##
**Modeling nonlinear dynamics and chaos: a review.**
*(English)*
Zbl 1180.37003

Summary: This paper reviews the major developments of modeling techniques applied to nonlinear dynamics and chaos. Model representations, parameter estimation techniques, data requirements, and model validation are some of the key topics that are covered in this paper, which surveys slightly over two decades since the pioneering papers on the subject appeared in the literature.

### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

PDF
BibTeX
XML
Cite

\textit{L. A. Aguirre} and \textit{C. Letellier}, Math. Probl. Eng. 2009, Article ID 238960, 35 p. (2009; Zbl 1180.37003)

### References:

[1] | H. Poincaré, “Sur les courbes définies par une équation différentielle,” Journal de Mathématiques Pures et Appliquées, vol. 4, no. 1, pp. 167-244, 1885. · JFM 17.0680.01 |

[2] | H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, vol. 3, Gauthier-Villard, Paris, France, 1899. · JFM 25.1847.03 |

[3] | A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators, Pergamon Press, Oxford, UK, 1966, Traduction anglaise par F. Immirzi, Dover, New York, NY, USA, 1937. · Zbl 0188.56304 |

[4] | E. N. Lorenz, “Deterministic nonperiodic flow,” Journal of the Atmospheric Sciences, vol. 20, no. 2, pp. 130-141, 1963. · Zbl 1417.37129 |

[5] | D. Ruelle and F. Takens, “On the nature of turbulence,” Communications in Mathematical Physics, vol. 20, pp. 167-192, 1971. · Zbl 0223.76041 |

[6] | R. M. May, “Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos,” Science, vol. 186, no. 4164, pp. 645-647, 1974. |

[7] | O. E. Rössler, “An equation for continuous chaos,” Physics Letters A, vol. 57, no. 5, pp. 397-398, 1976. · Zbl 1371.37062 |

[8] | T.-Y. Li and J. A. Yorke, “Period three implies chaos,” The American Mathematical Monthly, vol. 82, no. 10, pp. 985-992, 1975. · Zbl 0351.92021 |

[9] | N. H. Packard, J. P. Crutchfield, J. D. Farmer, and R. S. Shaw, “Geometry from a time series,” Physical Review Letters, vol. 45, no. 9, pp. 712-716, 1980. |

[10] | 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, Berlin, Germany, 1981. · Zbl 0513.58032 |

[11] | R. Mañé, “On the dimension of the compact invariant sets of certain nonlinear maps,” 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. 230-242, Springer, Berlin, Germany, 1981. · Zbl 0544.58014 |

[12] | M. Casdagli, S. Eubank, J. D. Farmer, and J. Gibson, “State space reconstruction in the presence of noise,” Physica D, vol. 51, no. 1-3, pp. 52-98, 1991. · Zbl 0736.62075 |

[13] | T. Sauer, J. A. Yorke, and M. Casdagli, “Embedology,” Journal of Statistical Physics, vol. 65, no. 3-4, pp. 579-616, 1991. · Zbl 0943.37506 |

[14] | J. D. Farmer, E. Ott, and J. A. Yorke, “The dimension of chaotic attractors,” Physica D, vol. 7, no. 1-3, pp. 153-180, 1983. · Zbl 0561.58032 |

[15] | J. Theiler, “Estimating fractal dimension,” Journal of the Optical Society of America A, vol. 7, no. 6, pp. 1055-1073, 1990. |

[16] | V. I. Oseledec, “A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems,” Transactions of the Moscow Mathematical Society, vol. 19, pp. 179-210, 1968. |

[17] | A. N. Kolmogorov, “A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces,” Doklady Akademii Nauk SSSR, vol. 119, no. 5, pp. 861-864, 1958. · Zbl 0083.10602 |

[18] | H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. Sh. Tsimring, “The analysis of observed chaotic data in physical systems,” Reviews of Modern Physics, vol. 65, no. 4, pp. 1331-1392, 1993. |

[19] | H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge, UK, 2nd edition, 2004. · Zbl 1050.62093 |

[20] | M. Casdagli, D. Des Jardins, S. Eubank, et al., “Nonlinear modeling of chaotic time series: theory and applications,” in Applied Chaos, J. H. Kim and J. Stringer, Eds., Wiley-Interscience Publication, chapter 15, pp. 335-380, John Wiley & Sons, New York, NY, USA, 1992. |

[21] | G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis: Forecasting and Control, Prentice-Hall, Englewood Cliffs, NJ, USA, 3rd edition, 1994. · Zbl 0858.62072 |

[22] | L. Ljung, System Identification: Theory for the User, Prentice-Hall, Englewood Cliffs, NJ, USA, 2nd edition, 1999. · Zbl 0949.93509 |

[23] | R. J. G. B. Campello, G. Favier, and W. C. do Amaral, “Optimal expansions of discrete-time Volterra models using Laguerre functions,” Automatica, vol. 40, no. 5, pp. 815-822, 2004. · Zbl 1050.93031 |

[24] | S. A. Billings, “Identification of nonlinear systems-a survey,” IEE Proceedings D, vol. 127, no. 6, pp. 272-285, 1980. |

[25] | I. J. Leontaritis and S. A. Billings, “Input-output parametric models for nonlinear systems-part II: stochastic nonlinear systems,” International Journal of Control, vol. 41, no. 2, pp. 329-344, 1985. · Zbl 0569.93012 |

[26] | S. A. Billings and Q. M. Zhu, “Rational model identification using an extended least-squares algorithm,” International Journal of Control, vol. 54, no. 3, pp. 529-546, 1991. · Zbl 0728.93084 |

[27] | K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Transactions on Neural Networks, vol. 1, no. 1, pp. 4-27, 1990. |

[28] | S. Chen, C. F. N. Cowan, and P. M. Grant, “Orthogonal least squares learning algorithm for radial basis function networks,” IEEE Transactions on Neural Networks, vol. 2, no. 2, pp. 302-309, 1991. |

[29] | L. Cao, Y. Hong, H. Fang, and G. He, “Predicting chaotic time series with wavelet networks,” Physica D, vol. 85, no. 1-2, pp. 225-238, 1995. · Zbl 0888.93005 |

[30] | L. Z. Guo, S. A. Billings, and D. Q. Zhu, “An extended orthogonal forward regression algorithm for system identification using entropy,” International Journal of Control, vol. 81, no. 4, pp. 690-699, 2008. · Zbl 1152.93507 |

[31] | S. A. Billings and H. L. Wei, “An adaptive orthogonal search algorithm for model subset selection and non-linear system identification,” International Journal of Control, vol. 81, no. 5, pp. 714-724, 2008. · Zbl 1152.93343 |

[32] | H.-L. Wei and S. A. Billings, “Model structure selection using an integrated forward orthogonal search algorithm assisted by squared correlation and mutual information,” International Journal of Modelling, Identification and Control, vol. 3, no. 4, pp. 341-356, 2008. |

