Physics Letters A 354 (2006) 119–125

www.elsevier.com/locate/pla

Antisynchronization in coupled chaotic oscillators

Weiqing Liu a,b, Xiaolan Qian a, Junzhong Yang a, Jinghua Xiao a,∗

a School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of Chinab School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, People’s Republic of China

Received 12 December 2005; received in revised form 9 January 2006; accepted 11 January 2006

Available online 30 January 2006

Communicated by A.R. Bishop

Abstract

The dynamics behaviors of coupled Saito’s oscillators are investigated intensively. Various phenomena are explored such as antisynchronization(AS), hysteresis, bistability, riddled basin and coexistence of chaotic AS and lag AS. With increasing coupling intensity, the coupled chaoticoscillators undergo a transition from phase synchronization to AS. With decreasing coupling intensity, they will transit from AS to antiphasesynchronization. The necessary condition for AS is explored and the stability of AS is studied.© 2006 Elsevier B.V. All rights reserved.

PACS: 05.45.-a; 05.45.Xt

Keywords: Antisynchronization; Saito’s oscillator; Hysteresis; Bistability; Riddled basin

1. Introduction

The synchronous phenomena of coupled identical chaoticsystems have received a great deal of interest since the pio-neering works by Fujisaka and Yamada [1], Afraimovich [2],and Pecora and Carroll [3]. Various synchronization phenom-ena are being reported as complete synchronization (CS) [3,4],phase synchronization (PS) [5], lag synchronization (LS) [6,7],generalized synchronization (GS) [8–10]. Among these syn-chronizations, CS is the strongest in the degree of correlationand describes the interaction of two identical systems, leadingto their trajectories remaining identical in the course of tempo-ral evolution, i.e., x1(t) = x2(t) as t → ∞. PS describes thatthe mismatch of the phase is locked within 2π of nonidenti-cal chaotic oscillators, whereas their amplitudes may remainchaotic and uncorrelated. LS has been proposed as the coin-cidence of the states of two coupled systems in which one ofthe system is delayed by a finite time τ , i.e., x1(t) = x2(t + τ).GS, as introduced for drive-response systems, is defined as thepresence of a functional relationship between the states of the

* Corresponding author.E-mail address: jhxiao@bupt.edu.cn (J. Xiao).

0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.01.039

responser and driver, i.e., x1(t) = F(x2(t)). However, antisyn-chronization (AS) is also an interesting phenomenon in coupledoscillators since the first observation of synchronization be-tween two pendulum clocks by Huygens in 17th century. ASis also observed depending on initial condition in the coupledLorenz system [11], Chua circuit [12], and coupled map [13].AS phenomena have even been observed experimentally in thecontext of self-synchronization, e.g., in salt–water oscillators[14] and laser systems [15]. However, AS is somewhat differ-ent from antiphase synchronization (APS) [16], since AS getsonly when the summation of two signals converge to zero, whileAPS means π phase-delay PS.

The main goal of this work is to investigate the AS phenom-ena occurred in a system of two diffusively coupled identicalSaito’s oscillators [19]. As couple switches on, the system un-dergoes a transition from PS to AS with increasing couple in-tensity, while transition from AS to APS with decreasing coupleintensity. Hysteresis, bistability, and first-order transition be-tween these two branches are observed. Moreover, coexistenceof chaotic AS and lag AS, the riddled basin [17,18] can be ob-served in the AS state. General theory for AS in coupled chaoticoscillators is offered as the necessary condition and the criteriafor the stability of AS. The Letter is organized as follows: in

120 W. Liu et al. / Physics Letters A 354 (2006) 119–125

Section 2, we give the necessary condition and the criteria forthe stability of AS. Section 3 explores the AS phenomena inthe coupled Saito’s oscillators. More interesting phenomena ashysteresis, bistability, riddled basin and coexistence of AS andlag AS are discussed in Section 4. Finally the conclusion anddiscussion are offered.

