)v + H.O.T.

Then, we have

u− v = f ′(0)(u− v)− 2K(u− v) + H.O.T.

= [f ′(0)− 2K](u− v) + H.O.T. (6)

Since the matrix [f ′(0) − 2K] is stable under con-dition (5) with K > 0, and since the Higher-Order-Terms H.O.T. are quadratic and higher so it

Uncertainty in Chaos Synchronization 1725

satisfies

lim‖u−v‖→0

‖H.O.T.‖‖u− v‖ = 0 ,

it follows from the Poincare–Lyapunov Theorem[Verhulst, 1996] that u − v → 0. When K = 0,the two subsystems in (3) are identical when theyhave the same initial conditions. This completesthe proof of the theorem. �

By the procedure similar to the above, it canbe proved that this result is also true for theunidirectionally-coupled system (4).

We refer to this behavior as the in-phase syn-chronization. This is the simplest and a very specialcase as it seldom occurs in practice.

2.2.2. Equal initial conditions withopposite signs

Next we examine the behavior in the mutually-coupled system (3) in the case of f1 = f2 andu(t = 0) = −v(t = 0).

Theorem B. Suppose that in system (3), f1 = f2 =f which has an odd symmetry. If there is a constantM > 0 such that

f ′(q) ≤ −MI < 0 (7)

uniformly for all q ∈ Rn, where I is the identitymatrix, then, with u(t = 0) = −v(t = 0) we haveu(t) = −v(t) for all t ≥ 0, independent of the valueof K ≥ 0.

Proof. In system (3), let w = −v, so that it canbe rewritten as

u = f(u) +K(−w− u) ,

−w = f(−w) +K(u + w) .

Adding the two equations together, with theodd symmetry of f , it yields that

u− w = f(u) + f(−w) = f(u)− f(w) ,

which has initial conditions u(t = 0) = w(t = 0)by assumption. The odd symmetry of f impliesf(0) = 0. Furthermore, notice that

u− w = f(u)− f(w) = f ′(q)(u −w)

for some q in between u and w. It follows undercondition (7) that: starting from initial condition

u(t = 0) −w(t = 0) = 0, we have u(t) −w(t) = 0or u(t) = −v(t) for all t ≥ 0. This completes theproof of the theorem. �

We refer to this type of dynamical behavioras anti-phase synchronization. It should be notedthat Theorem B does not hold in the case ofthe unidirectionally-coupled system (4), hence theanti-phase synchronization does not occur in suchsystems.

Depending on the coupling strength K, twotypes of anti-phase synchronization in the system(3) with two sets of initial conditions identical invalue but opposite in sign were observed. Thereexists a critical value (threshold) Kc of the cou-pling strength, such that for all K ≥ Kc, the os-cillations in the two subsystems are quenched witheach other, and each subsystem operates in a stablepoint attractor with anti-phase. Otherwise, a pro-cess of period-doubling bifurcation to chaos withanti-phase occurs in the two subsystems as the cou-pling strength K decreases. Below we shall presentthe bifurcation induced by the coupling decreasing.

Case (i). K ≥ Kc.Consider one subsystem of (3), say,

u = f1 +K(v − u) with u(0) = −v(0)

and u(t) = −v(t) .

It leads since u = −v that

u = f1 − 2Ku .

Suppose that there exists a threshold Kc suchthat u = 0. Then we have Kc = f1(ue)/2ue, whereue is the equilibrium. It yields that

u(t) = C1

at the equilibrium, where C1 is the integral constantmatrix.

Utilizing the procedure similar to the above foranother subsystem of (3)

v = f2 +K(u− v) with u(0) = −v(0)

and u(t) = −v(t) ,

it follows that

Kc =f2(ve)

2veand v(t) = C2 ,

where ve is the equilibrium, and C2 is the inte-gral constant matrix. Since u(t) = −v(t), it leadsthat C2 = −C1, which implies that two subsystems

1726 G.-Q. Zhong et al.

operate in constant states with anti-phase, respec-tively. We refer to this dynamical behavior asoscillation-quenching.

Case (ii). K < Kc.In this case a process of period-doubling bifur-

cation to chaos with anti-phase were observed inthe two subsystems as the coupling strength K isdecreasing.