[33] | L. Piroddi, “Simulation error minimisation methods for NARX model identification,” International Journal of Modelling, Identification and Control, vol. 3, no. 4, pp. 392-403, 2008. |

[34] | X. Hong, R. J. Mitchell, S. Chen, C. J. Harris, K. Li, and G. W. Irwin, “Model selection approaches for non-linear system identification: a review,” International Journal of Systems Science, vol. 39, no. 10, pp. 925-946, 2008. · Zbl 1233.93097 |

[35] | F. P. Pach, A. Gyenesei, and J. Abonyi, “MOSSFARM: model structure selection by fuzzy association rule mining,” Journal of Intelligent and Fuzzy Systems, vol. 19, no. 6, pp. 399-407, 2008. · Zbl 1168.68372 |

[36] | J. D. Farmer and J. J. Sidorowich, “Predicting chaotic time series,” Physical Review Letters, vol. 59, no. 8, pp. 845-848, 1987. |

[37] | J. P. Crutchfield and B. S. McNamara, “Equations of motion from a data series,” Complex Systems, vol. 1, no. 3, pp. 417-452, 1987. · Zbl 0675.58026 |

[38] | D. S. Broomhead and D. Lowe, “Multivariable functional interpolation and adaptive networks,” Complex Systems, vol. 2, no. 3, pp. 321-355, 1988. · Zbl 0657.68085 |

[39] | M. Casdagli, “Nonlinear prediction of chaotic time series,” Physica D, vol. 35, no. 3, pp. 335-356, 1989. · Zbl 0671.62099 |

[40] | J. Cremers and A. Hübler, “Construction of differential equations from experimental data,” Zeitschrift für Naturforschung A, vol. 42, no. 8, pp. 797-802, 1987. |

[41] | J. D. Farmer and J. J. Sidorowich, “Exploiting chaos to predict the future and reduce noise,” in Evolution, Learning and Cognition, Y. C. Lee, Ed., pp. 277-330, World Scientific, Teaneck, NJ, USA, 1988. |

[42] | M. Casdagli, “Chaos and deterministic versus stochastic nonlinear modelling,” Journal of the Royal Statistical Society. Series B, vol. 54, no. 2, pp. 303-328, 1992. |

[43] | H. D. I. Abarbanel, R. Brown, and J. B. Kadtke, “Prediction in chaotic nonlinear systems: methods for time series with broadband Fourier spectra,” Physical Review A, vol. 41, no. 4, pp. 1782-1807, 1990. |

[44] | M. B. Kennel and S. Isabelle, “Method to distinguish possible chaos from colored noise and to determine embedding parameters,” Physical Review A, vol. 46, no. 6, pp. 3111-3118, 1992. |

[45] | P. S. Linsay, “An efficient method of forecasting chaotic time series using linear interpolation,” Physics Letters A, vol. 153, no. 6-7, pp. 353-356, 1991. |

[46] | G. Xiaofeng and C. H. Lai, “Improvement of the local prediction of chaotic time series,” Physical Review E, vol. 60, no. 5, pp. 5463-5468, 1999. |

[47] | L. Cao, A. Mees, and K. Judd, “Dynamics from multivariate time series,” Physica D, vol. 121, no. 1-2, pp. 75-88, 1998. · Zbl 0933.62083 |

[48] | P. Grassberger, T. Schreiber, and C. Schaffrath, “Nonlinear time sequence analysis,” International Journal of Bifurcation and Chaos, vol. 1, no. 3, pp. 521-547, 1991. · Zbl 0874.58029 |

[49] | D. Kugiumtzis, O. C. Lingjærde, and N. Christophersen, “Regularized local linear prediction of chaotic time series,” Physica D, vol. 112, no. 3-4, pp. 344-360, 1998. · Zbl 0931.37037 |

[50] | L. Cao and A. S. Soofi, “Nonlinear deterministic forecasting of daily dollar exchange rates,” International Journal of Forecasting, vol. 15, no. 4, pp. 421-430, 1999. · Zbl 0921.90037 |

[51] | A. Lapedes and R. Farber, “Nonlinear signal processing using neural networks: prediction and system modelling,” submitted to Proceedings of the IEEE, Los Alamos Report LA-UR 87-2662, 1887. |

[52] | U. Parlitz, A. Hornstein, D. Engster, et al., “Identification of pre-sliding friction dynamics,” Chaos, vol. 14, no. 2, pp. 420-430, 2004. · Zbl 1080.70009 |

[53] | J.-S. Zhang and X.-C. Xiao, “Predicting chaotic time series using recurrent neural network,” Chinese Physics Letters, vol. 17, no. 2, pp. 88-90, 2000. |

[54] | J.-S. Zhang and X.-C. Xiao, “Predicting hyper-chaotic time series using adaptive higher-order nonlinear filter,” Chinese Physics Letters, vol. 18, no. 3, pp. 337-340, 2001. |

[55] | J. L. Breeden and A. Hübler, “Reconstructing equations of motion from experimental data with unobserved variables,” Physical Review A, vol. 42, no. 10, pp. 5817-5826, 1990. |

[56] | J. L. Breeden, F. Dinkelacker, and A. Hübler, “Noise in the modeling and control of dynamical systems,” Physical Review A, vol. 42, no. 10, pp. 5827-5836, 1990. |

[57] | G. Gouesbet, “Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series,” Physical Review A, vol. 43, no. 10, pp. 5321-5331, 1991. |

[58] | G. Gouesbet, “Reconstruction of vector fields: the case of the Lorenz system,” Physical Review A, vol. 46, no. 4, pp. 1784-1796, 1992. |

[59] | G. Gouesbet and J. Maquet, “Construction of phenomenological models from numerical scalar time series,” Physica D, vol. 58, no. 1-4, pp. 202-215, 1992. · Zbl 1194.37132 |

[60] | 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. |

[61] | M. Giona, F. Lentini, and V. Cimagalli, “Functional reconstruction and local prediction of chaotic time series,” Physical Review A, vol. 44, no. 6, pp. 3496-3502, 1991. |

[62] | R. Brown, N. F. Rulkov, and E. R. Tracy, “Modeling and synchronizing chaotic systems from time-series data,” Physical Review E, vol. 49, no. 5, pp. 3784-3800, 1994. |

[63] | L. Le Sceller, C. Letellier, and G. Gouesbet, “Structure selection for global vector field reconstruction by using the identification of fixed points,” Physical Review E, vol. 60, no. 2, pp. 1600-1606, 1999. |

[64] | L. A. Aguirre, U. S. Freitas, C. Letellier, and J. Maquet, “Structure-selection techniques applied to continuous-time nonlinear models,” Physica D, vol. 158, no. 1-4, pp. 1-18, 2001. · Zbl 1098.34508 |

[65] | 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, 2000. |

[66] | C. S. M. Lainscsek, C. Letellier, and F. Schürrer, “Ansatz library for global modeling with a structure selection,” Physical Review E, vol. 64, no. 1, Article ID 016206, 15 pages, 2001. |

