Phase Characterization in Experimental Chaotic Systems

Abstract:

In this work we present and discuss a method for measuring the phase of chaotic systems. This method has as input a scalar time series and operates by estimating a fundamental frequency for short segments, or windows, along the whole extension of the signal. It minimizes the mean square error of fitting a sinusoidal function to the series segment. This approach does not require following the trajectory on the attractor, works well over a wide range of adjustable parameters, is of easy implementation, and is particularly appealing for experimental settings with single signal outputs since there is no need of attractor reconstruction. We demonstrate the applicability of this method on experimental time series obtained from two coupled Chua circuits.

• Periodical: Applied Mechanics and Materials (Volume 706)
• Pages: 137-148
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.706.137
• Online since: December 2014
• Authors: R. Follmann*, E. Rosa, E.E.N. Macau, J.R.C. Piqueira
• Keywords: Chaotic Systems, Noise Systems, Noncoherence, Phase Synchronization

References

[1] S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, and C.S. Zhou. The synchronization of chaotic systems. Phys. Reports, 366: 1, January (2002).

DOI: 10.1016/s0370-1573(02)00137-0

[2] Andreas Bruns. Fourier-, hilbert- and wavelet-based signal analysis: are they really different approaches? Journal of Neuroscience Methods, 137(2): 321 - 332, (2004).

DOI: 10.1016/j.jneumeth.2004.03.002

[3] Leon O. Chua, Makoto Itoh, Ljupco Kocarev, and Kevin Eckert. Chaos synchronization in chua's circuit. J. of Circuits, Syst. and Comp., 3(1): 93-108, (1993).

DOI: 10.1142/s0218126693000071

[4] Rosangela Follmann, Elbert E. N. Macau, and Epaminondas Rosa. Phase detection of chaos. Phys. Rev. E, 83(1): 016209, Jan (2011).

[5] Rosangela Follmann, Elbert E.N. Macau, and Epaminondas Rosa Jr. Detecting phase synchronization between coupled non-phase-coherent oscillators. Phys. Let. A, 373(25): 2146 - 2153, (2009).

DOI: 10.1016/j.physleta.2009.04.037

[6] Rosangela Follmann, Epaminondas Rosa, Elbert E. N. Macau, and José Roberto Castilho Piqueira. Identifying phase synchronous regimes in non-coherent and multiple scroll attractor systems. Int. J. Bifurcation and Chaos, 23(11): 1350179, (2013).

DOI: 10.1142/s0218127413501794

[7] Alexander E. Hramov and Alexey A. Koronovskii. Time scale synchronization of chaotic oscillators. Physica D: Non. Phenon., 206(3-4): 252 - 264, (2005).

DOI: 10.1016/j.physd.2005.05.008

[8] Christiaan Huygens. Horologium Oscillatorium. Apud F. Muguet, Parisiis, 1673.

[9] M. P. Kennedy. Robust op amp realization of chua's circuit. Frequenz, 46(3-4): 66-80, (1992).

[10] Jürgen Kurths, M. Carmen Romano, Marco Thiel, Grigory V. Osipov, Mikhail V. Ivanchenko, István Z. Kiss, and John L. Hudson. Synchronization analysis of coupled noncoherent oscillators. Nonlinear Dynamics, 44(1): 135-149, (2006).

DOI: 10.1007/s11071-006-1957-x

[11] T. Matsumoto. A chaotic attractor from chua's circuit. IEEE Transactions on Circuit and Systems, 31(12): 1055-1058, December (1984).

DOI: 10.1109/tcs.1984.1085459

[12] Choongseok Park, Robert M. Worth, and Leonid L. Rubchinsky. Fine Temporal Structure of Beta Oscillations Synchronization in Subthalamic Nucleus in Parkinson's Disease. J Neurophysiol, 103(5): 2707-2716, (2010).

DOI: 10.1152/jn.00724.2009

[13] Luis M. Pecora and Thomas L. Carroll. Synchronization in chaotic systems. Phys. Rev. Lett., 64(8): 821, February (1990).

[14] T. Pereira, M. S. Baptista, and J. Kurths. General framework for phase synchronization through localized sets. Phys. Rev. E, 75(2): 026216, Feb (2007).

DOI: 10.1103/physreve.75.026216

[15] A. S. Pikovsky, M. G. Rosenblum, G. V. Osipov, and J. Kurths. Phase synchronization of chaotic oscillators by external driving. Physica D, 104: 219, (1997).

DOI: 10.1016/s0167-2789(96)00301-6

[16] M. C. Romano, M. Thiel, J. Kurths, I. Z. Kiss, and J. L. Hudson. Detection of synchronization for non-phase-coherent and non-stationary data. Europhys. Lett., 71: 237-329, (2005).

DOI: 10.1209/epl/i2005-10095-1

[17] E. Rosa Jr., E. Ott, and M. H. Hess. Transition to phase synchronization of chaos. Phys. Rev. Lett, 80(8): 1642, Feb (1998).

DOI: 10.1103/physrevlett.80.1642

[18] M. G. Rosenblum, A. S. Pikovsky, and J. Kurths. Phase synchronization of chaotic oscillatiors. Phys. Rev. Lett., 76(11): 1804, March (1996).

DOI: 10.1103/physrevlett.76.1804

[19] O E Rössler. An equation for continuous chaos. Phys. Lett. A, 57(5): 397, (1976).

[20] Carten Schäfer, M. G. Rosenblum, A. S. Pikovsky, and J�rgen Kurths. Heartbeat synchronized with ventilation. Nature, 392: 239, (1998).

DOI: 10.1038/32567

[21] C.J. Stam, T. Montez, B.F. Jones, S.A.R.B. Rombouts, Y. van der Made, Y.A.L. Pijnenburg, and Ph. Scheltens. Disturbed fluctuations of resting state eeg synchronization in alzheimer's disease. Clinical Neurophysiology, 116(3): 708- 715, (2005).

DOI: 10.1016/j.clinph.2004.09.022

[22] E. F. Stone. Frequency entrainment of phase coherent attractor. Phys. Lett. A, 163(5-6): 367- 374, (1992).

[23] Steven H. Strogatz and Ian Stewart. Coupled oscillators and biological synchronization. Scientific American, 269(6): 68, december (1993).

[24] Ming-Chya Wu and Chin-Kun Hu. Empirical mode decomposition and synchrogram approach to cardiorespiratory synchronization. Phys. Rev. E, 73: 051917, May (2006).

DOI: 10.1103/physreve.73.051917

[25] Tolga Yalcinkaya and Ying-Cheng Lai. Phase characterization of chaos. Phys. Rev. Lett., 79(20): 3885-3888, Nov (1997).

[26] Tomoji Yamada and Hirokazu Fujisaka. Stability theory of synchronized motion in coupledoscillator systems. Prog. of Theor. Phys., 72(5): 885, (1984).