Methods of Analysis of Chaotic Behavior of Measurement Data of a Complex Energy System
DOI:
https://doi.org/10.26408/103.08Keywords:
chaos, Poincare map, embedding dimension, Lyapunov exponentAbstract
The article presents an example of chaotic behavior in a selected piece of the power network modeled by nonlinear system of differential equations. This article also discusses the most important aspects of identifying chaotic behavior in a system, based on only the recorded time course of one state variable without knowing the real dimension of the phase space.References
Abarbanel, H.D.I., Brown, R., Kadtke, J.B., 1990, Prediction in Chaotic Nonlinear Systems: Methods for time series with broadband Fourier spectra, Physical Review A, vol. 41, no. 4, s. 1782–1807.
[2] Araujo, A.E., Marti, J.R., Soudack, A.C., 1993, Ferroresonance in power systems: chaotic behaviour, IEE Proceedings C, vol. 140, no. 3, s. 237–240.
[3] Baker, G.L., Gollub, J.P., 1998, Wstęp do dynamiki układów chaotycznych, Wydawnictwo Naukowe PWN, Warszawa.
[4] Eckmann, J.P., Kamphorst, S.O., Ruelle, D., 1987, Recurrence plots of dynamical systems, Europhysics Letters, vol. 4, no. 9, s. 973–977.
[5] Eckmann, J.P, Kamphorst, S.O., Ruelle, D., Ciliberto, S., 1986, Liapunov exponents from time series, Physical Review A, vol. 34, no. 6, s. 4971–4979.
[6] Fraser, A.M., Swinney, H.L., 1986, Independent coordinates for strange attractors from mutual information, Physical Review A, vol. 33, no. 2, s. 1134–1140.
[7] Grassberger, P., Procaccia I., 1983, Characterization of strange attractors, Physical Review Letters, vol. 50, no. 5, s. 346–349.
[8] Hirsch, M.W., Smale, S., Devaney, R.L., 2004, Differential equations, dynamical systems and an introduction to chaos, Series: Pure and applied mathematics, vol. 60, Academic Press, New York.
[9] Jankowski, P., 2015, Selected aspects of numerical solution of damped oscillator in ptc prime 3.0 environment, Proceedings of XXI IMEKO World Congress-Full papers, Prague, 22-24 September, s. 603–607.
[10] Kantz, H., 1994, A Robust Method to Estimate the Maximal Lyapunov Exponent of a Time Series, Physical Letters A, vol. 185(1), s. 77–87.
[11] Kantz, H., Schreiber, T., 2004, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge.
[12] Kennel, M.B., Brown, R., Abarbanel H.D.I., 1992, Determining embedding dimension for phase-space reconstruction using a geometrical construction, Physical Review A, vol. 45, no. 6, s. 3403–3411.
[13] Małyszko, O., 2016, Zastosowanie wykładników Lapunowa do badania stabilności sieci elektro¬energe-tycznej, www.astat.com.pl.
[14] Marwan, N., Romano, M.C., Thiel, M., Kurths, J., 2007, Recurrence plots for the analysis of complex systems, Physics Reports, vol. 438, no. 5–6, s. 237–329.
[15] Milnor, J., 1985, On the concept of attractor, Communications of Mathematical Physics, vol. 99, no. 2, s. 177–195.
[16] Milos, M., Schreiber, I., 1991, Chaotic Behaviour of Deterministic Dissipative systems, Cambridge University Press, Cambridge.
[17] Miśkiewicz-Nawrocka, M., 2012, Zastosowanie wykładników Lapunowa do analizy ekonomicznych szeregów czasowych, Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach, Katowice.
[18] Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S., 1980, Geometry from a time series, Physical Review Letters, vol. 45, no. 9, s. 712–716.
[19] Peitgen, H.O., Jürgens, H., Saupe, D., 1996, Granice chaosu. Fraktale, cz. 2, Wydawnictwo Naukowe PWN, Warszawa.
[20] Peters, E.E., 1997, Teoria chaosu a rynki kapitałowe, WiG_Press, Warszawa.
[21] Purczyński, J., 2000, Wybrane problemy numeryczne stosowania analizy R/S, Przegląd Statystyczny, XLVII-2, s. 17–21.
[22] Rosenstein, T., Collins, J.J., De Luca, C.J., 1993, A practical method for calculating largest Lyapunov exponents from small data sets, Physica D: Nonlinear Phenomena, vol. 65, no. 1, s. 117–134.
[23] Ruelle, D., 1989, Chaotic evolution and strange Attractors, Cambridge University Press, Cambridge.
[24] Schuster, H.G., Chaos deterministyczny, Wydawnictwo Naukowe PWN, Warszawa 1995.
[25] Wolf, A., Swift, J., Swinney, H., Vastano, J., 1985, Determining Lyapunov Exponents from a Time Series, Physica D, vol. 16, s. 285–317.
[26] Wysocki, H., 2012, Rekonstrukcja atraktora Monarchy Safye na podstawie szeregów czasowych, Zeszyty Naukowe Akademii Marynarki Wojennej w Gdyni, LIII, nr 1(188), s. 149–172.