[67] | M.-A. Boiron and J.-M. Malasoma, “Modélisation globale d’un système électrochimique,” in Compte-Rendus de la 8ème Rencontre du Non-Linéaire, Non-Linéaire, Orsay, France, 2005. |

[68] | M.-A. Boiron and J.-M. Malasoma, “Modélisation globale d’un système électrochimique,” in Proceedings of the 5th Colloque on Temporal Chaos and Spatiotemporal Chaos, Le Havre, France, December 2005. |

[69] | C. Letellier, L. Le Sceller, E. Maréchal, et al., “Global vector field reconstruction from a chaotic experimental signal in copper electrodissolution,” Physical Review E, vol. 51, no. 5, pp. 4262-4266, 1995. |

[70] | A. D. Irving and T. Dewson, “Determining mixed linear-nonlinear coupled differential equations from multivariate discrete time series sequences,” Physica D, vol. 102, no. 1-2, pp. 15-36, 1997. · Zbl 1009.37501 |

[71] | C. Letellier, G. Gouesbet, and N. F. Rulkov, “Topological analysis of chaos in equivariant electronic circuits,” International Journal of Bifurcation and Chaos, vol. 6, no. 12B, pp. 2531-2555, 1996. · Zbl 1298.94157 |

[72] | C. Letellier, L. Le Sceller, G. Gouesbet, F. Lusseyran, A. Kemoun, and B. Izrar, “Recovering deterministic behavior from experimental time series in mixing reactor,” AIChE Journal, vol. 43, no. 9, pp. 2194-2202, 1997. |

[73] | J. Timmer, H. Rust, W. Horbelt, and H. U. Voss, “Parametric, nonparametric and parametric modelling of a chaotic circuit time series,” Physics Letters A, vol. 274, no. 3-4, pp. 123-134, 2000. · Zbl 1055.37586 |

[74] | W. Horbelt, J. Timmer, M. J. Bünner, R. Meucci, and M. Ciofini, “Identifying physical properties of a CO2 laser by dynamical modeling of measured time series,” Physical Review E, vol. 64, no. 1, Article ID 016222, 7 pages, 2001. · Zbl 1033.37040 |

[75] | K. H. Chon and R. J. Cohen, “Linear and nonlinear ARMA model parameter estimation using an artificial neural network,” IEEE Transactions on Biomedical Engineering, vol. 44, no. 3, pp. 168-174, 1997. |

[76] | S. Haykin and J. Príncipe, “Making sense of a complex world,” IEEE Signal Processing Magazine, vol. 15, no. 3, pp. 66-81, 1998. |

[77] | L. A. Aguirre and E. C. Furtado, “Building dynamical models from data and prior knowledge: the case of the first period-doubling bifurcation,” Physical Review E, vol. 76, no. 4, Article ID 046219, 13 pages, 2007. |

[78] | A. M. Albano, A. Passamante, T. Hediger, and M. E. Farrell, “Using neural nets to look for chaos,” Physica D, vol. 58, no. 1-4, pp. 1-9, 1992. · Zbl 1194.68184 |

[79] | J. B. Elsner, “Predicting time series using a neural network as a method of distinguishing chaos from noise,” Journal of Physics A, vol. 25, no. 4, pp. 843-850, 1992. |

[80] | K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Physics Letters A, vol. 144, no. 6-7, pp. 333-340, 1990. |

[81] | R. Bakker, J. C. Schouten, C. L. Giles, F. Takens, and C. M. van den Bleek, “Learning chaotic attractors by neural networks,” Neural Computation, vol. 12, no. 10, pp. 2355-2383, 2000. |

[82] | H. M. Henrique, E. L. Lima, and J. C. Pinto, “A bifurcation study on neural network models for nonlinear dynamic systems,” Latin American Applied Research, vol. 28, no. 3, pp. 187-200, 1998. |

[83] | R. A. Adomaitis, R. M. Farber, J. L. Hudson, I. G. Kevrekidis, M. Kube, and A. S. Lapedes, “Application of neural nets to system identification and bifurcation analysis of real world experimental data,” in Neural Networks: Biological Computers or Electronic Brains, pp. 87-97, Springer, Paris, France, 1990. |

[84] | R. Gen\ccay and T. Liu, “Nonlinear modeling and prediction with feedforward and recurrent networks,” Physica D, vol. 108, no. 1-2, pp. 119-134, 1997. |

[85] | G. Boudjema and B. Cazelles, “Extraction of nonlinear dynamics from short and noisy time series,” Chaos, Solitons & Fractals, vol. 12, no. 11, pp. 2051-2069, 2001. · Zbl 0980.37034 |

[86] | H. J. Kim and K. S. Chang, “A method of model validation for chaotic chemical reaction systems based on neural networks,” Korean Journal of Chemical Engineering, vol. 18, no. 5, pp. 623-629, 2001. |

[87] | M. Small, Applied Nonlinear Time Series Analysis, vol. 52 of World Scientific Series on Nonlinear Science. Series A, World Scientific, Hackensack, NJ, USA, 2005. · Zbl 1186.62109 |

[88] | L. A. Smith, “Identification and prediction of low-dimensional dynamics,” Physica D, vol. 58, no. 1-4, pp. 50-76, 1992. · Zbl 1194.37142 |

[89] | S. Ogawa, T. Ikeguchi, T. Matozaki, and K. Aihara, “Nonlinear modeling by radial basis function networks,” IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E79-A, no. 10, pp. 1608-1617, 1996. |

[90] | T. Miyano, S. Kimoto, H. Shibuta, K. Nakashima, Y. Ikenaga, and K. Aihara, “Time series analysis and prediction on complex dynamical behavior observed in a blast furnace,” Physica D, vol. 135, no. 3-4, pp. 305-330, 2000. · Zbl 0985.37096 |

[91] | B. Pilgram, K. Judd, and A. Mees, “Modelling the dynamics of nonlinear time series using canonical variate analysis,” Physica D, vol. 170, no. 2, pp. 103-117, 2002. · Zbl 1019.37046 |

[92] | E. Bagarinao Jr., T. Nomura, K. Pakdaman, and S. Sato, “Generalized one-parameter bifurcation diagram reconstruction using time series,” Physica D, vol. 124, no. 1-3, pp. 258-270, 1998. · Zbl 0988.37095 |

[93] | E. Bagarinao Jr., K. Pakdaman, T. Nomura, and S. Sato, “Time series-based bifurcation diagram reconstruction,” Physica D, vol. 130, no. 3-4, pp. 211-231, 1999. · Zbl 0988.37096 |

[94] | E. Bagarinao Jr., K. Pakdaman, T. Nomura, and S. Sato, “Reconstructing bifurcation diagrams from noisy time series using nonlinear autoregressive models,” Physical Review E, vol. 60, no. 1, pp. 1073-1076, 1999. |

