We investigate the interplay between phase synchronization and amplitude synchronization in nonlinear dynamical systems. It is numerically found that phase synchronization intends to be established earlier than amplitude synchronization. Nevertheless, amplitude synchronization (or the state with large correlation between the amplitudes) is crucial for the maintenance of a high correlation between two time series. A breakdown of high correlation in amplitudes will lead to a desynchronization of two time series. It is shown that these unique features are caused essentially by the Hilbert transform. This leads to a deep concern and criticism on the current usage of phase synchronization.

1. Introduction

Synchronization of chaotic systems has been an important area in nonlinear dynamics [1]. “Synchronization” is defined as a complete coincidence of two variables (or time series) that are belonging to different systems [2] while the appearance of some functional relations between two variables is termed as “generalized synchronization” [3, 4]. Instead of focusing on the synchronization of time series, Rosenblum et al. introduced the concept of phase synchronization to describe how the coupled chaotic oscillators could present a nearly perfect locking of phase, whereas the amplitude remained chaotic [5]. In [5], the phase of a time series was defined based on the Hilbert transform [6]. This definition is also very attractive in characterization of chaos [7]. In a more explicit form, the Hilbert transform of a time series x(t) follows:(1)x^t=1πP.V.∫-∞∞xτt-τdτ,where P.V. means the Cauchy principal value for the integral. Thus, a new complex quantity ψ(t) can be introduced; that is,(2)ψt=xt+ix^t=AHteiθH(t),where AH(t) is the phase and θH(t) is the amplitude [5, 7] and they form a conjugate pair. One can also define the phase angle to be the projection of phase point on the x-y plane with the phase angle θT(t)=tan-1(y/x) and the amplitude AT(t)=x2+y2. Alternatively by using the Poincare section, one can also choose a phase:(3)θPt=2πt-tntn+1-tn+2nπ,where tn<t<tn+1 and tn denotes the nth crossing, but there is no conjugate amplitude for θP(t). In this comment, we reexamine the virtue of using the phase variable as the indicator of synchronization of the time series. We believe that phase synchronization may not be a good tracer to the actual synchronization of the time series. Then our work will explain the onset mechanism of phase synchronization and it leads to a deep concern on the current usage of phase synchronization.

2. Give an Example of Coupled Rossler Model

Let us recalculate the same coupled Rossler model as in [5] to explore the synchronization. The model follows:(4)x˙1,2=-ω1,2y1,2-z1,2+Cx2,1-x1,2,y˙1,2=ω1,2x1,2+0.15y1,2,z˙1,2=0.2+z1,2x1,2-10,where C is the strength of coupling, ω1,2=1±Δω, and Δω=0.015 which indicates that there is a frequency mismatch between two oscillators. Because of finite frequency mismatch, there is no exact synchronization in time series. Synchronization can be found for C not only in the range [0,1], as reported in [5], region I, but also in [2.9,3.45], region II. Thus, in such a case, “synchronization” only implies a high correlation between two variables, such as time series. By using the Hilbert transform, the variables xi have the phase θi and the amplitude Ai(i=1,2). It has been reported that θ1 and θ2 move together and get nearly synchronized, while A1 and A2 remain irregular and unrelated in a range of small C [5]. Since then, cited and extended works have been expanded dramatically [8–19]. However, it should be emphasized that as phase is introduced, the influence of its counterpart, that is, amplitude, and their interplay should not be ignored. Unfortunately, although the concept of phase synchronization has been extensively addressed, the correlation between phase synchronization and amplitude synchronization remains to be clarified.

3. Results and Discussion

Let us use a common measure, the mean square error, to quantify the degree of synchronization. For two time series, x1 and x2, the mean square error is defined as(5)σx=1T∫0Tx1t-x2t2dt,where the integration time T should be sufficiently long. For the phase and the amplitude deduced by the Hilbert transform, we denote their mean square errors to be Hθ and HA. For phase angle defined by projection (here xi and yi of the Rossler model), the corresponding mean square errors are Tθ and TA, while in the case of the Poincare section (here a typical section in the xi-yi plane) we use Pθ to denote it. Obviously, to be good indicators of the synchronization, variations of these quantities should faithfully reflect the true status of the coupled chaotic oscillators. As shown in Figure 1, where the values of the mean square errors in region I are plotted, the mean square errors of the phase variables are insensitive to the true state of synchronization in time series. In contrast, the mean square errors of the conjugate amplitudes reflect more faithfully to the status of oscillators in this case. Similar feature can also be found in region II as well as for different coupled oscillators.

The mean square error (mse) diagram of the coupled Rossler model. σx is the mse of the time series. Hθ, Tθ and Pθ are the mse for the phase variables while HA and TA are for the amplitudes.

4. Conclusion

The result shows that phase synchronization will intend to be established earlier than amplitude synchronization under the Hilbert transform. This unique feature is novel, but it is caused by the Hilbert transform. The transformation on the phase part is nonlinear. This nonlinear transform is the generic mechanism for the novel features that have been reported on the phase synchronization [5, 6, 8–19]. Therefore, we believe the role of “phase” and phase synchronization may not be a good indicator to the true synchronization in the time series. The relevance of phase variables to the synchronization of the original variables in nonlinear dynamical systems seems to be a mathematical consequence of the transform one used. Thus, the works on phase synchronization [6, 13–19] may be worthwhile to be reconsidered.

Conflict of Interests

There is no conflict of interests related to this paper.

Acknowledgment

The authors would like to thank the National Science Council, Taiwan, for financially supporting this research under contract no. NSC 102-2112-M-017-002-.

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