Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 265–269 c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect∗ LI Xiao-Wen and ZHENG Zhi-Gang Department of Physics and the Beijing-Hong-Kong-Singapore Joint Center for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China (Received April 30, 2006) Abstract Phase synchronization of two linearly coupled Rossler oscillators with parameter misfits is explored. It is found that depending on parameter mismatches, the synchronization of phases exhibits different manners. The synchronization regime can be divided into three regimes. For small mismatches, the amplitude-insensitive regime gives the phase-dominant synchronization; When the parameter misfit increases, the amplitudes and phases of oscillators are correlated, and the amplitudes will dominate the synchronous dynamics for very large mismatches. The lag time among phases exhibits a power law when phase synchronization is achieved. PACS numbers: 05.45.Xt Key words: phase synchronization, amplitude effect Synchronization is both a universal cooperative behavior and a fundamental mechanism in nature, and it has been extensively studied in relating to numerous phenomena in physics, chemistry, and biology.[1,2] Recently, chaos synchronization becomes one of the central topics in nonlinear science.[3−5] It has been found that though individual chaotic subsystems exhibit the exponential instability on initial conditions, they can follow each other and become synchronized when they are coupled together. Chaos synchronization was recently shown for different degrees such as complete synchronization,[6] generalized synchronization,[7] phase synchronization,[8] and so on.[5] These are all extensions of the traditional concept of synchronization of coupled periodic oscillators. Most interestingly, a closest analog in all these degrees of chaos synchronizations is the phase synchronization (PS), which has been numerically and experimentally observed in a variety of systems.[5] PS reveals an intrinsic order in collective behaviors of interacting chaotic oscillators. Much attention has been paid to the phase dynamics of chaotic oscillators. However, different degrees of freedom of chaotic motions are always related to each other. Therefore the amplitude may strongly affect the behavior of the phase. When chaotic oscillations are coupled to each other, the collective dynamics of phases may strongly depend on their amplitudes. The amplitude effect of coupled limit cycles was extensively investigated,[9,10] and various phenomena were revealed, such as amplitude death, complex spatiotemporal dynamics, and so on. In this paper, it is our task to explore the role of amplitudes of coupled chaotic oscillators relating to the synchronization of phases. We choose the Rossler system as our working model, and we study the phase synchronization behavior of two coupled Rossler units. We find that the phase ∗ The synchronous dynamics can be divided into three different regimes, i.e., the amplitude-insensitive regime, the amplitude-correlated regime, and the amplitude-dominant regime, depending on the misfit of parameters of two oscillators. This implies the important role played by the amplitude. The discussions and results obtained in this paper can be useful in understanding the collective phase behavior of chaotic motion. Let us start from the equations of motion of two linearly coupled Rossler oscillators, which can be written as ẋ1,2 = −ω1,2 y1,2 − z1,2 + ε(x2,1 − x1,2 ) , ẏ1,2 = ω1,2 x1.2 + ay1,2 , ż1,2 = f + z1,2 (x1,2 − c) , (1) where we adopt a = 0.165, f = 0.2, and c = 10 in the present work. The parameters ω1,2 = ω0 ± ∆ are different for different oscillators, where ∆ measures the mismatch of two oscillators.  is the coupling strength. For general chaotic systems, the definition of a phase is difficult due to the multiple centers of rotation.