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Dynamical effects of overparametrization in nonlinear models. (English) Zbl 0888.58060
Summary: This paper is concerned with dynamical reconstruction for nonlinear systems. The effects of the driving function and of the complexity of a given representation on the bifurcation pattern are investigated. It is shown that the use of different driving functions to excite the system may yield models with different bifurcation patterns. The complexity of the reconstructions considered is quantified by the embedding dimension and the number of estimated parameters. In this respect it appears that models which reproduce the original bifurcation behaviour are of limited complexity and that excessively complex models tend to induce ghost bifurcations and spurious dynamical regimes. Moreover, some results suggest that the effects of overparametrization on the global dynamical behaviour of a nonlinear model may be more deleterious than the presence of moderate noise levels. In order to precisely quantify the complexity of the reconstructions, global polynomials are used although the results are believed to apply to a much wider class of representations including neural networks.

37N99 Applications of dynamical systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
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[1] Billings, S. A.: Identification of nonlinear systems – a survey. IEE proceedigs pt. D 127, 272-285 (1980)
[2] Grassberger, P.; Schreiber, J.; Schaffrath, C.: Nonlinear time sequence analysis. Int. J. Bifurcation and chaos 1, No. 3, 521-547 (1991) · Zbl 0874.58029
[3] Kadtke, J. B.; Brush, J.; Holzfuss, J.: Global dynamical equations and Lyapunov exponents from noisy chaotic time series. Int. J. Bifurcation and chaos 3, No. 3, 607-616 (1993) · Zbl 0875.58025
[4] Mees, A. I.: Parsimonious dynamical reconstruction. Int. J. Bifurcation and chaos 3, No. 3, 669-675 (1993) · Zbl 0875.62426
[5] Packard, N. H.; Crutchfiled, J. P.; Farmer, J. D.; Shaw, R. S.: Geometry from a time series. Phys. rev. Lett. 45, No. 9, 712-716 (1980)
[6] Takens, F.: Detecting strange attractors in turbulence. Lecture notes in mathematics 898, 366-381 (1980)
[7] Sauer, T.; Yorke, J. A.; Casdagli, M.: Embedology. J. of statistical physics 65, No. 3/4, 579-616 (1991) · Zbl 0943.37506
[8] Farmer, J. D.; Sidorowich, J. J.: Predicting chaotic time series. Phys. rev. Lett. 59, No. 8, 845-848 (1987)
[9] Casdagli, M.: Chaos and deterministic versus stochastic nonlinear modelling. J.R.. stat. Soc. B 54, No. 2, 303-328 (1991)
[10] Abarbanel, H. D. I.; Brown, R.; Kadtke, J. B.: Prediction in chaotic nonlinear systems: methods for time series with broadband Fourier spectra. Phys. rev. A 41, No. 4, 1782-1807 (1990)
[11] Kennel, M. B.; Isabelle, S.: Method to distinguish possible chaos from coloured noise and to determine embedding parameters. Phys. rev. A 46, No. 6, 3111-3118 (1992)
[12] Linsay, P. S.: An efficient method of forecasting chaotic time series using linear regression. Phys. lett. A 153, No. 6,7, 353-356 (1991)
[13] Stokbro, L.; Umberger, D. K.: Forecasting with weighted maps. Nonlinear modeling and forecasting, 73-93 (1992)
[14] Cremers, J.; Hübler, A.: Construction of differential equations from experimental data. Z. naturforsch 42a, 797-802 (1987)
[15] Crutchfield, J. P.; Mcnamara, B. S.: Equations of motion from a data series. Complex systems 1, 417-452 (1987) · Zbl 0675.58026
[16] Aguirre, L. A.; Billings, S. A.: Discrete reconstruction of strange attractors in chuas circuit. Int. J. Bifurcation and chaos (1994) · Zbl 0900.70335
[17] Broomhead, D. S.; Lowe, D.: Multivariable functional interpolation and adaptive networks. Complex systems 2, 321-355 (1988) · Zbl 0657.68085
[18] Casdagli, M.: Nonlinear prediction of chaotic time series. Physica D 35, 335-356 (1989) · Zbl 0671.62099
[19] Principe, J. C.; Rathie, A.; Kuo, J. M.: Prediction of chaotic time series with neural networks and the issue of dynamic modeling. Int. J. Bifurcation and chaos 2, No. 4, 989-996 (1992) · Zbl 0900.62497
