Nonlinear Dynamics

, Volume 86, Issue 1, pp 227–234 | Cite as

Measuring phase synchronization in periodically driven Lü oscillator with a phase-incoherent attractor

  • Yao-Chen Hung
  • Tzu-Fang Hsu
Original Paper


A signal-based synchronization coefficient is proposed to quantify phase synchronization in the periodically driven Lü system, which is a phase-incoherent chaotic system. Phase synchronization is verified in advance using some traditional methods, and the relevant results are compared with those obtained using the proposed synchronization coefficient. The coefficient is determined by a series of scenarios, which are signal decomposition based on empirical mode decomposition, phase transformation based on the Hilbert transform, and degree quantification based on Shannon entropy. The route to phase synchronization is demonstrated to be well characterized without prior knowledge of the formulation or the attractor structure of Lü system.


Phase synchronization Phase-incoherent attractor Chaotic system 



We thank Dr. Ming-Chya Wu for helpful discussions on the EMD method.


  1. 1.
    Otsuka, K., Kawai, R., Hwong, S.-L., Ko, J.-Y., Chern, J.-L.: Synchronization of mutually coupled self-mixing modulated lasers. Phys. Rev. Lett. 84, 3049–3052 (2000)CrossRefGoogle Scholar
  2. 2.
    Hassel, J., Grönberg, L., Helistö, P., Seppä, H.: Self-synchronization in distributed Josephson junction arrays studied using harmonic analysis and power balance. Appl. Phys. Lett. 89, 072503 (2006)CrossRefGoogle Scholar
  3. 3.
    Bag, B.C., Petrosyan, K.G., Hu, C.-K.: Influence of noise on the synchronization of the stochastic Kuramoto model. Phys. Rev. E 76, 056210 (2007)CrossRefGoogle Scholar
  4. 4.
    Rajesh, S., Sinha, S., Sinha, S.: Synchronization in coupled cells with activator-inhibitor pathways. Phys. Rev. E 75, 011906 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Pyragienė, T., Pyragas, K.: Anticipating spike synchronization in nonidentical chaotic neurons. Nonlinear Dyn. 74, 297–306 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Schäfer, C., Rosenblum, M.G., Kurths, J., Abel, H.-H.: Heartbeat synchronized with ventilation. Nature (London) 392, 239–240 (1998)CrossRefGoogle Scholar
  7. 7.
    Amritkar, R.E., Rangarajan, G.: Spatially synchronous extinction of species under external forcing. Phys. Rev. Lett. 96, 258102 (2006)CrossRefGoogle Scholar
  8. 8.
    Kiss, I.Z., Lv, Q., Hudson, J.L.: Synchronization of non-phase-coherent chaotic electrochemical oscillations. Phys. Rev. E 71, 035201(R) (2005)CrossRefGoogle Scholar
  9. 9.
    Pikovsky, A.S., Rosenblum, M.G., Kurths, J.: Synchronization: A Univeral Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2001)CrossRefMATHGoogle Scholar
  10. 10.
    Pecora, L.M., Carrol, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kaneko, K.: Globally coupled chaos violates the law of large numbers but not the central-limit theorem. Phys. Rev. Lett. 65, 1391–1394 (1990)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ho, M.C., Hung, Y.C.: Synchronization of two different systems by using generalized active control. Phys. Lett. A 301, 424–428 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Luo, Y.P., Hung, Y.C.: Control synchronization and parameters identification of two different chaotic systems. Nonlinear Dyn. 73, 1507–1513 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Pikovsky, A., Zaks, M., Rosenblum, M., Osipov, G., Kurths, J.: Phase synchronization of chaotic oscillations in terms of periodic orbits. Chaos 7, 680–687 (1997)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Ho, M.C., Hung, Y.C., Jiang, I.M.: Phase synchronization in inhomogeneous globally coupled map lattices. Phys. Lett. A 324, 450–457 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Mahmoud, G.M., Mahmoud, E.E.: Phase and antiphase synchronization of two identical hyperchaotic complex nonlinear systems. Nonlinear Dyn. 61, 141–152 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Das, S., Srivastava, M., Leung, A.Y.T.: Hybrid phase synchronization between identical and nonidentical three-dimensional chaotic systems using the active control method. Nonlinear Dyn. 73, 2261–2272 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Parlitz, U., Junge, L., Kocarev, L.: Subharmonic entrainment of unstable period orbits and generalized synchronization. Phys. Rev. Lett. 79, 3158–3161 (1997)CrossRefGoogle Scholar
  19. 19.
    Hung, Y.C., Huang, Y.T., Ho, M.C., Hu, C.K.: Paths to globally generalized synchronization in scale-free networks. Phys. Rev. E 77, 016202 (2008)CrossRefGoogle Scholar
  20. 20.
    Blasius, B., Huppert, A., Stone, L.: Complex dynamics and phase synchronization in spatially extended ecological systems. Nature 399, 354–359 (1999)CrossRefGoogle Scholar
  21. 21.