2. Necessary condition and stability analysis of AS

The model of two diffusively coupled identical oscillators isdescribed as

X1 = f(X1) + εΓ (X2 − X1),

(1)X2 = f(X2) + εΓ (X1 − X2),

where Xi ∈ RN (i = 1,2), f :RN → RN is nonlinear and ca-pable of exhibiting rich dynamic such as chaos, ε is couplingstrength, and Γ describes coupling scheme. If Eq. (1) pos-sesses the property of AS, and there exists an anti-synchronousmanifold (ASM), M = {X1 = −X2 = X∗}, which satisfies theequation below:

X∗ = f(X∗) − 2εΓ X∗,

(2)−X∗ = f(−X∗) + 2εΓ X∗.

To keep the compatibility between two equations in Eq. (2),the necessary condition for AS can be given as: the nonlinearfunction f(x) is an odd function of x, i.e., f(−x) = −f(x). How-ever, the noticeable thing is that the AS state is not the solutionof isolated oscillators any more, which is quite different withthe state of complete synchronous manifold. The stability ofthe state of AS can be determined by letting Xi = X∗ + ηi , andlinearizing Eq. (1) about X∗(t). This leads to

(3)

(η1η2

)=

(Df(X∗) 0

0 Df(−X∗)

)(η1η2

)+ εΓ B

(η1η2

),

where Df(X∗) is the Jacobian of f on X∗, and matrix

B =(−1 1

1 −1

).

Since f(x) possesses odd function property, we haveDf(−X∗) = Df(X∗) and linear stability equations can be diag-onalized by expanding into the eigenvectors of B , η = ∑

δiφi .

Carrying this out gives δi = [Df(X∗) + ελiΓ ]δi (i = 1,2)

where λi = 0,−2 is the eigenvalues of B , which indicates thatASM coincides with the subspace spanned by the eigenvectorof B with eigenvalue λ = −2. The λ = 0 mode governs themotion transversal to ASM. This mode has Lyapunov expo-nents Λ

(0)1 � Λ

(0)2 � · · · � Λ

(0)n . Therefore AS is stable if and

only if Λ(0)1 < 0. While the dynamics on ASM is determined

by λ = −2 mode and it is possible to observe rich dynamics forthe state of AS no matter how the isolated oscillator behaves.

3. AS in coupled Saito’s oscillators

In our discussion, we let Xi = (xi, yi, zi) (i = 1,2), and fo-cus on the Saito’s oscillator, which gained wide popularity asa classical hysteretic double-screw chaotic oscillator, and con-tains a nonmonotone current-controlled hysteresis resistor R,and inductor L, a capacitor C and a linear current-controllednegative resistor NR , in addition to the transit inductance L0(Fig. 1(a)) modeled and analyzed in detail [19] and was usedas the units of State Controlled Cellular Neural Network (SC-CNN), which can generate the hyperchaotic signals for the se-cure communication [20]. It is modified by Elwakil [21] andmodeled as

x = μ(y − x − z),

y = (1 − 1/α)y − x,

z = β(x − f (z)

),

(4)f (z) = α1z + α2(|z + m| − |z − m|).

Oscillator described by Eq. (4) has chaotic attractor as shownin Fig. 1(b) with parameter μ = 1/2, α = 2.5, α1 = 10, α2 =−7.5, β = 5,m = 1. Since it owns odd function property, ASis possible according to the analysis in Section 2. Let couplingscheme to be

Γ =(0 0 0

0 0 01 0 0

).

Fig. 2(a) gives the bifurcation of x1 + x2 versus the couplingconstant, the dots in the graph is the local maximum of x1 + x2and initial conditions are got from the state of last parameteradding proper noise. The system gets to AS as x1 +x2 converge

Fig. 1. (a) The circuit model of Saito’s oscillator. (b) The chaotic attractor of single modified Saito’s oscillator described in Eq. (4).

W. Liu et al. / Physics Letters A 354 (2006) 119–125 121

to zero in the area of ε = 0.244–0.264 and ε = 0.815–0.94. Andthe result is confirmed by the largest conditional Lyapunov ex-ponent of mode λ = 0 in Fig. 2(b), since in the area of ε =0.244–0.264 and ε = 0.815–0.94 the Lyapunov exponent Λ(0)

is negative, which indicates the stable AS state. Λ(−2) guaran-tees the state of ASM, and ASM is chaotic only if Λ(−2) > 0,periodical if Λ(−2) = 0. As shown in Fig. 2(b), Λ(−2) transits

from zero to positive with negative Λ(0) in the parameter areaof ε = 0.244–0.264. Therefore, AS state can be periodical orchaotic as shown in Fig. 3(a), (b), respectively. To explore theAS with time series of two subsystems, we define variable

(5)〈s〉 = 1

Tlim

T →∞

T∫0

(x1 + x2)2 + (y1 + y2)

2 + (z1 + z2)2 dt,

Fig. 2. (a) Bifurcation of the variable x1 +x2 versus couple constant ε, when the coupled system gets to AS, x1 +x2 will converge to zero. (b) The largest conditionalLyapunov exponents for the λ = 0 (Λ(0) solid line) and λ = −2 (Λ(−2) dashed line) modes against ε are plotted.