2.2.3. Otherwise

Apart from the two typical classes of initial con-ditions discussed above, we shall examine the be-havior relating to the problem of synchronizationwhen the initial conditions are excluded from theabove two cases in Secs. 2.2.1 and 2.2.2. To evaluatequantitatively the sensitivity to initial conditions inchaos synchronization, the analysis can be done byevaluating the states of the systems through inte-gration of (3) and (4) with different initial condi-tions. We adopted the following definition of syn-chronization for the study.

Definition. The two systems u and v of (3) and(4) are in the state of synchronization if there ex-ists a sufficiently small value ε ≥ 0 such that thedistance ρ = ‖u− v‖ → ε as t→∞.

Using the distance ρ to assess the states in thesystems (3) and (4), it has been found that the stateof synchronization is sensitive to the initial condi-tions of the systems, and the basins of attraction areriddling due to the sensitive dependence on initialconditions.

3. Observations in Coupled Chua’sCircuits

We adopted Chua’s circuit [Zhong & Ayrom, 1985;Kennedy, 1992; Chua, 1994], shown in Figs. 1(a)and 1(b), as a vehicle to demonstrate the dynami-cal behaviors described in Sec. 2. For simplicity, thefollowing dimensionless state equations of Chua’scircuit [Chua, 1994] were used for the investigation:

x = α(y − x− g(x))

y = x− y + z

z = −βy(8)

where

g(x) = bx+1

2(a− b)[|x+ 1| − |x− 1|] . (9)

(a)

(b)

Fig. 1. (a) Chua’s circuit. (b) v–i characteristic of nonlinearresistor NR.

Note that the above equations are of odd sym-metry, and the nonlinear function in the equationssatisfies the theorem conditions mentioned in theprevious section globally.

We chose and fixed the following parameter val-ues for the Chua’s circuit throughout the investiga-tion: α = 10, β = 14.87, a = −1.27, b = −0.68, sothat the Chua’s circuit operated in a typical chaoticregime with a Double-Scroll attractor.

Based on the Chua’s circuit described above,the mutually-coupled system (3) in the case off1 = f2 can be written in the following form

x = α(y − x− g(x)) + kx(x− x)

y = x− y + z + ky(y − y)

z = −βy + kz(z − z)˙x = α(y − x− g(x)) + kx(x− x)

(10)

Uncertainty in Chaos Synchronization 1727

˙y = x− y + z + ky(y − y)

˙z = −βy + kz(z − z)

where

g(x) = bx+1

2(a− b)[|x+ 1| − |x− 1|] , (11)

g(x) = bx+1

2(a− b)[|x+ 1| − |x− 1|] , (12)

and kj(j = x, y, z) are the coupling factors.Similarly, the unidirectionally-coupled system

(4) in the case of f1 = f2 becomes

x = α(y − x− g(x))

y = x− y + z

z = −βy˙x = α(y − x− g(x)) + kx(x− x)

˙y = x− y + z + ky(y − y)

˙z = −βy + kz(z − z)

(13)

where

g(x) = bx+1

2(a− b)[|x+ 1| − |x− 1|] , (14)

g(x) = bx+1

2(a− b)[|x+ 1| − |x− 1|] , (15)

and kj(j = x, y, z) are the coupling factors.For simplicity, identical kj(j = x, y, z) = k are

selected throughout our discussion hereafter whenmore than one pair of corresponding state variablesare coupled, where k is a real non-negative num-ber. When kj = 0, it implies that the correspondingstate variables are not coupled with each other.

3.1. In-phase synchronization

When two sets of identical initial conditions aretaken, the systems (10) and (13) having two identi-cal Chua’s circuits coupled always exhibit identicalbehaviors, and the systems operate in a synchro-nization mode, regardless of the coupling strengthand the coupling form. We refer to this performanceof the system as the in-phase synchronization. Thephase plot in the phase plane x versus x is a straightline crossing the origin located in the first and thirdquadrants, as shown in Fig. 2.

3.2. Anti-phase synchronization

Anti-phase synchronization can be demonstrated bythe mutually-coupled Chua’s circuit system (10).

Fig. 2. Phase plot in x versus x plane of two coupled iden-tical Chua’s circuits with identical initial conditions.