[11] A typical definition of a time series s(t) is to introduce a principal integral,[1] Z +∞ 1 s(τ ) s̃(t) = P dτ , (2) π t −τ −∞ where P means that the integral is taken in the sense of the Cauchy principal value. A phase θ(t) of this signal s(t) thus can be defined by using ψ(t) = s(t) + i s̃(t) = A(t) exp[iθ(t)]. For a Rossler oscillator, s(t) ≈ x(t), s̃(t) ≈ y(t) . Therefore x(t) and y(t) can be considered as ideal variables to define the phase of the oscillator by h y (t) i 1,2 θ1,2 (t) = tan−1 . (3) x1,2 (t) To study the self-organized dynamics in phases of oscillators, we may introduce the average winding number as project supported in part by National Natural Science Foundation of China under Grant Nos. 70431002 and 10575010, the FANEDD, and the TRAPOYT in Higher Education Institutions of MOE 266 LI Xiao-Wen and ZHENG Zhi-Gang the temporal average of the phase velocity, Z 1 T θ̇1,2 (t)dt . Ω1,2 = lim T →∞ T 0 (4) The synchronization between different oscillators in the sense of the time average, i.e., Ω1 = Ω2 , can be observed when  > c , although the strict phase-locking condition ∆θ̇i (t) = 0 is broken. We first are interested in the phase dynamics near the synchronous threshold. Usually for coupled limit cycles, a power law can be observed in the vicinity of the synchronous threshold,[12] ∆Ω ∝ (c − )1/2 , (5) which is a natural consequence of the saddle-node bifurcation near the critical point. For both coupled limit cycles and coupled chaotic Rossler oscillators with small parameter mismatches, it has been found that one zero Lyapunov exponent becomes negative at c .[13] However, the critical behavior of other quantities may be different. In Fig. 1(a), the synchronization bifurcation tree of two Rossler oscil- Vol. 47 lators for a small misfit ∆ = 0.02 is plotted. It can be found that when  is far from c , two averaged frequencies approach each other via the similar power law to Eq. (5). This can be clearly seen in the right panel of Fig. 1(a), where the difference ∆Ω = Ω2 − Ω1 against ω is plotted. Obviously ∆Ω ∝ (0 − )1/2 when  is far from c , where 0 6= c . This is in agreement with the case of coupled limit cycles. However, one can obviously find that the scaling law for chaotic oscillators at the synchronization threshold is different from that for coupled limit cycles. The scaling relation for chaotic systems near the critical point, as shown in Fig. 1(a), is ∆Ω ∝ (0 − )2 . This indicates that the bifurcation at the critical point is not the saddle-node type. The reason for this typical difference is that chaoticity plays an important role in governing the phase dynamics. Chaotic motion leads to disorderness of the phase evolution. Chaos plays a role of “noise”. It has been shown that for coupled limit cycles with the action of noises, the scaling behavior near the synchronous threshold deviates from the saddle-node law.[1] Fig. 1 The synchronization bifurcation trees Ω1,2 ∼  for different parameter mismatches. The right panels are corresponding differences ∆Ω = Ω2 − Ω1 against the coupling. (a) ∆ = 0.02, (b) ∆ = 0.06, and (c) ∆ = 0.15. Different behaviors near the synchronization threshold can be found for different ∆. When the parameter mismatch is increased, one no longer observes the above scaling law. In Fig. 1(b), the averaged frequency vs. the coupling strength near c is plotted. It is interesting to find the following two phenomena. (i) The oscillator with a higher winding number dominates the other oscillator, i.e., Ω1 keeps nearly constant when  increases; (ii) Ω2 approaches Ω1 in a stepwise manner, and several steps can be observed, as indicated in Fig. 1(b) with dashed lines. This is also clearly indicated in the right panel of Fig. 1(b) by plotting the difference of winding numbers. These steps suggest that the transition to phase synchronization for moderate parameter mismatches is not continuous. It has been shown that for coupled chaotic oscillators, some metastable states can exist in a regime of coupling.