Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 720190, 31 pages
doi:10.1155/2010/720190
Review Article
Chaotic Time Series Analysis
Institute of Theoretical Physics and Department of Physics, East China Normal University, Shanghai 200062, China
Received 25 December 2009; Accepted 7 February 2010
Academic Editor: Ming Li
Copyright © 2010 Zonghua Liu. 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.
Linked References
- E. Ott, Chaos in Dynamical Systems, Cambridge University, Cambridge, UK, 1993.
- 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. View at MathSciNet
- 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. View at MathSciNet
- A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, vol. 38 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. View at MathSciNet
- L. E. Reichl, The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations, Institute for Nonlinear Science, Springer, New York, NY, USA, 1992. View at MathSciNet
- S. N. Elaydi, Discrete Chaos, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2000. View at MathSciNet
- H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University, Cambridge, UK, 2000.
- H. D. I. Abarbanel, Analysis of Observed Chaotic Data, Springer, New York, NY, USA, 1996. View at MathSciNet
- T. S. Parker and L. O. Chua, Practical Numerical Algorithms for Chaotic Systems, Springer, New York, NY, USA, 1689. View at MathSciNet
- 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.
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D, vol. 9, no. 1-2, pp. 189–208, 1983. View at Zentralblatt MATH · View at MathSciNet · View at Scopus
- P. Grassberger and I. Procaccia, “Characterization of strange attractors,” Physical Review Letters, vol. 50, no. 5, pp. 346–349, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
- G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press, San Diego, Calif, USA, 4th edition, 1995. View at MathSciNet
- 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. View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Scopus
- G. Gouesbet and J. Maquet, “Construction of phenomenological models from numerical scalar time series,” Physica D, vol. 58, no. 2, pp. 202–215, 1992. View at Scopus
- G. Gouesbet and C. Letellier, “Global vector-field reconstruction by using a multivariate polynomial approximation on nets,” Physical Review E, vol. 49, no. 6, pp. 4955–4972, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
- 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. View at Scopus
- 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. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
- S. F. Farmer, “Rhythmicity, synchronization and binding in human and primate motor systems,” Journal of Physiology, vol. 509, no. 1, pp. 3–14, 1998. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at MathSciNet
- L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
- 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 Scopus
- 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 Scopus
- 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
- 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. View at Scopus
- 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 · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Scopus
- 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 Scopus
- 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
- K. Pyragas, “Conditional lyapunov exponents from time series,” Physical Review E, vol. 56, no. 5, pp. 5183–5188, 1997. View at Scopus
- Z. Liu and S. Chen, “General method of synchronization,” Physical Review E, vol. 55, no. 6, pp. 6651–6655, 1997. View at Scopus
- R. Brown, “Approximating the mapping between systems exhibiting generalized synchronization,” Physical Review Letters, vol. 81, no. 22, pp. 4835–4838, 1998. View at Scopus
- 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. View at Scopus
- 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. View at Scopus
- K. Pyragas, “Weak and strong synchronization of chaos,” Physical Review E, vol. 54, no. 5, pp. R4508–R4511, 1996. View at Scopus
- Z. Liu, “Measuring the degree of synchronization from time series data,” Europhysics Letters, vol. 68, no. 1, pp. 19–25, 2004. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
- C. W. J. Granger, “Investigating causal relations by econometric models and cross-spectral methods,” Econometrica, vol. 37, no. 3, p. 424, 1969.
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at MathSciNet
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Scopus
- 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. View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at PubMed · View at Scopus
- Z. Liu and Y.-C. Lai, “Coherence resonance in coupled chaotic oscillators,” Physical Review Letters, vol. 86, no. 21, pp. 4737–4740, 2001. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
- M. Li and W. Zhao, “Representation of a stochastic traffic bound,” to appear in IEEE Transactions on Parallel and Distributed Systems. View at Publisher · View at Google Scholar
- M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- 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. View at Publisher · View at Google Scholar
- M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584–2594, 2008. View at Publisher · View at Google Scholar · View at Scopus
- B. A. Huberman and R. M. Lukose, “Social dilemmas and Internet congestion,” Science, vol. 277, no. 5325, pp. 535–537, 1997. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Scopus
- 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. View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus
- 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. View at Publisher · View at Google Scholar · View at Scopus