[27] Zeug-Żebro, K., 2000, Badanie wpływu redukcji szumu na identyfikację dynamiki chaotycznej na przykładzie finansowych szeregów czasowych, Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach, Katowice.
Remove [1] Abarbanel, H.D.I., Brown, R., Kadtke, J.B., 1990, Prediction in Chaotic Nonlinear Systems: Methods for time series with broadband Fourier spectra, Physical Review A, vol. 41, no. 4, s. 1782–1807.
[2] Araujo, A.E., Marti, J.R., Soudack, A.C., 1993, Ferroresonance in power systems: chaotic behaviour, IEE Proceedings C, vol. 140, no. 3, s. 237–240.
[3] Baker, G.L., Gollub, J.P., 1998, Wstęp do dynamiki układów chaotycznych, Wydawnictwo Naukowe PWN, Warszawa.
[4] Eckmann, J.P., Kamphorst, S.O., Ruelle, D., 1987, Recurrence plots of dynamical systems, Europhysics Letters, vol. 4, no. 9, s. 973–977.
[5] Eckmann, J.P, Kamphorst, S.O., Ruelle, D., Ciliberto, S., 1986, Liapunov exponents from time series, Physical Review A, vol. 34, no. 6, s. 4971–4979.
[6] Fraser, A.M., Swinney, H.L., 1986, Independent coordinates for strange attractors from mutual information, Physical Review A, vol. 33, no. 2, s. 1134–1140.
[7] Grassberger, P., Procaccia I., 1983, Characterization of strange attractors, Physical Review Letters, vol. 50, no. 5, s. 346–349.
[8] Hirsch, M.W., Smale, S., Devaney, R.L., 2004, Differential equations, dynamical systems and an introduction to chaos, Series: Pure and applied mathematics, vol. 60, Academic Press, New York.
[9] Jankowski, P., 2015, Selected aspects of numerical solution of damped oscillator in ptc prime 3.0 environment, Proceedings of XXI IMEKO World Congress-Full papers, Prague, 22-24 September, s. 603–607.
[10] Kantz, H., 1994, A Robust Method to Estimate the Maximal Lyapunov Exponent of a Time Series, Physical Letters A, vol. 185(1), s. 77–87.
[11] Kantz, H., Schreiber, T., 2004, Nonlinear Time Series Analysis, Cambridge University Press, Cambridge.
[12] Kennel, M.B., Brown, R., Abarbanel H.D.I., 1992, Determining embedding dimension for phase-space reconstruction using a geometrical construction, Physical Review A, vol. 45, no. 6, s. 3403–3411.
[13] Małyszko, O., 2016, Zastosowanie wykładników Lapunowa do badania stabilności sieci elektro¬energe-tycznej, www.astat.com.pl.
[14] Marwan, N., Romano, M.C., Thiel, M., Kurths, J., 2007, Recurrence plots for the analysis of complex systems, Physics Reports, vol. 438, no. 5–6, s. 237–329.
[15] Milnor, J., 1985, On the concept of attractor, Communications of Mathematical Physics, vol. 99, no. 2, s. 177–195.
[16] Milos, M., Schreiber, I., 1991, Chaotic Behaviour of Deterministic Dissipative systems, Cambridge University Press, Cambridge.
[17] Miśkiewicz-Nawrocka, M., 2012, Zastosowanie wykładników Lapunowa do analizy ekonomicznych szeregów czasowych, Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach, Katowice.
[18] Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S., 1980, Geometry from a time series, Physical Review Letters, vol. 45, no. 9, s. 712–716.
[19] Peitgen, H.O., Jürgens, H., Saupe, D., 1996, Granice chaosu. Fraktale, cz. 2, Wydawnictwo Naukowe PWN, Warszawa.
[20] Peters, E.E., 1997, Teoria chaosu a rynki kapitałowe, WiG_Press, Warszawa.
[21] Purczyński, J., 2000, Wybrane problemy numeryczne stosowania analizy R/S, Przegląd Statystyczny, XLVII-2, s. 17–21.
[22] Rosenstein, T., Collins, J.J., De Luca, C.J., 1993, A practical method for calculating largest Lyapunov exponents from small data sets, Physica D: Nonlinear Phenomena, vol. 65, no. 1, s. 117–134.
[23] Ruelle, D., 1989, Chaotic evolution and strange Attractors, Cambridge University Press, Cambridge.
[24] Schuster, H.G., Chaos deterministyczny, Wydawnictwo Naukowe PWN, Warszawa 1995.
[25] Wolf, A., Swift, J., Swinney, H., Vastano, J., 1985, Determining Lyapunov Exponents from a Time Series, Physica D, vol. 16, s. 285–317.
[26] Wysocki, H., 2012, Rekonstrukcja atraktora Monarchy Safye na podstawie szeregów czasowych, Zeszyty Naukowe Akademii Marynarki Wojennej w Gdyni, LIII, nr 1(188), s. 149–172.
[27] Zeug-Żebro, K., 2000, Badanie wpływu redukcji szumu na identyfikację dynamiki chaotycznej na przykładzie finansowych szeregów czasowych, Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach, Katowice.
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