Each Chua’s circuit of the system with the param-eter values listed above operated in a Double-Scrollattractor before being coupled. For the system, inwhich the three state variables x, y, z were all cou-pled with the same strength, i.e. kx = ky = kz =k > 0 in (10), and two sets of initial conditionswith identical values but opposite signs, {x(0), y(0),z(0), x(0), y(0), z(0) : x(0) = 0.5, y(0) = 0.2,z(0) = 0.05, x = −0.5, y = −0.2, z = −0.05}, wereselected, the critical value kc = 0.116 was obtainedin this case.

For all k ≥ kc, the oscillations in the two Chua’scircuits were quenched with each other, and eachChua’s circuit operated in a stable point attractorwith anti-phase. The phase portraits of the attrac-tors and the time series are presented in Figs. 3(a)and 3(b), respectively. Note that both phase por-traits in y versus x plane and y versus x plane arestable points with anti-phase, and the time seriesare constant horizontal lines above and below hori-zontal axes, respectively. This effect was found forall forms of state variable coupling.

As for the case when k < kc, the following pro-cess of bifurcation appeared in the system as thecoupling strength k was decreased.

When k was decreased to 0.1, the two Chua’scircuits operated in identical period-1 orbits withanti-phase, as shown in Fig. 3(c). A period-doubling bifurcation route to Double-Scroll attrac-tor occurred successively as the coupling factor kwas further decreased, as shown in Figs. 3(d)–3(g).The trajectories of two Chua’s circuits were foundto be identical but in anti-phase, as shown by thephase plots in x versus x planes given in thesefigures.

1728 G.-Q. Zhong et al.

(a)

(b)

Fig. 3. Performance in two mutually-coupled identical Chua’s circuits with opposite identical initial conditions (0.5, 0.2, 0.05;−0.5, −0.2, −0.05) as k varies. Parameter: α = 10, β = 14.87, a = −1.27, b = −0.68. (a and b) Phase portrait of a stablepoint attractor (top) and time waveform (bottom). k = 0.15. Vertical axis: (a) y (top) and x (bottom), (b) y (top) and x

(bottom); Horizontal axis: (a) x (top) and t (bottom), (b) x (top) and t. (c–g) Phase portrait of the attractor (left) and phaseplot (right). Vertical axis: y (left) and x (right); Horizontal axis: x. (c) k = 0.1. Period-1 attractor. (d) k = 0.095. Period-2attractor. (e) k = 0.091. Period-4 attractor. (f) k = 0.089. Spiral attractor. (g) k = 0.05. Double-Scroll attractor.

Uncertainty in Chaos Synchronization 1729

(c)

(d)

(e)

Fig. 3. (Continued )

1730 G.-Q. Zhong et al.

(f)

(g)

Fig. 3. (Continued )

Table 1. Performance in coupled Chua’s circuits with opposite initial conditions.

Coupling Performance in UnidirectionalStrength Performance in Mutual Coupled Chua’s Circuits Coupled Chua’s Circuits

k ≥ kc k ≥ 0.116 stable points w/anti-phase, Figs. 3(a) and 3(b) synchronization

δ = 0.10 P-1 orbits w/anti-phase, Fig. 3(c)

0.095 P-2 orbits w/anti-phase, Fig. 3(d)

k < kc 0.091 P-4 orbits w/anti-phase, Fig. 3(e) asynchronization

0.089 spiral attrs. w/anti-phase, Fig. 3(f)

0.05 D. S. attrs. w/anti-phase, Fig. 3(g)

The above said quenching phenomenon and bifur-cation process can also be demonstrated by the bi-furcation diagram with respect to coupling factor k,as shown in Figs. 4(a) and 4(b). One can see thatthere exists a period-doubling route to chaos anda period-3 window, similar to those observed in an

uncoupled Chua’s circuit. Note from Fig. 4(a) thatthe oscillation stopped at about k = 0.116.

