Chaotic time series analysis. (English) Zbl 1191.37046

Summary: Chaotic dynamical systems are ubiquitous in nature and most of them does not have an explicit dynamical equation and can be only understood through the available time series. We here briefly review the basic concepts of time series and its analytic tools, such as dimension, Lyapunov exponent, Hilbert transform, and attractor reconstruction. Then we discuss its applications in a few fields such as the construction of differential equations, identification of synchronization and coupling direction, coherence resonance, and traffic data analysis in Internet.


37M10 Time series analysis of dynamical systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N99 Applications of dynamical systems
68M11 Internet topics
Full Text: DOI EuDML


[1] E. Ott, Chaos in Dynamical Systems, Cambridge University, Cambridge, UK, 1993. · Zbl 0792.58014
[2] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, Textbooks in Mathematical Sciences, Springer, New York, NY, USA, 1997. · Zbl 0867.58043
[3] J. Guckenheimer and P. Holmes, Nonlinear Oscillators, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. · Zbl 0515.34001
[4] A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, vol. 38 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. · Zbl 0506.70016
[5] L. E. Reichl, The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations, Institute for Nonlinear Science, Springer, New York, NY, USA, 1992. · Zbl 0776.70003
[6] S. N. Elaydi, Discrete Chaos, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2000. · Zbl 0945.37010
[7] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University, Cambridge, UK, 2000. · Zbl 1050.62093
[8] H. D. I. Abarbanel, Analysis of Observed Chaotic Data, Springer, New York, NY, USA, 1996. · Zbl 0890.93006
[9] T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer, New York, NY, USA, 1689. · Zbl 0692.58001
[10] Y.-C. Lai, Z. Liu, N. Ye, and T. Yalcinkaya, “Nonlinear time series analysis,” in The Handbook of Data Mining, N. Ye, Ed., Lawrence Erlbaum Associates, Mahwah, NJ, USA, 2003.
[11] L. A. Aguirre and C. Letellier, “Modeling nonlinear dynamics and chaos: a review,” Mathematical Problems in Engineering, vol. 2009, Article ID 238960, 35 pages, 2009. · Zbl 1180.37003
[12] F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence , Warwick 1980 (Coventry, 1979/1980), D. A. Rand and L.-S. Young, Eds., vol. 898 of Lecture Notes in Mathematics, pp. 366-381, Springer, London, UK, 1981. · Zbl 0513.58032
[13] P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D, vol. 9, no. 1-2, pp. 189-208, 1983. · Zbl 0593.58024
[14] P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Physical Review Letters, vol. 50, no. 5, pp. 346-349, 1983. · Zbl 0593.58024
[15] J.-P. Eckmann and D. Ruelle, “Ergodic theory of chaos and strange attractors,” Reviews of Modern Physics, vol. 57, no. 3, pp. 617-656, 1985. · Zbl 0989.37516
[16] J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, and S. Ciliberto, “Liapunov exponents from time series,” Physical Review A, vol. 34, no. 6, pp. 4971-4979, 1986.
[17] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press, San Diego, Calif, USA, 4th edition, 1995. · Zbl 0970.00005
[18] C. G. Schroer, T. Sauer, E. Ott, and J. A. Yorke, “Predicting chaos most of the time from embeddings with self-intersections,” Physical Review Letters, vol. 80, no. 7, pp. 1410-1413, 1998.
[19] A. M. Fraser and H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Physical Review A, vol. 33, no. 2, pp. 1134-1140, 1986. · Zbl 1184.37027
[20] J. M. Martinerie, A. M. Albano, A. I. Mees, and P. E. Rapp, “Mutual information, strange attractors, and the optimal estimation of dimension,” Physical Review A, vol. 45, no. 10, pp. 7058-7064, 1992.
[21] D. Kugiumtzis, “State space reconstruction parameters in the analysis of chaotic time series-the role of the time window length,” Physica D, vol. 95, no. 1, pp. 13-28, 1996. · Zbl 0914.62063
[22] G. Gouesbet and J. Maquet, “Construction of phenomenological models from numerical scalar time series,” Physica D, vol. 58, no. 2, pp. 202-215, 1992. · Zbl 1194.37132
[23] G. Gouesbet and C. Letellier, “Global vector-field reconstruction by using a multivariate polynomial L2 approximation on nets,” Physical Review E, vol. 49, no. 6, pp. 4955-4972, 1994.
