Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article
Special Issue

Macroscopic/Mesoscopic Computational Materials Science Modeling and Engineering

View this Special Issue

Research Article | Open Access

Volume 2015 |Article ID 640107 | 3 pages | https://doi.org/10.1155/2015/640107

Phase Synchronization Is the Amplified Result by the Hilbert Transform

Academic Editor: Mo Li
Received30 Sep 2014
Accepted26 Dec 2014
Published12 Oct 2015

Abstract

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 follows:where P.V. means the Cauchy principal value for the integral. Thus, a new complex quantity can be introduced; that is,where is the phase and 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 and the amplitude . Alternatively by using the Poincare section, one can also choose a phase:where and denotes the th crossing, but there is no conjugate amplitude for . 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:where is the strength of coupling, , and 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 not only in the range , as reported in [5], region I, but also in , 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 have the phase and the amplitude   . It has been reported that and move together and get nearly synchronized, while and remain irregular and unrelated in a range of small C [5]. Since then, cited and extended works have been expanded dramatically [819]. 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, and , the mean square error is defined aswhere the integration time should be sufficiently long. For the phase and the amplitude deduced by the Hilbert transform, we denote their mean square errors to be and . For phase angle defined by projection (here and of the Rossler model), the corresponding mean square errors are and , while in the case of the Poincare section (here a typical section in the plane) we use 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.

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, 819]. 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, 1319] 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-.

References

  1. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, pp. 821–824, 1990. View at: Publisher Site | Google Scholar
  2. L. Kocarev and U. Parlitz, “General approach for chaotic synchronization with applications to communication,” Physical Review Letters, vol. 74, no. 25, pp. 5028–5031, 1995. View at: Publisher Site | Google Scholar
  3. N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, “Generalized synchronization of chaos in directionally coupled chaotic systems,” Physical Review E, vol. 51, no. 2, pp. 980–994, 1995. View at: Publisher Site | Google Scholar
  4. L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Physical Review Letters, vol. 76, no. 11, pp. 1816–1819, 1996. View at: Publisher Site | Google Scholar
  5. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Physical Review Letters, vol. 76, no. 11, pp. 1804–1807, 1996. View at: Publisher Site | Google Scholar
  6. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,” Physical Review Letters, vol. 78, no. 22, pp. 4193–4196, 1997. View at: Publisher Site | Google Scholar
  7. T. Yalçınkaya and Y.-C. Lai, “Phase characterization of chaos,” Physical Review Letters, vol. 79, no. 20, pp. 3885–3888, 1997. View at: Publisher Site | Google Scholar
  8. A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths, “Phase synchronization of chaotic oscillators by external driving,” Physica D, vol. 104, no. 3-4, pp. 219–238, 1997. View at: Publisher Site | Google Scholar
  9. P. Tass, J. Kurths, M. G. Rosenblum, G. Gauasti, and H. Hefer, “Delay-induced transitions in visually guided movements,” Physical Review E, vol. 54, Article ID R2224, 1996. View at: Publisher Site | Google Scholar
  10. D. Y. Tang and N. R. Heckenberg, “Synchronization of mutually coupled chaotic systems,” Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, vol. 55, no. 6, pp. 6618–6623, 1997. View at: Google Scholar
  11. P. S. Landa, A. A. Zaikin, M. G. Rosenblum, and J. Kurths, “On-off intemittency phenomena in a pendulum with a randomly vibrating suspension axis,” Chaos, Solitons & Fractals, vol. 9, no. 1-2, pp. 157–169, 1998. View at: Publisher Site | Google Scholar
  12. M. Palus, “Detecting phase synchronization in noisy systems,” Physics Letters A, vol. 235, no. 4, pp. 341–351, 1997. View at: Publisher Site | Google Scholar | MathSciNet
  13. T. F. Hsu, K. H. Jao, and Y. C. Hung, “Phase synchronization in a two-mode solid state laser: periodic modulations with the second relaxation oscillation frequency of the laser output,” Physics Letters A, vol. 378, no. 44, pp. 3269–3273, 2014. View at: Publisher Site | Google Scholar
  14. Y. Kawamura, “Collective phase dynamics of globally coupled oscillators: noise-induced anti-phase synchronization,” Physica D, vol. 270, pp. 20–29, 2014. View at: Publisher Site | Google Scholar | MathSciNet
  15. K. J. Lee, Y. Kwak, and T. K. Lin, “Phase jumps near a phase synchronization transition in systems of two coupled chaotic oscillators,” Physical Review Letters, vol. 81, no. 2, pp. 321–324, 1998. View at: Publisher Site | Google Scholar
  16. E. Rosa Jr., E. Ott, and M. H. Hess, “Transition to phase synchronization of chaos,” Physical Review Letters, vol. 80, no. 8, pp. 1642–1645, 1998. View at: Publisher Site | Google Scholar
  17. R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, “Synchronous behavior of two coupled biological neurons,” Physical Review Letters, vol. 81, no. 25, pp. 5692–5695, 1998. View at: Publisher Site | Google Scholar
  18. R. Ma, G. S. Klindt, I. H. Riedel-Kruse, F. Jülicher, and B. M. Friedrich, “Active phase and amplitude fluctuations of flagellar beating,” Physical Review Letters, vol. 113, Article ID 048101, 2014. View at: Publisher Site | Google Scholar
  19. F. Pessacg, A. Taitz, G. Patterson, P. Fierens, and D. Grosz, “Experimental demonstration of a noise-tunable delay line with applications to phase synchronization,” Communications in Nonlinear Science and Numerical Simulation, vol. 22, no. 1–3, pp. 872–876, 2015. View at: Publisher Site | Google Scholar

Copyright © 2015 Ming-Chi Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

625 Views | 308 Downloads | 1 Citation
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19.