However, the quenching phenomenon and thebifurcation process mentioned above will not ap-pear in the unidirectionally-coupled system (13).Instead, for all k < kc the system operates in an

Uncertainty in Chaos Synchronization 1731

(a)

(b)

Fig. 4. (a) Bifurcation diagram of x for the system (10) with respect to coupling factor k = kj (j = x, y, z), correspondingto the performance presented in Fig. 3. (b) Blow-up of inset A in (a).

asynchronization mode; otherwise, the synchroniza-

tion is achievable.

The above dynamical behaviors due to the op-

posite initial conditions can be summarized and

tabulated in Table 1.

3.3. Otherwise

The error σx = |x(ta) − x(ta)|, ta > 0 was used asthe criterion to assess the states in the systems (10)and (13), and the process previous to ta was ignoredas the transient.

1732 G.-Q. Zhong et al.

For the demonstration, a set of initial condi-tions {x(0), y(0), z(0)} were chosen and fixed forthe subsystem {x, y, z}, and the initial conditionsfor another subsystem {x, y, z} were selected by useof the following simple rule:

x(0) = [x(0)−∆x(0); x(0) + ∆x(0)]

y(0) = [y(0)−∆y(0); y(0) + ∆y(0)]

z(0) = z(0) or − z(0) ,

(16)

or

y(0) = [y(0)−∆y(0); y(0) + ∆y(0)]

z(0) = [z(0) −∆z(0); z(0) + ∆z(0)]

x(0) = x(0) or − x(0) .

(17)

The dynamics of mutually-coupled two identi-cal Chua’s circuits (10) with following distinct cou-pling approaches, namely, full and partial couplingrespectively, in the case of k > kc, or k′z < kz < k′′zfor the system with z only coupled, were exam-ined. The results are presented in Figs. 5(a)–5(f).

The white dots in these figures correspond to initialconditions attracted to the invariant submanifold{x, y, z, x, y, z : x = x, y = y, z = z} in the phasespace R6 of (10), i.e. the two Chua’s circuits operatein a synchronization regime, while the black onescorrespond to an asynchronization regime. Thosegray or deep gray dots correspond to initial condi-tions for which the trajectories of the two coupledsubsystems diverge (unbounded) synchronously orasynchronously. In practice, the diverging trajec-tory becomes a large limit cycle due to the eventualpassivity in physical systems.

It can be noted, from the figures, that a varietyof riddled basins of attraction, due to the sensitivedependence on initial conditions of chaos synchro-nization, were observed in the systems with distinctcoupling forms.

(i) Case of full coupling. Figure 5(a) shows the be-havior in x(0) versus y(0) plane for the full mutualcoupled Chua’s circuits when kx = ky = kz = 1.5

(a) (b)

Fig. 5. Behavior showing the dependence of synchronization on initial conditions in two mutually-coupled identical Chua’scircuits. Parameter: α = 10, β = 14.87, a = −1.27, b = −0.68. The criterion is ∆x = |x(ta) − x(ta)| = 10−6, ta is thenormalized time. The synchronization regions are in white, the asynchronization ones are in black, while the gray regionsand deep gray regions are for the cases that the trajectories of two coupled subsystems diverge (unbounded) synchronouslyand asynchronously, respectively. (a) Behavior in x(0) − y(0) plane when kx = ky = kz = 1.5 and x(0) = 0.2, y(0) = 0.1,z(0) = 0.01, ∆x(0) = ∆y(0) = 0.6, z(0) = −z(0), ta = 5. (b) Behavior in x(0) − y(0) plane when kx = 0, ky = kz = 0.6and x(0) = 0.01, y(0) = 0.01, z(0) = −0.01, ∆x(0) = ∆y(0) = 0.6, z(0) = −z(0), ta = 50. (c) Behavior in y(0)− z(0) planewhen kx = 0, ky = kz = 0.6 and x(0) = 0.01, y(0) = 0.01, z(0) = −0.01, x(0) = −0.2, ∆y(0) = 0.6, ∆z(0) = 0.06, ta = 50.(d) Behavior in x(0)− y(0) plane when kx = 8, ky = kz = 0 and x(0) = 0.1, y(0) = 0.1, z(0) = −0.01, ∆x(0) = ∆y(0) = 0.6,z(0) = −z(0), ta = 50. (e) Behavior in x(0)−y(0) plane when kx = kz = 0, ky = 5 and x(0) = 0.01, y(0) = 0.01, z(0) = −0.01,∆x(0) = ∆y(0) = 0.6, z(0) = −z(0), ta = 10. (f) Behavior in x(0)− y(0) plane when kx = ky = 0, kz = 0.52 and x(0) = 0.01,y(0) = 0.01, z(0) = 0.001, ∆x(0) = ∆y(0) = 0.6, z(0) = z(0), ta = 50.