[95] | E. Bagarinao Jr., K. Pakdaman, T. Nomura, and S. Sato, “Reconstructing bifurcation diagrams of dynamical systems using measured time series,” Methods of Information in Medicine, vol. 39, no. 2, pp. 146-149, 2000. |

[96] | M. Barahona and C.-S. Poon, “Detection of nonlinear dynamics in short, noisy time series,” Nature, vol. 381, no. 6579, pp. 215-217, 1996. |

[97] | K. H. Chon, J. K. Kanters, R. J. Cohen, and N.-H. Holstein-Rathlou, “Detection of chaotic determinism in time series from randomly forced maps,” Physica D, vol. 99, no. 4, pp. 471-486, 1997. · Zbl 0913.62080 |

[98] | L. A. Aguirre, B. O. S. Teixeira, and L. A. B. Tôrres, “Using data-driven discrete-time models and the unscented Kalman filter to estimate unobserved variables of nonlinear systems,” Physical Review E, vol. 72, no. 2, Article ID 026226, 12 pages, 2005. |

[99] | N. Wessel, H. Malberg, R. Bauernschmitt, A. Schirdewan, and J. Kurths, “Nonlinear additive autoregressive model-based analysis of short-term heart rate variability,” Medical and Biological Engineering and Computing, vol. 44, no. 4, pp. 321-330, 2006. |

[100] | D. A. Vaccari and H.-K. Wang, “Multivariate polynomial regression for identification of chaotic time series,” Mathematical and Computer Modelling of Dynamical Systems, vol. 13, no. 4, pp. 395-412, 2007. · Zbl 1117.62099 |

[101] | D. J. de Oliveira, C. Letellier, M. E. D. Gomes, and L. A. Aguirre, “The use of synthetic input sequences in time series modeling,” Physics Letters A, vol. 372, no. 32, pp. 5276-5282, 2008. · Zbl 1223.60026 |

[102] | M. C. S. Coelho, E. M. A. M. Mendes, and L. A. Aguirre, “Testing for intracycle determinism in pseudoperiodic time series,” Chaos, vol. 18, no. 2, Article ID 023125, p. 12, 2008. · Zbl 06417136 |

[103] | M. Lei and G. Meng, “The influence of noise on nonlinear time series detection based on Volterra-Wiener-Korenberg model,” Chaos, Solitons & Fractals, vol. 36, no. 2, pp. 512-516, 2008. |

[104] | E. Floriani, T. Dudok de Wit, and P. Le Gal, “Nonlinear interactions in a rotating disk flow: from a Volterra model to the Ginzburg-Landau equation,” Chaos, vol. 10, no. 4, pp. 834-847, 2000. · Zbl 1014.76094 |

[105] | M. E. D. Gomes, A. V. P. Souza, H. N. Guimarães, and L. A. Aguirre, “Evidences of determinism in HRV signals,” Chaos, vol. 10, no. 2, pp. 398-410, 2000. · Zbl 0990.92006 |

[106] | L. A. Aguirre, C. Letellier, and J. Maquet, “Forecasting the time series of sunspot numbers,” Solar Physics, vol. 249, no. 1, pp. 103-120, 2008. |

[107] | S. A. Billings and S. Chen, “Identification of nonlinear rational systems using a prediction-error estimation algorithm,” International Journal of Systems Science, vol. 20, no. 3, pp. 467-494, 1989. · Zbl 0674.93066 |

[108] | Q. M. Zhu and S. A. Billings, “Recursive parameter estimation for nonlinear rational models,” Journal of Systems Engineering, vol. 1, pp. 63-76, 1991. |

[109] | Q. M. Zhu and S. A. Billings, “Parameter estimation for stochastic nonlinear rational models,” International Journal of Control, vol. 57, no. 2, pp. 309-333, 1993. · Zbl 0782.62084 |

[110] | M. V. Corrêa, L. A. Aguirre, and E. M. A. M. Menues, “Modeling chaotic dynamics with discrete nonlinear rational models,” International Journal of Bifurcation and Chaos, vol. 10, no. 5, pp. 1019-1032, 2000. |

[111] | O. Ménard, C. Letellier, and G. Gouesbet, “Map modeling by using rational functions,” Physical Review E, vol. 62, no. 5, pp. 6325-6331, 2000. |

[112] | D. Allingham, M. West, and A. I. Mees, “Wavelet reconstruction of nonlinear dynamics,” International Journal of Bifurcation and Chaos, vol. 8, no. 11, pp. 2191-2201, 1998. · Zbl 0984.37102 |

[113] | S. A. Billings and D. Coca, “Discrete wavelet models for identification and qualitative analysis of chaotic systems,” International Journal of Bifurcation and Chaos, vol. 9, no. 7, pp. 1263-1284, 1999. · Zbl 0955.93502 |

[114] | H. L. Wei and S. A. Billings, “Identification and reconstruction of chaotic systems using multiresolution wavelet decompositions,” International Journal of Systems Science, vol. 35, no. 9, pp. 511-526, 2004. · Zbl 1085.93022 |

[115] | R. J. G. B. Campello and W. C. do Amaral, “A relational approach for complex system identification,” Controle & Automa, vol. 10, no. 3, pp. 139-148, 1999. |

[116] | M. C. M. Teixeira and S. H. Zak, “Stabilizing controller design for uncertain nonlinear systems using fuzzy models,” IEEE Transactions on Fuzzy Systems, vol. 7, no. 2, pp. 133-142, 1999. |

[117] | S. Guillaume, “Designing fuzzy inference systems from data: an interpretability-oriented review,” IEEE Transactions on Fuzzy Systems, vol. 9, no. 3, pp. 426-443, 2001. |

[118] | R. Ballini and F. Gomide, “Heuristic learning in recurrent neural fuzzy networks,” Journal of Intelligent & Fuzzy Systems, vol. 13, no. 2-4, pp. 63-74, 2002. · Zbl 1110.68432 |

[119] | H. L. Hiew and C. P. Tsang, “An adaptive fuzzy system for modeling chaos,” Information Sciences, vol. 81, no. 3-4, pp. 193-212, 1994. · Zbl 0827.58039 |

[120] | A. A. P. Santos, N. C. A. da Costa Jr., and L. dos Santos Coelho, “Computational intelligence approaches and linear models in case studies of forecasting exchange rates,” Expert Systems with Applications, vol. 33, no. 4, pp. 816-823, 2007. |

[121] | L. Le Sceller, C. Letellier, and G. Gouesbet, “Global vector field reconstruction including a control parameter dependence,” Physics Letters A, vol. 211, no. 4, pp. 211-216, 1996. · Zbl 1060.37502 |

[122] | M. Casdagli, “A dynamical systems approach to modeling input-output systems,” in Nonlinear Modeling and Forecasting, M. Casdagli and S. Eubank, Eds., pp. 265-281, Addison Wesley, New York, NY, USA, 1992. |