    Ticos, C.M., Rosa, E., Pardo, W.B., Walkenstein, J.A., Monti, M.: Experimental real-time phase synchronization of a paced chaotic plasma discharge. Phys. Rev. Lett. 85, 2929–2932 (2000)CrossRefMATHGoogle Scholar
  22. 22.
    Maza, D., Vallone, A., Mancini, H., Boccaletti, S.: Experimental phase synchronization of a chaotic convective flow. Phys. Rev. Lett. 85, 5567–5570 (2001)CrossRefMATHGoogle Scholar
  23. 23.
    Ahlborn, A., Parlitz, U.: Experimental observation of chaotic phase synchronization of a periodically modulated frequency-doubled Nd:YAG laser. Opt. Lett. 34, 2754–2756 (2009)CrossRefGoogle Scholar
  24. 24.
    Hsu, T.F., Jao, K.H., Hung, Y.C.: Phase synchronization in a two-mode solid state laser: periodic modulations with the second relaxation oscillation frequency of the laser output. Phys. Lett. A 378, 3269–3273 (2014)CrossRefMATHGoogle Scholar
  25. 25.
    Chen, D.Y., Wu, C., Liu, C.F., Ma, X.Y., You, Y.J., Zhang, R.F.: Synchronization and circuit simulation of a new double-wing chaos. Nonlinear Dyn. 67, 1481–1504 (2012)CrossRefMATHGoogle Scholar
  26. 26.
    Osipov, G.V., Hu, B., Zhou, C.T., Ivanchenko, M.V., Kurths, J.: Three types of transitions to phase synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 91, 024101 (2003)CrossRefGoogle Scholar
  27. 27.
    Anishchenko, V.S., Silchenko, A.N., Khovanov, I.A.: Synchronization of switching processes in coupled Lorenz systems. Phys. Rev. E 57, 316–322 (1998)CrossRefGoogle Scholar
  28. 28.
    Pereira, T., Baptista, M.S., Kurths, J.: Detecting phase synchronization by localized maps: application to neural networks. Europhys. Lett. 77, 40006 (2007)CrossRefGoogle Scholar
  29. 29.
    Jan, H., Ho, M.C., Kuo, C.T., Jiang, I.M.: Detecting weak phase locking in chaotic system with dual attractors and ill-defined phase structure. Phys. Rev. E 79, 067202 (2009)CrossRefGoogle Scholar
  30. 30.
    Jan, H., Tsai, K.T., Kuo, L.W.: Phase locking route behind complex periodic windows in a forced oscillator. Chaos 23, 033126 (2013)CrossRefGoogle Scholar
  31. 31.
    Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.C., Tung, C.C., Liu, H.H.: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. Ser. A 454, 903–995 (1998)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Lü, J., Chen, G., Zhang, S.: Dynamical analysis of a new chaotic attractor. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 1001–1015 (2002)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Leonov, G.A., Kuznetsov, N.V.: On differences and similarities in the analysis of Lorenz, Chen and Lu systems. Appl. Math. Comput. 256, 334–343 (2015)MathSciNetMATHGoogle Scholar
  34. 34.
    Zaks, M.A., Park, E.-H., Rosenblum, M.G., Kurths, J.: Alternating locking ratios in imperfect phase synchronization. Phys. Rev. Lett. 82, 4228 (1999)CrossRefGoogle Scholar
  35. 35.
    Breban, R.: Phase synchronization of chaotic attractors with prescribed periodic signals. Phys. Rev. E 68, 047201 (2003)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Leonov, G.A., Kuznetsov, N.V.: Time-varying linearization and the perron effects. Int. J. Bifurc. Chaos 17, 1079–1107 (2007)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Kuznetsov, N.V., Leonov, G.A.: On stability by the first approximation for discrete systems. In: 2005 International Conference on Physics and Control, Proceedings Volume 2005, pp. 596–599 (2005)Google Scholar
  38. 38.
    Wu, M.C., Huang, M.C., Yu, H.C., Chiang, T.C.: Phase distribution and phase correlation of financial time series. Phys. Rev. E 73, 016118 (2006)CrossRefGoogle Scholar
  39. 39.
    Wu, M.C.: Phase correlation of foreign exchange time series. Phys. A 375, 633–642 (2007)CrossRefGoogle Scholar
  40. 40.
    Wu, M.C.: Damped oscillations in the ratios of stock market indices. Europhys. Lett. 97, 48009 (2012)CrossRefGoogle Scholar
  41. 41.
    Wu, M.C., Hu, C.K.: Empirical mode decomposition and synchrogram approach to cardiorespiratory synchronization. Phys. Rev. E 73, 051917 (2006)Google Scholar
  42. 42.
    Wu, M.-C., Watanabe, E., Struzik, Z.R., Hu, C.-K., Yamamoto, Y.: Phase statistics approach to human ventricular fibrillation. Phys. Rev. E 80, 051917 (2009)CrossRefGoogle Scholar
  43. 43.
    Romano, M.C., Thiel, M., Kurths, J., Kiss, I.Z., Hudson, J.L.: Detection of synchronization for non-phase-coherent and non-stationary data. Europhys. Lett. 71, 466–472 (2005)CrossRefGoogle Scholar
  44. 44.
    Tokuda, I.T., Kurths, J., Kiss, I.Z., Hudson, J.L.: Predicting phase synchronization of non-phase-coherent chaos. Europhys. Lett 83, 50003 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Physics Teaching and Research CenterFeng Chia UniversityTaichungTaiwan
  2. 2.Department of Applied PhysicsNational Pingtung UniversityPingtungTaiwan

Personalised recommendations