Uncertainty in Chaos Synchronization 1733

(c) (d)

(e) (f)

Fig. 5. (Continued )

and x(0) = 0.2, y(0) = 0.1, z(0) = 0.01, ∆x(0) =∆y(0) = 0.6, z(0) = −z(0). It can be noted that thebasin of attraction is riddled. Regions attracted tothe invariant submanifold {x, y, z, x, y, z : x = x,y = y, z = z} in the phase space R6 of (10)actually contain regions attracted to another at-tractor, which corresponds to an asynchronizationregime. Hence, the riddling can lead to the loss ofsynchronization.

(ii) Case of partial coupling. For the system of twoidentical Chua’s circuits mutually-coupled via vari-ables y and z, there exist some regions in x(0) versus

y(0) plane, in which the trajectories of the two sub-systems diverge synchronously. A positive measureset of points goes to infinity. In other words, thebasin of attraction has globally riddling property,and thus is a fat fractal [Kapitaniak et al., 1998],as shown in Fig. 5(b), where kx = 0, ky = kz = 0.6and x(0) = y(0) = 0.01, z(0) = −0.01, ∆x(0) =∆y(0) = 0.6, z(0) = −z(0).

To explore the sensitivity to initial state z(0),a basin of attraction for the initial conditions in aslice x = −0.2 of the phase space R6 is depictedin y(0) versus z(0) plane, as shown in Fig. 5(c). It

1734 G.-Q. Zhong et al.

may seem that the behaviors in the system are stat-istically less sensitive to the initial state of the vari-able z.

(iii) Case of one variable coupling. We examinedthe system (10) in the cases of kx > 0 only, ky > 0only, and kz > 0 only, respectively. The riddledbasins of attraction are presented in Figs. 5(d)–5(f).One can see, from Fig. 5(d), that there is a valley-like area in x(0) versus y(0) plane for x only coupledsystem, where the synchronization state is almostindependent of the initial conditions x(0), while thebehavior shown in Fig. 5(e) for the y only coupledsystem is chaotic.

The basins of attraction shown in Fig. 5(f) forthe z only coupled system is strongly riddling. Mostareas of the x(0) versus y(0) plane is irregularly oc-cupied by deep gray dots and black dots in this case.It indicates that the synchronizing state in the sys-tem is extremely sensitive to the initial conditionsof the system.

It should be noted that the detailed riddledcharacteristics of the picture shown in Figs. 5(a)–5(f) will be changed if different initial conditionsare chosen at random, or if different computationswith different round-off algorithms and precisionare used.

As for the unidirectionally-coupled system (13),similar results to those described above have beenobtained, even though the patterns in the two pa-rameter planes may be different. However, there areno initial conditions which give rise to the trajecto-ries of the subsystems diverging in a synchronous orasynchronous manner; namely, there exists no glob-ally riddled basins of attraction for unidirectionally-coupled systems.

4. Concluding Remarks

In this work, using two identical Chua’s circuitscoupled as the framework, we have investigated theprocess of synchronization both in mutual and uni-directional coupling forms. The results have shownthat the behaviors of the coupled chaotic systemsare strongly affected by the initial states and thecoupling strength. This effect gives rise to the rid-dling of basins of attraction and, consequently, theoccurrence of on–off intermittency in chaos synchro-nization, which is closely related to the robustnessof synchronization state.

The observations suggest that the synchroniza-tion state in x only coupled system is the most

robust, whereas the z coupling approach shouldnot be recommended for the purpose of chaossynchronization.

On the other hand, the findings also revealsome interesting phenomena on chaos synchroniza-tion, namely, in-phase synchronization, anti-phasesynchronization, and oscillation-quenching, inducedby the sensitive dependence on initial conditions.We referred to the phenomena as the uncertaintyof chaos synchronization. Because of their uniquefeatures, these behaviors may have potential appli-cations in secure communication, control and anti-control of chaos.