[123] | N. F. Hunter, “Application of nonlinear time-series models to driven systems,” in Nonlinear Modeling and Forecasting, M. Casdagli and S. Eubank, Eds., pp. 467-491, Addison Wesley, New York, NY, USA, 1992. |

[124] | I. J. Leontaritis and S. A. Billings, “Input-output parametric models for nonlinear systems-part I: deterministic non-linear systems,” International Journal of Control, vol. 41, no. 2, pp. 303-328, 1985. · Zbl 0569.93011 |

[125] | L. A. Aguirre and S. A. Billings, “Retrieving dynamical invariants from chaotic data using NARMAX models,” International Journal of Bifurcation and Chaos, vol. 5, no. 2, pp. 449-474, 1995. · Zbl 0886.58100 |

[126] | J. Lu, J. Lü, J. Xie, and G. Chen, “Reconstruction of the Lorenz and Chen systems with noisy observations,” Computers & Mathematics with Applications, vol. 46, no. 8-9, pp. 1427-1434, 2003. · Zbl 1046.37016 |

[127] | E. M. A. M. Mendes and S. A. Billings, “A note on discretization of nonlinear differential equations,” Chaos, vol. 12, no. 1, pp. 66-71, 2002. |

[128] | E. M. A. M. Mendes and C. Letellier, “Displacement in the parameter space versus spurious solution of discretization with large time step,” Journal of Physics A, vol. 37, no. 4, pp. 1203-1218, 2004. · Zbl 1036.37031 |

[129] | R. Reed, “Pruning algorithms-a survey,” IEEE Transactions on Neural Networks, vol. 4, no. 5, pp. 740-747, 1993. |

[130] | A. C. Tsoia and A. Back, “Discrete time recurrent neural network architectures: a unifying review,” Neurocomputing, vol. 15, no. 3-4, pp. 183-223, 1997. · Zbl 0879.68092 |

[131] | R. de Albuquerque Teixeira, A. P. Braga, R. H. C. Takahashi, and R. R. Saldanha, “Improving generalization of MLPs with multi-objective optimization,” Neurocomputing, vol. 35, no. 1-4, pp. 189-194, 2000. · Zbl 1003.68627 |

[132] | M. Small and C. K. Tse, “Minimum description length neural networks for time series prediction,” Physical Review E, vol. 66, no. 6, Article ID 066701, 12 pages, 2002. · Zbl 1008.37048 |

[133] | D. A. G. Vieira, J. A. Vasconcelos, and W. M. Caminhas, “Controlling the parallel layer perceptron complexity using a multiobjective learning algorithm,” Neural Computing & Applications, vol. 16, no. 4-5, pp. 317-325, 2007. · Zbl 05192940 |

[134] | B. Feil, J. Abonyi, and F. Szeifert, “Model order selection of nonlinear input-output models-a clustering based approach,” Journal of Process Control, vol. 14, no. 6, pp. 593-602, 2004. |

[135] | J. Madár, J. Abonyi, and F. Szeifert, “Genetic programming for the identification of nonlinear input-output models,” Industrial and Engineering Chemistry Research, vol. 44, no. 9, pp. 3178-3186, 2005. |

[136] | E. Baake, M. Baake, H. G. Bock, and K. M. Briggs, “Fitting ordinary differential equations to chaotic data,” Physical Review A, vol. 45, no. 8, pp. 5524-5529, 1992. |

[137] | G. L. Baker, J. P. Gollub, and J. A. Blackburn, “Inverting chaos: extracting system parameters from experimental data,” Chaos, vol. 6, no. 4, pp. 528-533, 1996. · Zbl 1055.37529 |

[138] | V. S. Anishchenko, A. N. Pavlov, and N. B. Janson, “Global reconstruction in the presence of a priori information,” Chaos, Solitons & Fractals, vol. 9, no. 8, pp. 1267-1278, 1998. · Zbl 0972.37549 |

[139] | G. Rowlands and J. C. Sprott, “Extraction of dynamical equations from chaotic data,” Physica D, vol. 58, no. 1-4, pp. 251-259, 1992. · Zbl 1194.37141 |

[140] | A. I. Mees, “Parsimonious dynamical reconstruction,” International Journal of Bifurcation and Chaos, vol. 3, no. 3, pp. 669-675, 1993. · Zbl 0875.62426 |

[141] | J. B. Kadtke, J. Brush, and J. Holzfuss, “Global dynamical equations and Lyapunov exponents from noisy chaotic time series,” International Journal of Bifurcation and Chaos, vol. 3, no. 3, pp. 607-616, 1993. · Zbl 0875.58025 |

[142] | L. A. Aguirre and S. A. Billings, “Dynamical effects of overparametrization in nonlinear models,” Physica D, vol. 80, no. 1-2, pp. 26-40, 1995. · Zbl 0888.58060 |

[143] | E. M. A. M. Mendes, Identification of nonlinear discrete systems with intelligent structure detection, Ph.D. thesis, University of Sheffield, Sheffield, UK, 1995. |

[144] | L. A. Aguirre and E. M. A. M. Mendes, “Global nonlinear polynomial models: structure, term clusters and fixed points,” International Journal of Bifurcation and Chaos, vol. 6, no. 2, pp. 279-294, 1996. · Zbl 0870.58091 |

[145] | E. M. A. M. Mendes and S. A. Billings, “On identifying global nonlinear discrete models from chaotic data,” International Journal of Bifurcation and Chaos, vol. 7, no. 11, pp. 2593-2601, 1997. · Zbl 0976.93501 |

[146] | E. M. A. M. Mendes and S. A. Billings, “On overparametrization of nonlinear discrete systems,” International Journal of Bifurcation and Chaos, vol. 8, no. 3, pp. 535-556, 1998. · Zbl 0933.37055 |

[147] | E. M. A. M. Mendes and S. A. Billings, “An alternative solution to the model structure selection problem,” IEEE Transactions on Systems, Man, and Cybernetics, Part A, vol. 31, no. 6, pp. 597-608, 2001. |

[148] | L. A. Aguirre, “Some remarks on structure selection for nonlinear models,” International Journal of Bifurcation and Chaos, vol. 4, no. 6, pp. 1707-1714, 1994. · Zbl 0875.93030 |

[149] | L. A. Aguirre and S. A. Billings, “Improved structure selection for nonlinear models based on term clustering,” International Journal of Control, vol. 62, no. 3, pp. 569-587, 1995. · Zbl 0837.93009 |

[150] | R. Brown, V. In, and E. R. Tracy, “Parameter uncertainties in models of equivariant dynamical systems,” Physica D, vol. 102, no. 3-4, pp. 208-226, 1997. · Zbl 0890.58037 |

[151] | L. A. Aguirre, R. A. M. Lopes, G. F. V. Amaral, and C. Letellier, “Constraining the topology of neural networks to ensure dynamics with symmetry properties,” Physical Review E, vol. 69, no. 2, Article ID 026701, 11 pages, 2004. |