[20] Billings, S. A.; Chen, S.: Neural networks and system identification. Neural networs for systems and control, 181-205 (1992) · Zbl 0939.68761
[21] Adomaitis, R. A.; Farber, R. M.; Hudson, J. L.; Kevrekidis, I. G.; Kube, M.; Lapedes, A. S.: Application of neural nets to system identification and bifurcation analysis of real world experimental data. Neural networks: biological computers or electronic brains, 87-97 (1990)
[22] Masri, S. F.; Chassiakos, A. G.; Caughey, T. K.: Identification of nonlinear dynamic systems using neural networks. Transactions of the ASME, J. Appl. mech. 60, 123-133 (1993)
[23] Leontaritis, I. J.; Billings, S. A.: Input-output parametric models for nonlinear systems part I: Deterministic nonlinear systems. Int. J. Control 41, No. 2, 303-328 (1985) · Zbl 0569.93011
[24] Leontaritis, I. J.; Billings, S. A.: Input-output parametric models for nonlinear systems part II: Stochastic nonlinear systems. Int. J. Control 41, No. 2, 329-344 (1985) · Zbl 0569.93012
[25] Casdagli, M.: A dynamical systems approach to modeling input-output systems. Nonlinear modeling and forecasting, 265-281 (1992)
[26] Billings, S. A.; Chen, S.; Korenberg, M. J.: Identification of MIMO nonlinear systems using a forward-regression orthogonal estimator. Int. J. Control 49, No. 6, 2157-2189 (1989) · Zbl 0683.93074
[27] Aguirre, L. A.; Billings, S. A.: Validating identified nonlinear models with chaotic dynamics. Int. J. Bifurcation and chaos 4, No. 1, 109-125 (1994) · Zbl 0876.58028
[28] Ueda, Y.: Steady motions exhibited by Duffing s equation: A picture book of regular and chaotic motions. New approaches to nonlinear problems in dynamics, 311-322 (1980)
[29] Kawakami, H.: Strange attractors in Duffing s equation. Technical report (1986)
[30] Moon, F. C.: Chaotic vibrations - an introduction for applied scientists and engineers. (1987) · Zbl 0745.58003
[31] Haynes, B. R.: A qualitative approach to the global analysis of nonlinear systems with application to system identification. Phd thesis (1989)
[32] Leontaritis, I. J.; Billings, S. A.: Experimental design and identifiability for nonlinear systems. Int. J. Systems sci. 18, No. 1, 189-202 (1987) · Zbl 0619.93066
[33] Hunter, N. F.: Application of nonlinear time-series models to driven systems. Nonlinear modeling and forecasting, 467-491 (1992)
[34] Ueda, Y.; Akamatsu, N.: Chaotically transitional phenomena in the forced negative-resistance oscillator. IEEE trans. Circuits syst. 28, No. 3, 217-224 (1981)
[35] Aguirre, L. A.; Mendes, E. M.; Billings, S. A.: Smoothing data with local instabilities for the identification of chaotic systems. (1994) · Zbl 0847.93025
[36] Akaike, H.: A new look at the statistical model identification. IEEE trans. Automat. contr. 19, No. 6, 716-723 (1974) · Zbl 0314.62039
[37] Hannan, E. J.; Quinn, B. G.: The determination of the order of an autoregression. J. royal statist. Soc. B 41, No. 2, 190-195 (1979) · Zbl 0408.62076
[38] Kashyap, R. L.: A Bayesian comparison of different classes of dynamica models using empirical data. IEEE trans. Automat. contr. 22, No. 5, 715-727 (1977) · Zbl 0369.62007
[39] Chua, L. O.: The genesis of Chua s circuit. Archiv für elektronik und übertragungstechnik 46, No. 4, 250-257 (1992)
[40] Chua, L. O.; Hasker, M.: Speciall issue on chaos in nonlinear electronic circuits. IEEE trans. Circuits syst. 40, 10-11 (1993)
[41] Fraser, A. M.; Swinney, H. L.: Independent coordinates for strange attractors from mutual information. Phys. rev. A 33, No. 2, 1134-1140 (1986) · Zbl 1184.37027
[42] Broomhead, D. S.; Jones, R.; King, G. P.: Topological dimension and local coordinates for time series data. J. phys. A 20, L563-L569 (1987) · Zbl 0644.58030
[43] Aleksic, Z.: Estimating the embedding dimension. Physica D 52, 362-368 (1991) · Zbl 0856.54043
[44] Kennel, M. B.; Brown, R.; Abarbanel, H. D. I.: Determining embedding dimensions for phase-space reconstruction using geometrical construction. Phys. rev. A 45, No. 6, 3403-3411 (1992)
[45] Marteau, P. F.; Abarbanel, H. D. I.: Noise reduction in chaotic times series using scaled probabilistic methods. J. nonlinear sci. 1, 313-343 (1991) · Zbl 0799.60039
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