[24] B. P. Bezruchko and D. A. Smirnov, “Constructing nonautonomous differential equations from experimental time series,” Physical Review E, vol. 63, no. 1, Article ID 016207, 7 pages, 2001. · Zbl 1052.34502
[25] R. Hegger, H. Kantz, F. Schmuser, M. Diestelhorst, R.-P. Kapsch, and H. Beige, “Dynamical properties of a ferroelectric capacitor observed through nonlinear time series analysis,” Chaos, vol. 8, no. 3, pp. 727-736, 1998. · Zbl 0971.37039
[26] F. Sorrentino and E. Ott, “Using synchronization of chaos to identify the dynamics of unknown systems,” Chaos, vol. 19, no. 3, Article ID 033108, 2009. · Zbl 1317.34124
[27] S. F. Farmer, “Rhythmicity, synchronization and binding in human and primate motor systems,” Journal of Physiology, vol. 509, no. 1, pp. 3-14, 1998.
[28] J. Arnhold, P. Grassberger, K. Lehnertz, and C. E. Elger, “A robust method for detecting interdependences: application to intracranially recorded EEG,” Physica D, vol. 134, no. 4, pp. 419-430, 1999. · Zbl 0976.92011
[29] F. Mormann, K. Lehnertz, P. David, and C. E. Elger, “Mean phase coherence as a measure for phase synchronization and its application to the EEG of epilepsy patients,” Physica D, vol. 144, no. 3, pp. 358-369, 2000. · Zbl 0962.92020
[30] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Science, vol. 12 of Cambridge Nonlinear Science Series, Cambridge University, Cambridge, UK, 2001. · Zbl 0993.37002
[31] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821-824, 1990. · Zbl 0938.37019
[32] 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. · Zbl 0896.60090
[33] 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.
[34] 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. · Zbl 0898.70015
[35] W. Wang, Z. Liu, and B. Hu, “Phase order in chaotic maps and in coupled map lattices,” Physical Review Letters, vol. 84, no. 12, pp. 2610-2613, 2000.
[36] 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.
[37] L. M. Pecora, T. L. Carroll, and J. F. Heagy, “Statistics for mathematical properties of maps between time series embeddings,” Physical Review E, vol. 52, no. 4, pp. 3420-3439, 1995.
[38] H. D. I. Abarbanel, N. F. Rulkov, and M. M. Sushchik, “Generalized synchronization of chaos: the auxiliary system approach,” Physical Review E, vol. 53, no. 5, pp. 4528-4535, 1996.
[39] 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.
[40] B. R. Hunt, E. Ott, and J. A. Yorke, “Differentiable generalized synchronization of chaos,” Physical Review E, vol. 55, no. 4, pp. 4029-4034, 1997.
[41] K. Pyragas, “Conditional lyapunov exponents from time series,” Physical Review E, vol. 56, no. 5, pp. 5183-5188, 1997.
[42] Z. Liu and S. Chen, “General method of synchronization,” Physical Review E, vol. 55, no. 6, pp. 6651-6655, 1997.
[43] R. Brown, “Approximating the mapping between systems exhibiting generalized synchronization,” Physical Review Letters, vol. 81, no. 22, pp. 4835-4838, 1998.
[44] C. L. Goodridge, L. M. Pecora, T. L. Carroll, and F. J. Rachford, “Detecting functional relationships between simultaneous time series,” Physical Review E, vol. 64, no. 2, Article ID 026221, 2001.
[45] S. Boccaletti, L. M. Pecora, and A. Pelaez, “Unifying framework for synchronization of coupled dynamical systems,” Physical Review E, vol. 63, no. 6, Article ID 066219, 2001.
[46] K. Pyragas, “Weak and strong synchronization of chaos,” Physical Review E, vol. 54, no. 5, pp. R4508-R4511, 1996.
[47] Z. Liu, “Measuring the degree of synchronization from time series data,” Europhysics Letters, vol. 68, no. 1, pp. 19-25, 2004.
[48] Z. Liu, H. U. Bambi, and L. D. Iasemidis, “Detection of phase locking from non-stationary time series,” Europhysics Letters, vol. 71, no. 2, pp. 200-206, 2005.
[49] C. W. J. Granger, “Investigating causal relations by econometric models and cross-spectral methods,” Econometrica, vol. 37, no. 3, p. 424, 1969. · Zbl 1366.91115
[50] W. Wang, B. T. Anderson, R. K. Kaufmann, and R. B. Myneni, “The relation between the north Atlantic oscillation and SSTs in the north Atlantic basin,” Journal of Climate, vol. 17, no. 24, pp. 4752-4759, 2004.
[51] B. Schelter, M. Winterhalder, M. Eichler, et al., “Testing for directed influences among neural signals using partial directed coherence,” Journal of Neuroscience Methods, vol. 152, no. 1-2, pp. 210-219, 2006.