Fig. 3. Time series of x1 and x2: (a) time series of periodical AS for ε = 0.25; (b) time series of chaotic AS for ε = 0.84.

122 W. Liu et al. / Physics Letters A 354 (2006) 119–125

Fig. 4. (a) 〈s〉 versus ε are plotted as ε increases from zero (solid line) and decreases from 1 (dotted line), respectively. Multi-state can be achieved with parameterε = 0.19–0.27,0.76–0.85; (b) enlargement of left branch of multi-state parameter; (c) enlargement of the right branch of multi-state one. The dotted vertical linesin (b) and (c) are the traces of hopping between three states given random initial condition for various ε. 〈s〉 is calculated with integral constant T = 105. Transitionfrom AS (〈s〉 = 0) to non-AS (〈s〉 ≈ 1) is obvious at ε = 0.245 and ε = 0.815.

〈s〉 = 0 means AS, and 〈s〉 > 0 when AS losts its stability. InFig. 4(a) 〈s〉 versus ε is plotted and 〈s〉 = 0 is reached in twointervals of parameter ε = 0.244–0.264 and ε = 0.815–0.94,which indicates that AS exists in the parameter interval men-tioned above. Moreover, AS state in diffusively coupled Saito’soscillators is not sensitive to the parameter of single oscillator,AS can be also obtained at α = 2.4 and α = 2.3 in Eq. (4) (de-tailed results are not shown here).

As mentioned above, AS reported in this Letter relies onthe condition of the mirror symmetry of two coupled systemsand odd property of subsystem. Generally, the subsystem hastwo centrosymmetric branches of attractors. With proper cou-pling constant, the mutual interactions between two subsystemswill keep each other staying on the opposite branch. Moreover,The mechanism of AS is similar to that of CS, that is, AS (CS)achieved only when the coupled systems converge to the ASM(SM) even in noise condition.

4. Coexistence of AS and PS with riddled basinphenomenon

However, besides couple induced AS, various interestingphenomena occur such as the transition from PS to AS andfrom AS to APS, hysteresis, bistability, riddled basin in two in-tervals of parameter ε = 0.244–0.264 and ε = 0.815–0.94. Todescribe those phenomena, 〈s〉 is calculated according to quasi-stationary way by taking the last state added proper noise as the

initial condition of the next parameter with ε increasing fromzero and decreasing from 1, respectively. Let us firstly fix ourattention to Fig. 4(b). By quasi-stationary way, the system willstay in the upper branch with 〈s〉 ≈ 10 as ε decreases from 0.28to zero, and switch to the middle branch with 〈s〉 ≈ 6 as ε in-creases from zero. The system will drop down to the lowestbranch (〈s〉 = 0) if larger perturbation is added at ε = 0.265.The lower branch will transit from the AS state (〈s〉 = 0) toAPS state (〈s〉 ≈ 1) at ε = 0.245 as ε decreases. However, ifrandom initial condition is given for each parameter ε, the sys-tem will hop between those three states, i.e., it hops between thestates of 〈s〉 = 0, 〈s〉 ≈ 6,10 in the interval of ε = 0.245–0.265and transition to the situation of hopping between the states of〈s〉 ≈ 1,6,10 at ε = 0.245. The dotted vertical lines indicate thehopping traces when the coupled system is given random initialvalue for each ε as ε increases. To explore the three states indetail, we present the time series of x1 and x2 when the systemis on the states of 〈s〉 ≈ 10,6,1, respectively with parameterε = 0.208. As shown in Fig. 5(a)–(c), the state of 〈s〉 ≈ 10,6 arePS with different time delays τc (τc = T/12,2T/7) and 〈s〉 ≈ 1is in the state of APS (with T/2 time shift), where T is the pe-riod of x1. Moreover, different time shifts τc result to differentsummations of x1 + x2 as shown in Fig. 6 (a)–(c).

