Abstract
In the case called “grey box” in Sect. 5.2, a researcher has partial knowledge about the structure of model equations \({{\mathbf{x}}_{n + 1}} = {\mathbf{f}}({{\mathbf{x}}_n},{\mathbf{c}})\) or \({{{\mathrm{d}}{\mathbf{x}}} \mathord{\left/ {\vphantom {{{\mathrm{d}}{\mathbf{x}}} {{\mathrm{d}}t}}} \right. \kern-\nulldelimiterspace} {{\mathrm{d}}t}} = {\mathbf{f}}({\mathbf{x}},{\mathbf{c}})\). More concretely, some components of the function f are unknown. Then, the problem gets more complicated, than just parameter estimation discussed in Chap. 8, and more interesting from a practical viewpoint.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Anishchenko, V.S., Janson, N.B., Pavlov, A.N.: Global reconstruction in the presence of a priori information. Chaos, Solitons Fractals. 9(8), 1267–1278 (1998)
Bezruchko, B.P., Dikanev, T.V., Smirnov, D.A.: Role of transient processes for reconstruction of model equations from time series. Phys. Rev. E. 64, 036210 (2001a)
Bezruchko, B.P., Karavaev, A.S., Ponomarenko, V.I., Prokhorov, M.D.: Reconstruction of time-delay systems from chaotic time series. Phys. Rev. E. 64, 056216 (2001b)
Bezruchko, B.P., Seleznev, Ye.P., Smirnov, D.A.: Reconstructing equations of a non-autonomous nonlinear oscillator from time series: models and experiment. Izvestiya VUZ. Appl. Nonlinear Dynamics (ISSN 0869-6632). 7(1), 49–67, (in Russian) (1999a)
Bezruchko, B.P., Smirnov, D.A.: Constructing nonautonomous differential equations from a time series. Phys. Rev. E. 63, 016207, (2001)
Boekhoff-Falk, G.: Hearing in Drosophila: development of Johnston’s organ and emerging parallels to vertebrate ear development. Dev. Dyn. 232, 550–558 (2005)
Bünner, M.J., Ciofini, M., Giaquinta, A., et al. Reconstruction of systems with delayed feedback. Eur. Phys. J. D. 10, 165–185 (2000)
Bünner, M.J., Popp, M., Meyer, Th., et al.: Tool to recover scalar time-delay systems from experimental time series. Phys. Rev. E. 54, 3082–3085 (1996)
Dallos, P., Popper, A.N., Fay, R.R. (eds.): The Cochlea. Springer Handbook of Auditory Research. Springer, Berlin (1996)
Friedrich, R., Siegert, S., Peinke, J., Luck St., Siefert, M., Lindemann, M., Raethjen, J., Deuschl, G., Pfister, G.: Extracting model equations from experimental data. Phys. Lett. A. 271, 217–222 (2000)
Goepfert, M.C., Humpfries, A.D.L., Albert, J.T., Robert, D., Hendrich, O.: Power gain exhibited by motile neurons in Drosophila ears. Proc. Natl. Acad. Sci. USA. 102, 325–330 (2005)
Goepfert, M.C., Robert, D.: Active auditory mechanics in mosquitoes. Proc. R. Soc. Lond. B. 268, 333–339 (2001)
Goepfert, M.C., Robert, D.: Motion generation by Drosophila mechanosensory neurons. Proc. Natl. Acad. Sci. USA. 100, 5514–5519 (2003)
Goepfert, M.C., Robert, D.: Nanometer-range acoustic sensitivity in male and female mosquitoes. Proc. R. Soc. Lond. B. 267, 453–457 (2000)
Gold, T.: Hearing. II. The physical basis of the action of the cochlea. Proc. R. Soc. Lond. B. 135, 492–498 (1948)
Hegger, R., Kantz, H., Schmuser, F., et al. Dynamical properties of a ferroelectric capacitors observed through nonlinear time series analysis. Chaos. 8, 727–754 (1998)
Horbelt, W., Timmer, J., Voss, H.U.: Parameter estimation in nonlinear delayed feedback systems from noisy data. Phys. Lett. A. 299, 513–521 (2002)
Kern, A., Stoop, R.: Essential role of couplings between hearing nonlinearities. Phys. Rev. Lett. 91, 128101 (2003)
Ponomarenko, V.I., Prokhorov, M.D., Karavaev, A.S., Bezruchko, B.P.: Recovery of parameters of delayed feedback systems from chaotic time series. J. Exp. Theor. Phys. 100(3), 457–467 (2005)
Ponomarenko, V.I., Prokhorov, M.D.: Coding and recovery of information masked by the chaotic signal of a time-delay system. J. Commun. Technol. Electron. 49(9), 1031–1037 (2004)
Prokhorov, M.D., Ponomarenko, V.I., Karavaev, A.S., Bezruchko, B.P.: Reconstruction of time-delayed feedback systems from time series. Phys. D. 203, 209–223 (2005)
Ragwitz, M., Kantz, H.: Indispensable Finite time corrections for Fokker-Planck equations from time series data. Phys. Rev. Lett. 87, 254501 (2001)
Robert, D., Goepfert, M.C.: Novel schemes for hearing and orientation in insects. Curr. Opin. Neurobiol. 12, 715–720 (2002)
Robles, L., Ruggero, M.A.: Mechanics of the mammalian cochlea. Physiol. Rev. 81, 1305–1352 (2001)
Siefert, M., Kittel, A., Friedrich, R., Peinke, J.: On a quantitative method to analyze dynamical and measurement noise. Europhys. Lett. 61, 466–472 (2003)
Siegert, S., Friedrich, R., Peinke, J.: Analysis of data sets of stochastic systems. Phys. Lett. A. 243, 275–280 (1998)
Smirnov, D.A., Sysoev, I.V., Seleznev Ye.P., Bezruchko, B.P.: Reconstructing nonautonomous system models with discrete spectrum of external action. Tech. Phys. Lett. 29(10), 824–828 (2003)
Stoop, R., Kern, A., Goepfert, M.C., Smirnov, D.A., Dikanev, T.V., Bezrucko, B.P.: A generalization of the van-der-Pol oscillator underlies active signal amplification in Drosophila hearing. Eur. Biophys. J. 35, 511–516 (2006)
Sysoev, I.V., Smirnov, D.A., Seleznev Ye.P., Bezruchko, B.P.: Reconstruction of nonlinear characteristics and equivalent parameters from experimental time series. Proc. 2nd IEEE Int. Conf. Circuits and Systems for Communications. Paper No. 140. Moscow (2004)
Voss, H.U., Kurths, J.: Reconstruction of non-linear time delay models from data by the use of optimal transformations. Phys. Lett. A. 234, 336–344 (1997)
Voss, H.U., Kurths, J.: Reconstruction of nonlinear time delay models from optical data. Chaos, Solitons Fractals. 10, 805–809 (1999)
Voss, H.U., Schwache, A., Kurths, J., Mitschke, F.: Equations of motion from chaotic data: A driven optical fiber ring resonator. Phys. Lett. A. 256, 47–54 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bezruchko, B.P., Smirnov, D.A. (2010). Model Equations: Restoration of Equivalent Characteristics. In: Extracting Knowledge From Time Series. Springer Series in Synergetics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12601-7_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-12601-7_9
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-12600-0
Online ISBN: 978-3-642-12601-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)