[152] | J. D. Bomberger and D. E. Seborg, “Determination of model order for NARX models directly from input-output data,” Journal of Process Control, vol. 8, no. 5-6, pp. 459-468, 1998. |

[153] | J. Rissanen, Stochastic Complexity in Statistical Inquiry, vol. 15 of Series in Computer Science, World Scientific, Singapore, 1989. · Zbl 0800.68508 |

[154] | K. Judd and A. Mees, “On selecting models for nonlinear time series,” Physica D, vol. 82, no. 4, pp. 426-444, 1995. · Zbl 0888.58034 |

[155] | S. A. Billings, S. Chen, and M. J. Korenberg, “Identification of MIMO nonlinear systems using a forward-regression orthogonal estimator,” International Journal of Control, vol. 49, no. 6, pp. 2157-2189, 1989. · Zbl 0683.93074 |

[156] | L. A. Aguirre and S. A. Billings, “Identification of models for chaotic systems from noisy data: implications for performance and nonlinear filtering,” Physica D, vol. 85, no. 1-2, pp. 239-258, 1995. · Zbl 0888.93015 |

[157] | L. A. Aguirre, P. F. Donoso-Garcia, and R. Santos-Filho, “Use of a priori information in the identification of global nonlinear models-a case study using a buck converter,” IEEE Transactions on Circuits and Systems I, vol. 47, no. 7, pp. 1081-1085, 2000. |

[158] | C. Letellier and L. A. Aguirre, “Investigating nonlinear dynamics from time series: the influence of symmetries and the choice of observables,” Chaos, vol. 12, no. 3, pp. 549-558, 2002. · Zbl 1080.37600 |

[159] | C. Letellier, L. A. Aguirre, and J. Maquet, “How the choice of the observable may influence the analysis of nonlinear dynamical systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 11, no. 5, pp. 555-576, 2006. · Zbl 1099.37521 |

[160] | C. J. Cellucci, A. M. Albano, and P. E. Rapp, “Comparative study of embedding methods,” Physical Review E, vol. 67, no. 6, Article ID 066210, 13 pages, 2003. · Zbl 1046.37510 |

[161] | D. A. Smirnov, B. P. Bezruchko, and Y. P. Seleznev, “Choice of dynamical variables for global reconstruction of model equations from time series,” Physical Review E, vol. 65, no. 2, Article ID 026205, 7 pages, 2002. · Zbl 1049.37501 |

[162] | K. Judd and A. Mees, “Embedding as a modeling problem,” Physica D, vol. 120, no. 3-4, pp. 273-286, 1998. · Zbl 0965.37061 |

[163] | U. Parlitz, “Estimating model parameters from time series by autosynchronization,” Physical Review Letters, vol. 76, no. 8, pp. 1232-1235, 1996. |

[164] | A. Maybhate and R. E. Amritkar, “Use of synchronization and adaptive control in parameter estimation from a time series,” Physical Review E, vol. 59, no. 1, pp. 284-293, 1999. |

[165] | C. Tao, Y. Zhang, G. Du, and J. J. Jiang, “Fitting model equations to time series using chaos synchronization,” Physics Letters A, vol. 332, no. 3-4, pp. 197-206, 2004. · Zbl 1123.37325 |

[166] | U. S. Freitas, E. E. N. Macau, and C. Grebogi, “Using geometric control and chaotic synchronization to estimate an unknown model parameter,” Physical Review E, vol. 71, no. 4, Article ID 047203, 4 pages, 2005. |

[167] | E. J. Kostelich, “Problems in estimating dynamics from data,” Physica D, vol. 58, no. 1-4, pp. 138-152, 1992. · Zbl 1194.37134 |

[168] | L. Jaeger and H. Kantz, “Unbiased reconstruction of the dynamics underlying a noisy chaotic time series,” Chaos, vol. 6, no. 3, pp. 440-450, 1996. |

[169] | H. U. Voss, J. Timmer, and J. Kurths, “Nonlinear dynamical system identification from uncertain and indirect measurements,” International Journal of Bifurcation and Chaos, vol. 14, no. 6, pp. 1905-1933, 2004. · Zbl 1129.93545 |

[170] | J. M. Fullana, M. Rossi, and S. Zaleski, “Parameter identification in noisy extended systems: a hydrodynamic case,” Physica D, vol. 103, no. 1-4, pp. 564-575, 1997. · Zbl 1194.65123 |

[171] | P. Perona, A. Porporato, and L. Ridolfi, “On the trajectory method for the reconstruction of differential equations from time series,” Nonlinear Dynamics, vol. 23, no. 1, pp. 13-33, 2000. · Zbl 0966.34009 |

[172] | P. Connally, K. Li, and G. W. Irwin, “Prediction- and simulation-error based perceptron training: solution space analysis and a novel combined training scheme,” Neurocomputing, vol. 70, no. 4-6, pp. 819-827, 2007. |

[173] | J. L. Breeden and N. P. Packard, “A learning algorithm for optimal representation of experimental data,” International Journal of Bifurcation and Chaos, vol. 4, no. 2, pp. 311-326, 1994. · Zbl 0811.90018 |

[174] | J. Timmer, “Modeling noisy time series: physiological tremor,” International Journal of Bifurcation and Chaos, vol. 8, no. 7, pp. 1505-1516, 1998. · Zbl 0937.92006 |

[175] | H. U. Voss, A. Schwache, J. Kurths, and F. Mitschke, “Equations of motion from chaotic data: a driven optical fiber ring resonator,” Physics Letters A, vol. 256, no. 1, pp. 47-54, 1999. |

[176] | B. P. Bezruchko, A. S. Karavaev, V. I. Ponomarenko, and M. D. Prokhorov, “Reconstruction of time-delay systems from chaotic time series,” Physical Review E, vol. 64, no. 5, Article ID 056216, 6 pages, 2001. · Zbl 1079.93025 |

[177] | D. A. Smirnov, V. S. Vlaskin, and V. I. Ponomarenko, “Estimation of parameters in one-dimensional maps from noisy chaotic time series,” Physics Letters A, vol. 336, no. 6, pp. 448-458, 2005. · Zbl 1136.37361 |

[178] | E. G. Nepomuceno, R. H. C. Takahashi, G. F. V. Amaral, and L. A. Aguirre, “Nonlinear identification using prior knowledge of fixed points: a multiobjective approach,” International Journal of Bifurcation and Chaos, vol. 13, no. 5, pp. 1229-1246, 2003. · Zbl 1056.37091 |

[179] | L. A. Aguirre, E. C. Furtado, and L. A. B. Tôrres, “Evaluation of dynamical models: dissipative synchronization and other techniques,” Physical Review E, vol. 74, no. 6, Article ID 066203, 16 pages, 2006. |

[180] | L. A. Aguirre and S. A. Billings, “Validating identified nonlinear models with chaotic dynamics,” International Journal of Bifurcation and Chaos, vol. 4, no. 1, pp. 109-125, 1994. · Zbl 0876.58028 |