Now let us focus on Fig. 4(c). Similar phenomena realizedas that in Fig. 4(b). As the parameter ε increases, the sys-tem will transit from 〈s〉 ≈ 10 state to AS (〈s〉 = 0) state atε = 0.85. However it will transition from AS (〈s〉 = 0) state

W. Liu et al. / Physics Letters A 354 (2006) 119–125 123

Fig. 5. Time series of x1 (solid line) and x2 (dashed line) with ε = 0.208 ((a)–(c)), x1 is lag to x2 with time delay τc (time interval between the vertical solid lineand dotted line). The period of x1 is T (time interval between two vertical solid lines). (a) PS state with time delay τc = T/12 as 〈s〉 ≈ 10; (b) PS state with timedelay τc = 2T/7 as 〈s〉 ≈ 6; (c) APS state with time delay τc = T/2 as 〈s〉 ≈ 1.

Fig. 6. Time series of x + x (a)–(c) corresponding to Fig. 5 (a)–(c).

1 2

to APS (〈s〉 ≈ 1) at ε = 0.815 as ε decreases. The system isjust hopping between APS and AS given random initial con-dition in the parameter interval of ε = 0.815–0.85. Define thepossibility of AS attraction basin as P(ε) = NAS

Ntol, where NAS

denotes the number of initial conditions with which the systemdiverges to AS by random given initial value of two oscilla-

tors within 0–1, and Ntol denotes the total number of initialvalues (here Ntol = 104). Three states including non-AS state,coexistence of AS and non-AS, AS state can be described byP(ε) = 0,P (ε) ∈ (0,1) and P(ε) = 1, respectively. P(ε) ver-sus ε is plotted in Fig. 7(a). Large P(ε) indicates that thecoupled oscillators stay on AS state with more chances. Tran-

124 W. Liu et al. / Physics Letters A 354 (2006) 119–125

Fig. 7. (a) P(ε) versus ε, possibility of AS state; (b) enlarged view of the squared area in (a); (c) the attractor basin of the AS state (white dot) and PS (black dot)depend on the initial values of (X1,X2) with Xi = (xi , yi , zi ), and here we arbitrary take xi = yi = zi (i = 1,2), ε = 0.83; (d) enlarged view of the squared area in(c) fractal structure.

Fig. 8. (a) Time series of x1(t) and x2(t) with time delay τc , i.e., x1(t + τc) = −x2(t) as ε = 0.892; (b) the similarity function σ(τ) versus τ . σ(τc) = 0 means lagAS with time shift of τc (here τc = 14.5).

sition from non-AS to AS is obvious at ε = 0.814. We enlargethe parameter area of dotted box area in Fig. 7(a), as shownin Fig. 7(b), P(ε) increases dramatically at ε = 0.81416. Werandomly give initial condition, and record the initial valueas the system stays away from the ASM. PS and AS attrac-tors are effectively riddled in interval of ε = 0.245–0.265 andε = 0.815–0.95. The attractor basin shows the riddling phe-nomenon. These pictures can be supported by the section ofthe attractor basin in Fig. 7(c) and (d), where the white dots isAS attractor basin, in which fine details of a fractal structure arerevealed by the enlarged square region of Fig. 7(c) as shown inFig. 7(d). From the analysis presented above, it is obvious that

the AS state in diffusively coupled Saito’s oscillators can coex-ist with PS with different time shifts. Moreover, the AS and PSattractors are effectively riddled with fractal structure.

What’s more, the coexistence of AS and lag AS can be ob-served in the interval of ε = 0.875–0.892. In this interval, thesystem seems to keep in AS state with 〈s〉 = 0 in Fig. 4(a). How-ever, given random initial condition, the system will hop to non-AS state with rare possibility as shown by the vertical dottedline in Fig. 4(c). Which means AS also coexists with non-AS inthis interval. To explore what the non-AS state is, the time seriesof x1(t) and x2(t) are presented in Fig. 8(a), where lag AS statewith time delay of τc is obvious. Some time ago, Rosenblum