[14] The steps observed in Fig. 1(b) are just metastable states, and they can coexist. When the mismatch is very large, the oscillator with higher winding number still dominates, while the winding number of the other oscillator approaches it continuously. It is interesting to find from the right panel of Fig. 1(c) that ∆Ω ∝ (0 − )1/2 , which is the same as that found for coupled limit cycles. However, the synchronous dynamics No. 2 Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect of these two types of systems are completely different. For two coupled limit cycles, the averaged frequencies of two oscillators approach each other in a symmetric way, and one has the Ω1,2 − Ωc ∝ ±(c − )1/2 , where Ωc is the winding frequency at the critical c . For chaotic systems, one oscillator remains a constant frequency of oscillation Ω1 = const., while the other one approaches it with a power law Ω2 −Ω1 ∝ −(0 −)1/2 . In this case, the chaotic oscillator with faster oscillation may slave the slower one, which is analogous to the drive-response limit cycles: θ̇1 = ω1 , (6) θ̇2 = ω2 +  sin(θ2 − θ1 ) . (7) By introducing the phase difference ∆θ = θ2 − θ1 , one gets ∆θ̇ = (ω2 − ω1 ) +  sin(∆θ). Therefore the dynamical behavior of the second limit cycle in the vicinity of c = |ω2 − ω1 | where phase locking happens is the typical saddle-node type. This relation can be understood as the dominant role of phases. By introducing phase and amplitude variables, equations (1) can be expressed by coupled phase and amplitude equations. This can implemented by inserting xi (t) = Ai (t) cos[θi (t)], yi (t) = Ai (t) sin[θi (t)] into Eqs. (1): z1,2 θ̇1,2 = ω1,2 + a sin θ1,2 cos θ1,2 + sin θ1,2 A1,2 − 2K sin θ1,2 (A2,1 cos θ2,1 − A1,2 cos θ1,2 ) , A1,2 Ȧ1,2 = aA1,2 sin2 θ1,2 − z1,2 cos θ1,2 + 2K cos θ1,2 (A2,1 cos θ2,1 − A1,2 cos θ1,2 ) , ż1,2 = f − cz1,2 + A1,2 z1,2 cos θ1,2 . The above discussions reveal the significant difference for different parameter misfits. In Fig. 2, we compute the dependence of the average winding frequencies on both the parameter mismatch ∆ and the coupling strength . The result is given by the contour format, where the grey area represents larger values of ∆Ω, and the white area denotes the synchronous regime. Therefore the -∆ plane is divided into the non-synchronous and synchronous regimes by the critical line c = c (∆). It is very interesting that the critical line is composed of three different segments. When ∆ < ∆1 ≈ 0.047, the critical line is a straight line and can be written as c = 2∆ . (8) (9) The amplitudes A1,2 (t) are slow variables as compared with the phases θ1,2 (t) and can be adiabatically eliminated. Moreover, the z component of Rossler oscillators keeps nearly 0 for most of the time. Therefore one can obtain the evolution of the phase-difference ∆θ = θ2 − θ1 , which is just the equation of the dc-force driven overdamped pendulum:   A2 A1  ∆θ̇ = 2∆ − + sin(∆θ) . (10) 2 A1 A2 Thus the critical coupling is c = Fig. 2 The phase diagram of synchronous and nonsynchronous regimes in the ε-∆ plane. The boundary between these two regimes shows different behaviors for different ∆. The AIR critical line is c = 2∆, the ACR line is c ≈ 0.055 + ∆, and the ADR line is constant:  = 0 ≈ 0.155. 267 4∆A1 A2 . A21 + A22 (11) For small mismatches, one has A1 ≈ A2 , thus the critical coupling c ' 2∆. This is just the result given by Fig. 2. This agreement indicates that amplitudes of oscillators are not crucial in governing the phase dynamics, i.e., the correlation between two variables is very weak. Therefore we call this synchronous area the amplitudeinsensitive regime (AIR). When ∆ increases, another kind of behavior of the critical line can be observed from Fig. 2, which can be well fitted by the following straight line in the range 0.047 < ∆ < 0.1: c ≈ 0.055 + ∆ . (12) This behavior shows that the phase dynamics can be strongly affected by amplitudes, i.e., the correlation between amplitudes and phases cannot be neglected. We call this segment the amplitude-correlated regime (ACR). In this regime, as shown in Fig. 1(b), two oscillators become phase synchronized in a discontinuous way. Due to the strong correlation between phase and amplitudes, the dynamics of the coupled system becomes less chaotic when phase synchronization is achieved. Typically, the system may experience a transition from chaos to quasiperiodicity and periodicity. This is shown in Fig. 3, where