Acknowledgments

The authors are grateful to Dr. Guanrong Chen forthe helpful discussions with him. This paper waspartially supported by the strategic grant projectnumber: 7000956 of City University of Hong Kong.

ReferencesAfraimovich, V. S., Verichev, N. N. & Rabinovich, M. I.

[1986] “Stochastic synchronization of oscillations indissipative systems,” Izv. VUZ. Rasiofiz. RPQAEC.29, 795–803.

Ashwin, P., Buescu, J. & Stewart, I. [1994] “Bubbling ofattractors and synchronization of chaotic oscillators,”Phys. Lett. A193, 126–139.

Astakhov, V., Shabunin, A., Kapitaniak, T. &Anishchenko, V. [1997] “Loss of chaos synchronizationthrough the sequence of bifurcations of saddle periodicorbits,” Phys. Rev. Lett. 79(6), 1014–1017.

Chua, L. O., Itoh, M., Kocarev, Lj. & Eckert, K. [1993]“Chaos synchronization in Chua’s circuit,” J. CircuitsSyst. Comput. 3(1), 93–108.

Chua, L. O. [1994] “Chua’s circuit: An overview ten yearslater,” J. Circuits Syst. Comput. 4(2), 117–159.

Chua, L. O., Yang, T., Zhong, G. Q. & Wu, C. W. [1996]“Adaptive synchronization of Chua’s oscillators,” Int.J. Bifurcation and Chaos 6(1), 189–201.

Kapitaniak, T., Chua, L. O. & Zhong, G. Q. [1994] “Ex-perimental synchronization of chaos using continuouscontrol,” Int. J. Bifurcation and Chaos 4(2), 483–488.

Kapitaniak, T., Maistrenko, Yu., Stefanski, A. &Brindley, J. [1998] “Bifurcations from locally to glob-ally riddled basins,” Phys. Rev. E57(6), 6253–6256.

Kennedy, M. P. [1992] “Robust op amp realization ofChua’s circuit,” Frequenz 46, 66–80.

Kim, C.-M. [1997] “Mechanism of chaos synchroniza-tion and on–off intermittency,” Phys. Rev. E56(3),3697–3700.

Maistrenko, Yu. L., Maistrenko, V. L., Popovich, A.

Uncertainty in Chaos Synchronization 1735

& Mosekilde, E. [1998] “Role of the absorbing areain chaotic synchronization,” Phys. Rev. Lett. 80(8),1638–1641.

Nakajima, H. & Ueda, Y. [1997] “A simple model of a rid-dled basin in chaotic synchronization,” Electron. Com-mun. Jpn. Part 3, 80(11), 112–119.

Ott, E., Sommerer, J. C., Alexander, J. C., Kan, I.& Yorke, L. A. [1993] “Scaling behavior of chaoticsystems with riddled basins,” Phys. Rev. Lett. 71,4134–4137.

Ott, E. & Sommerer, J. C. [1994] “Blowout bifurcation:The occurrence of riddled basins and on–off intermit-tency,” Phys. Lett. A188, 39–47.

Pecora, L. M. & Carroll, T. L. [1990] “Synchronizationin chaotic systems,” Phys. Rev. Lett. 64, 821–824.

Pecora, L. M. & Carroll, T. L. [1991] “Driving systemswith chaotic signals,” Phys. Rev. A44(4), 2374–2383.

Verhulst, F. [1990, 1996] Nonlinear Differential Equa-tions and Dynamical Systems, 2nd edition (Springer-Verlag, Berlin–Heidelberg).

Wu, C. W. & Chua, L. O. [1993] “A simple way tosynchronize chaotic systems with applications to se-cure communication systems,” Int. J. Bifurcation andChaos 3(6), 1619–1627.

Wu, C. W., Zhong, G. Q. & Chua, L. O. [1996] “Synchro-nizing nonautonomous chaotic systems without phase-locking,” J. Circuits Syst. Comput. 6(3), 227–241.

Zhong, G. Q. & Ayrom, F. [1985] “Experimental confir-mation of chaos from Chua’s circuit,” Int. J. CircuitTh. Appl. 13(1), 93–98.