[181] | E. Bagarinao and S. Sato, “Algorithm for vector autoregressive model parameter estimation using an orthogonalization procedure,” Annals of Biomedical Engineering, vol. 30, no. 2, pp. 260-271, 2002. |

[182] | A. Garulli, C. Mocenni, A. Vicino, and A. Tesi, “Integrating identification and qualitative analysis for the dynamic model of a lagoon,” International Journal of Bifurcation and Chaos, vol. 13, no. 2, pp. 357-374, 2003. · Zbl 1058.92044 |

[183] | K. Rodríguez-Vázquez and P. J. Fleming, “Evolution of mathematical models of chaotic systems based on multiobjective genetic programming,” Knowledge and Information Systems, vol. 8, no. 2, pp. 235-256, 2005. · Zbl 02224727 |

[184] | 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. |

[185] | R. Brown, N. F. Rul’kov, and E. R. Tracy, “Modeling and synchronizing chaotic systems from experimental data,” Physics Letters A, vol. 194, no. 1-2, pp. 71-76, 1994. |

[186] | G. Gouesbet, S. Meunier-Guttin-Cluzel, and O. Ménard, “Global reconstructions of equations of motion from data series, and validation techniques, a review,” in Chaos and Its Reconstruction, G. Gouesbet, S. Meunier-Guttin-Cluzel, and O. Ménard, Eds., pp. 1-160, Nova Science, New York, NY, USA, 2003. |

[187] | D. Coca and S. A. Billings, “Identification of coupled map lattice models of complex spatio-temporal patterns,” Physics Letters A, vol. 287, no. 1-2, pp. 65-73, 2001. · Zbl 0971.37022 |

[188] | S. Ishii and M.-A. Sato, “Reconstruction of chaotic dynamics by on-line EM algorithm,” Neural Networks, vol. 14, no. 9, pp. 1239-1256, 2001. · Zbl 02022188 |

[189] | L. A. Aguirre and Á. V. P. Souza, “Stability analysis of sleep apnea time series using identified models: a case study,” Computers in Biology and Medicine, vol. 34, no. 4, pp. 241-257, 2004. |

[190] | J. Maquet, C. Letellier, and L. A. Aguirre, “Scalar modeling and analysis of a 3D biochemical reaction model,” Journal of Theoretical Biology, vol. 228, no. 3, pp. 421-430, 2004. |

[191] | O. Ménard, C. Letellier, J. Maquet, L. Le Sceller, and G. Gouesbet, “Analysis of a nonsynchronized sinusoidally driven dynamical system,” International Journal of Bifurcation and Chaos, vol. 10, no. 7, pp. 1759-1772, 2000. · Zbl 1090.34557 |

[192] | C. Letellier, L. Le Sceller, P. Dutertre, G. Gouesbet, Z. Fei, and J. L. Hudson, “Topological characterization and global vector field reconstruction of an experimental electrochemical system,” Journal of Physical Chemistry, vol. 99, no. 18, pp. 7016-7027, 1995. |

[193] | N. B. Tufillaro, P. Wyckoff, R. Brown, T. Schreiber, and T. Molteno, “Topological time-series analysis of a string experiment and its synchronized model,” Physical Review E, vol. 51, no. 1, pp. 164-174, 1995. |

[194] | C. Letellier and G. Gouesbet, “Topological characterization of reconstructed attractors modding out symmetries,” Journal de Physique II, vol. 6, no. 11, pp. 1615-1638, 1996. |

[195] | M. Small and K. Judd, “Comparisons of new nonlinear modeling techniques with applications to infant respiration,” Physica D, vol. 117, no. 1-4, pp. 283-298, 1998. · Zbl 1041.62523 |

[196] | J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, “Testing for nonlinearity in time series: the method of surrogate data,” Physica D, vol. 58, no. 1-4, pp. 77-94, 1992. · Zbl 1194.37144 |

[197] | J. Timmer, “Power of surrogate data testing with respect to nonstationarity,” Physical Review E, vol. 58, no. 4, pp. 5153-5156, 1998. |

[198] | J. P. Barnard, C. Aldrich, and M. Gerber, “Identification of dynamic process systems with surrogate data methods,” AIChE Journal, vol. 47, no. 9, pp. 2064-2075, 2001. |

[199] | C. Letellier, T. D. Tsankov, G. Byrne, and R. Gilmore, “Large-scale structural reorganization of strange attractors,” Physical Review E, vol. 72, no. 2, Article ID 026212, 12 pages, 2005. |

[200] | P. E. McSharry and L. A. Smith, “Consistent nonlinear dynamics: identifying model inadequacy,” Physica D, vol. 192, no. 1-2, pp. 1-22, 2004. · Zbl 1059.37064 |

[201] | R. Brown, N. F. Rul’kov, and E. R. Tracy, “Modeling and synchronizing chaotic systems from time-series data,” Physical Review E, vol. 49, no. 5, pp. 3784-3800, 1994. |

[202] | C. Letellier, O. Ménard, and L. A. Aguirre, “Validation of selected global models,” in Modeling and Forecasting Financial Data: Techniques of Nonlinear Dynamics, A. S. Soofi and L. Cao, Eds., pp. 283-302, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002. |

[203] | C. Sarasola, F. J. Torrealdea, A. d’Anjou, and M. Graña, “Cost of synchronizing different chaotic systems,” Mathematics and Computers in Simulation, vol. 58, no. 4-6, pp. 309-327, 2002. · Zbl 0995.65134 |

[204] | R. Gilmore and M. Lefranc, The Topology of Chaos: Alice in Stretch and Squeezelan, Wiley-Interscience, New York, NY, USA, 2002. · Zbl 1019.37016 |

[205] | C. Letellier, J. Maquet, L. Le Sceller, G. Gouesbet, and L. A. Aguirre, “On the non-equivalence of observables in phase-space reconstructions from recorded time series,” Journal of Physics A, vol. 31, no. 39, pp. 7913-7927, 1998. · Zbl 0936.81014 |

[206] | H. J. A. F. Tulleken, “Grey-box modelling and identification using physical knowledge and Bayesian techniques,” Automatica, vol. 29, no. 2, pp. 285-308, 1993. · Zbl 0800.93080 |

[207] | E. Eskinat, S. H. Johnson, and W. L. Luyben, “Use of auxiliary information in system identification,” Industrial and Engineering Chemistry Research, vol. 32, no. 9, pp. 1981-1992, 1993. |

[208] | T. A. Johansen, “Identification of non-linear systems using empirical data and prior knowledge-an optimization approach,” Automatica, vol. 32, no. 3, pp. 337-356, 1996. · Zbl 0851.93024 |

[209] | M. V. Corrêa, L. A. Aguirre, and R. R. Saldanha, “Using steady-state prior knowledge to constrain parameter estimates in nonlinear system identification,” IEEE Transactions on Circuits and Systems I, vol. 49, no. 9, pp. 1376-1381, 2002. |