[52] E. Sitnikova, T. Dikanev, D. Smirnov, B. Bezruchko, and G. Luijtelaar, “Granger causality: cortico-thalamic interdependencies during absence seizures in WAG/Rij rats,” Journal of Neuroscience Methods, vol. 170, no. 2, pp. 245-254, 2008.
[53] D. A. Smirnov and I. I. Mokhov, “From Granger causality yo long-term causality: application to climatic data,” Physical Review E, vol. 80, no. 1, Article ID 016208, 2009.
[54] K. Hlavackova-Schindler, M. Palus, M. Vejmelka, and J. Bhattacharya, “Causality detection based on information-theoretic approaches in time series analysis,” Physics Reports, vol. 441, no. 1, pp. 1-46, 2007.
[55] S. J. Schiff, P. So, T. Chang, R. E. Burke, and T. Sauer, “Detecting dynamical interdependence and generalized synchrony through mutual prediction in a neural ensemble,” Physical Review E, vol. 54, no. 6, pp. 6708-6724, 1996.
[56] R. Q. Quiroga, J. Arnhold, and P. Grassberger, “Learning driver-response relationships from synchronization patterns,” Physical Review E, vol. 61, no. 5, pp. 5142-5148, 2000.
[57] R. Q. Quiroga, A. Kraskov, T. Kreuz, and P. Grassberger, “Performance of different synchronization measures in real data: a case study on electroencephalographic signals,” Physical Review E, vol. 65, no. 4, Article ID 041903, 2002.
[58] M. G. Rosenblum and A. S. Pikovsky, “Detecting direction of coupling in interacting oscillators,” Physical Review E, vol. 64, no. 4, Article ID 045202, 2001.
[59] B. Bezruchko, V. Ponomarenko, M. G. Rosenblum, and A. S. Pikovsky, “Characterizing direction of coupling from experimental observations,” Chaos, vol. 13, no. 1, pp. 179-184, 2003.
[60] D. A. Smirnov, M. B. Bodrov, J. L. Perez Velazquez, R. A. Wennberg, and B. P. Bezruchko, “Estimation of coupling between oscillators from short time series via phase dynamics modeling: limitations and application to EEG data,” Chaos, vol. 15, no. 2, Article ID 024102, 10 pages, 2005.
[61] Z. Liu and Y.-C. Lai, “Coherence resonance in coupled chaotic oscillators,” Physical Review Letters, vol. 86, no. 21, pp. 4737-4740, 2001.
[62] L. Zhu, Y.-C. Lai, Z. Liu, and A. Raghu, “Can noise make nonbursting chaotic systems more regular?” Physical Review E, vol. 66, no. 1, Article ID 015204, 2002.
[63] H. Gang, T. Ditzinger, C. Z. Ning, and H. Haken, “Stochastic resonance without external periodic force,” Physical Review Letters, vol. 71, no. 6, pp. 807-810, 1993.
[64] A. S. Pikovsky and J. Kurths, “Coherence resonance in a noise-driven excitable system,” Physical Review Letters, vol. 78, no. 5, pp. 775-778, 1997. · Zbl 0961.70506
[65] M. Li and W. Zhao, “Representation of a stochastic traffic bound,” to appear in IEEE Transactions on Parallel and Distributed Systems.
[66] M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. · Zbl 1191.37002
[67] M. Li and P. Borgnat, “Forward for the special issue on traffic modeling, its computations and applications,” Telecommunication Systems, vol. 43, no. 3-4, pp. 181-195, 2009. · Zbl 05803250
[68] M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584-2594, 2008.
[69] B. A. Huberman and R. M. Lukose, “Social dilemmas and Internet congestion,” Science, vol. 277, no. 5325, pp. 535-537, 1997.
[70] X.-Y. Zhu, Z.-H. Liu, and M. Tang, “Detrended fluctuation analysis of traffic data,” Chinese Physics Letters, vol. 24, no. 7, pp. 2142-2145, 2007.
[71] C.-K. Peng, S. Havlin, H. E. Stanley, and A. L. Goldberger, “Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series,” Chaos, vol. 5, no. 1, pp. 82-87, 1995.
[72] Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H. E. Stanley, “Statistical properties of the volatility of price fluctuations,” Physical Review E, vol. 60, no. 2, pp. 1390-1400, 1999.
[73] Z. Chen, K. Hu, P. Carpena, P. Bernaola-Galvan, H. E. Stanley, and P. Ch. Ivanov, “Effect of nonlinear filters on detrended fluctuation analysis,” Physical Review E, vol. 71, no. 1, Article ID 011104, 2005.
[74] M. Tang and Z. Liu, “Detrended fluctuation analysis of particle condensation on complex networks,” Physica A, vol. 387, no. 5-6, pp. 1361-1368, 2008.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.