the bifurcation diagram of two coupled Rossler oscillators with 268 LI Xiao-Wen and ZHENG Zhi-Gang ∆ = 0.05 is plotted against the coupling strength . It can be found that with the increase of the coupling strength, coupled systems exhibit a bifurcation to period-1 motion, where phase synchronization has been achieved. Then a period-doubling bifurcation sequence to chaotic motion can be found when  is further increased. This behavior implies the suppression of chaos in the presence of larger parameter mismatches. One cannot observe this phenomenon in the AIR. Vol. 47 when the phases are locked. This interesting scenario is in agreement with those found in coupled limit cycles with amplitude variables.[10] Fig. 4 The rescaled averaged amplitude difference ∆A = hA2 i − hA1 i against the coupling  for misfits ∆ = 0.02 (diamond), 0.05 (triangle), 0.1 (square), and 0.15 (circle). Fig. 3 The bifurcation diagram χ1 - of two coupled Rossler oscillators with ∆ = 0.05. The transition from chaos to quasiperiodicity and periodicity can be observed with the increase of the coupling . The period-doubling to chaotic motion can be found in the phase-synchronous region. For a very large parameter mismatch ∆, the synchronous behavior is fully dominated by the amplitude. Here one may find from Fig. 2(c) that the critical coupling  remains unchanged for different misfits, i.e.,  = 0 ≈ 0.155. We call this segment the amplitude-dominated regime (ADR). In this regime, the oscillator with the lower winding frequency may be slaved by the faster one. Dynamics in the ADR is similar to that in the ACR, while an oscillation death can be found when phase synchronization is achieved. One can also study the changes in the amplitudes with increasing the coupling strength . In Fig. 4, the temporal averaged amplitude differences ∆A = hA2 i − hA1 i (rescaled by the largest average amplitude) against the coupling  for different misfits are plotted. It can be found that when the misfit ∆ is small (see lines corresponding to ∆ = 0.02 and 0.05 in Fig. 4), ∆A keeps very small, i.e., the amplitudes of both oscillators are weakly correlated. When ∆ increases, as seen from lines in Fig. 4 for ∆ = 0.1 and 0.15, ∆A becomes rather high at the onset of phase synchronization, indicating a strong correlation between phases and amplitudes. The peaks also show that the process of phase synchronization is accompanied by the inhomogeneity process of amplitudes, i.e., the amplitudes of two oscillators become distinctively different Fig. 5 The relation between the delay time τ and the coupling  when  > c . The scaling relation τ ∝ −1 can be observed. Finally, we would like to mention that when two oscillators become phase-synchronized, their phases may satisfy a lag relation, i.e., θ2 (t + τ ) = θ1 (t). The relation between the delay time τ and the coupling  when  > c is shown in Fig. 5 for different misfits ∆. It can be found that for misfits in both the AIR, ACR and the ADR the delay time satisfies the same scaling law: τ ∝ −1 . This result indicates that although amplitudes strongly affect the synchronization threshold and the behavior near the threshold, phases between oscillators can still build a good lag relation. To summarize, in this paper we explored the effect of amplitudes on phase synchronization dynamics by using coupled Rossler oscillators. It is found that the phase No. 2 Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect synchronous dynamics can be divided into three different regimes depending on parameter misfits: the AIR, the ACR, and the ADR, depending on the misfit of parameters. This implies the important role played by the References [1] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge (2001). [2] Z. Zheng, Spatiotemporal Dynamics and Collective Behaviors in Coupled Nonlinear Systems, Higher Education Press, Beijing (2004). [3] S. Boccaletti, C. Grebogi, Y.C. Lai, H. Mancini, and D.Maza, Phys. Rep. 329 (2000) 103. [4] S. Boccaletti, J. Kurths, G. Osipov, et al., Phys. Rep. 366 (2002) 1. [5] G. Hu, J. Xiao, and Z. 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