W. Liu et al. / Physics Letters A 354 (2006) 119–125 125

et al. [5] have introduced the notion of the similarity functionfor characterizing the LS as a time-averaged difference betweenthe variables x1 and x2 (with mean values being subtracted)taken with time shift τ . Here we define the similarity function

by σ 2(τ ) = 〈[x1(t+τ)+x2(t)]2〉[〈x2

1 (t)〉〈x22 (t)〉]1/2 . Then a minimum of σ 2(τ ), that is,

σ 2(τ ) = 0, indicates that there exists a time shift τc between thetwo signals x1(t) and x2(t) such that x2(t + τc) = −x1(t), i.e.,lag AS. In Fig. 8(b), the similarity function σ(τ) versus timedelay τ is presented with ε = 0.892, where σ(τc) = 0 meanslag AS with time shift of τc, here τc = 14.5.

5. Conclusion

In this Letter, we explored the dynamics of diffusively cou-pled Saito’s oscillators. AS state of the coupled system is an-alyzed theoretically and numerically. And the necessary con-dition for AS and its stability criteria are presented. Besidescouple induced AS, various interesting phenomena occur. Suchas the transition from PS to AS and from AS to APS, hys-teresis, bistability, riddled basin and coexistence of AS and lagAS. Moreover, AS state, just like CS state, exists widely onthe coupled periodical or chaotic systems, such as Chua circuit,Lorenz system, even in the coupled chaotic map. AS could alsobe observed in laser systems experimentally since the possibleequivalence between laser system and Lorenz oscillator [22].Though AS has the same mechanism as CS, the interactiontends to drag two formerly independent systems to the anti-synchronous manifold (ASM) or synchronous manifold (SM)and keep them firmly free from noise. However, the SM is righton the single oscillator, while ASM is on the modified oscillatoras described in Eq. (2). Finally, AS has potential application inpractice fields such as secure communications and digital com-munication [13].

Acknowledgements

This work was supported by the Grant Nos. 10575016,10405004 and 70431002 from Chinese Natural Science Foun-dation.

References

[1] H. Fujisaka, T. Yamada, Prog. Theor. Phys. 69 (1983) 32.[2] V.S. Afraimovich, N.N. Verichev, M.I. Rabinovich, Izv. Vyssh. Uchebn.

Zaved. Radiofiz. 29 (1986) 795.[3] L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821.[4] A.S. Pikovsky, Z. Phys. B: Condens. Matter 55 (1984) 149.[5] M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 76 (1996)

1804.[6] M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78 (1997)

4193.[7] S. Taherion, Y.C. Lai, Phys. Rev. E 59 (1999) R6247.[8] N.F. Rulkov, et al., Phys. Rev. E 51 (1995) 980.[9] K. Pyragas, Phys. Rev. E 54 (1996) R4508.

[10] H.D. Henry, I. Abarbanel, N.F. Rulkov, M. Sushchik, Phys. Rev. E 53(1996) 4528.

[11] C.M. Kim, S. Rim, W.H. Kye, Phys. Lett. A 320 (2003) 39.[12] V. Astakhov, A. Shabunin, V. Anishchenko, Int. J. Bifur. Chaos 10 (4)

(2000) 849.[13] L. Cao, Y.C. Lai, Phys. Rev. E 58 (1998) 382.[14] S. Nakata, T. Miyata, N. Ojima, K. Yoshikawa, Physica D 115 (1998) 313.[15] K. Otsuka, Y. Sato, Phys. Rev. A 54 (1996) 4464.[16] G. Hu, Y. Zhang, H.A. Cerdeira, S. Chen, Phys. Rev. Lett. 85 (2000) 3377.[17] J.C. Alexander, J.A. Yorke, I. Kan, Int. J. Bifur. Chaos 2 (1992) 795.[18] Y.C. Lai, C. Grebogi, J.A. Yorke, Phys. Rev. Lett. 77 (1996) 55.[19] T. Saito, S. Nakagawa, Philos Trans. R. Soc. 353 (1995) 47.[20] R. Caponetto, L. Fortuna, L. Occhipinti, Int. J. Neural System 13 (6)

(2003) 461.[21] A.S. Elwakil, M.P. Kennedy, IEICE Trans. Fundamentals E82-A (9)

(1999) 1769.[22] H. Haken, Phys. Lett. A 53 (1975) 77.