[210] | L. A. Aguirre, M. F. S. Barroso, R. R. Saldanha, and E. M. A. M. Mendes, “Imposing steady-state performance on identified nonlinear polynomial models by means of constrained parameter estimation,” IEE Proceedings: Control Theory and Applications, vol. 151, no. 2, pp. 174-179, 2004. |

[211] | P. E. Rapp, T. I. Schmah, and A. I. Mees, “Models of knowing and the investigation of dynamical systems,” Physica D, vol. 132, no. 1-2, pp. 133-149, 1999. · Zbl 0940.37021 |

[212] | G. Byrne, R. Gilmore, and C. Letellier, “Distinguishing between folding and tearing mechanisms in strange attractors,” Physical Review E, vol. 70, no. 5, Article ID 056214, 9 pages, 2004. |

[213] | G. F. V. Amaral, C. Letellier, and L. A. Aguirre, “Piecewise affine models of chaotic attractors: the Rössler and Lorenz systems,” Chaose, vol. 16, no. 1, Article ID 013115, 14 pages, 2006. · Zbl 1144.37311 |

[214] | R. Gilmore, “Summary of the second workshop on measures of complexity and chaos,” International Journal of Bifurcation and Chaos, vol. 3, no. 3, pp. 491-524, 1993. · Zbl 0870.58002 |

[215] | R. M. May, “Deterministic models with chaotic dynamics,” Nature, vol. 256, no. 5514, pp. 165-166, 1975. |

[216] | R. M. May, “Simple mathematical models with very complicated dynamics,” Nature, vol. 261, no. 5560, pp. 459-467, 1976. · Zbl 1369.37088 |

[217] | M. Hénon, “A two-dimensional mapping with a strange attractor,” Communications in Mathematical Physics, vol. 50, no. 1, pp. 69-77, 1976. · Zbl 0576.58018 |

[218] | M. Xu, G. Chen, and Y.-T. Tian, “Identifying chaotic systems using Wiener and Hammerstein cascade models,” Mathematical and Computer Modelling, vol. 33, no. 4-5, pp. 483-493, 2001. · Zbl 0992.37074 |

[219] | K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Optics Communications, vol. 30, no. 2, pp. 257-261, 1979. |

[220] | O. E. Rössler, “An equation for continuous chaos,” Physics Letters A, vol. 57, no. 5, pp. 397-398, 1976. · Zbl 1371.37062 |

[221] | M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science, vol. 197, no. 4300, pp. 287-289, 1977. · Zbl 1383.92036 |

[222] | B. van der Pol and M. van der Mark, “The heartbeat considered as a relaxation oscillation, and an electrical model of the heart,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 7, vol. 6, pp. 763-775, 1928, Pl 10-12. |

[223] | N. A. Gershenfeld and A. A. Weigend, “The future of time series: learning and understanding,” in Time Series Prediction: Forecasting the Future and Understanding the Past, A. A. Weigend and N. A. Gershenfeld, Eds., pp. 1-70, Addison-Wesley, New York, NY, USA, 1994. |

[224] | D. R. Rigney, A. L. Goldberger, W. C. Ocasio, Y. Ichimaru, G. B. Moody, and R. G. Mark, “Multi-channel physiological data: description and analysis,” in Time Series Prediction: Forecasting the Future and Understanding the Past, A. A. Weigend and N. A. Gershenfeld, Eds., pp. 105-129, Addison-Wesley, New York, NY, USA, 1994. |

[225] | A. A. Weigend and N. A. Gershenfeld, Time Series Prediction: Forecasting the Future and Understanding the Past, Addison-Wesley, New York, NY, USA, 1994. |

[226] | M. Casdagli and A. A. Weigend, “Exploring the continuum between deterministic and stochastic modeling,” in Time Series Prediction: Forecasting the Future and Understanding the Past, A. A. Weigend and N. A. Gershenfeld, Eds., pp. 347-366, Addison-Wesley, New York, NY, USA, 1994. |

[227] | D. Kaplan, “A geometrical statistic for detecting deterministic dynamics,” in Time Series Prediction: Forecasting the Future and Understanding the Past, A. A. Weigend and N. A. Gershenfeld, Eds., pp. 415-428, Addison-Wesley, New York, NY, USA, 1994. |

[228] | M. Palu\vs, “Detecting nonlinearity in multivariate time series,” Physics Letters A, vol. 213, no. 3-4, pp. 138-147, 1996. · Zbl 0972.82548 |

[229] | L. A. Aguirre, V. C. Barros, and Á. V. P. Souza, “Nonlinear multivariable modeling and analysis of sleep apnea time series,” Computers in Biology and Medicine, vol. 29, no. 3, pp. 207-228, 1999. |

[230] | L. Cao and A. Mees, “Deterministic structure in multichannel physiological data,” International Journal of Bifurcation and Chaos, vol. 10, no. 12, pp. 2767-2780, 2000. · Zbl 0985.37095 |

[231] | C. S. M. Lainscsek, F. Schürrer, and J. B. Kadtke, “A general form for global dynamical data models for three-dimensional systems,” International Journal of Bifurcation and Chaos, vol. 8, no. 5, pp. 899-914, 1998. · Zbl 0941.37056 |

[232] | C. Letellier, L. A. Aguirre, J. Maquet, and R. Gilmore, “Evidence for low dimensional chaos in sunspot cycles,” Astronomy & Astrophysics, vol. 449, no. 1, pp. 379-387, 2006. |

[233] | J. Maquet, C. Letellier, and L. A. Aguirre, “Global models from the Canadian lynx cycles as a direct evidence for chaos in real ecosystems,” Journal of Mathematical Biology, vol. 55, no. 1, pp. 21-39, 2007. · Zbl 1145.92036 |

[234] | N. F. Rul’kov, A. R. Volkovskiĭ, A. Rodríguez-Lozano, E. del Río, and M. G. Velarde, “Mutual synchronization of chaotic self-oscillators with dissipative coupling,” International Journal of Bifurcation and Chaos, vol. 2, no. 3, pp. 669-676, 1992. · Zbl 0875.94137 |

[235] | T. Matsumoto, L. O. Chua, and M. Komuro, “The double scroll,” IEEE Transactions on Circuits and Systems, vol. 32, no. 8, pp. 797-818, 1985. · Zbl 0578.94023 |

[236] | L. A. Aguirre, G. G. Rodrigues, and E. M. A. M. Mendes, “Nonlinear identification and cluster analysis of chaotic attractors from a real implementation of Chua’s circuit,” International Journal of Bifurcation and Chaos, vol. 7, no. 6, pp. 1411-1423, 1997. · Zbl 0965.93036 |

[237] | B. Cannas, S. Cincotti, M. Marchesi, and F. Pilo, “Learning of Chua’s circuit attractors by locally recurrent neural networks,” Chaos, Solitons & Fractals, vol. 12, no. 11, pp. 2109-2115, 2001. · Zbl 0981.68135 |

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.