Advertisement

Synchronization-Based Parameter Estimation in Chaotic Dynamical Systems

  • Igor TrpevskiEmail author
  • Daniel Trpevski
  • Lasko Basnarkov
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

We examine a method of estimating unknown parameters in models of chaotic dynamical systems by synchronizing the model with the time series measured as output of the system. The method drives the model’s parameters by a set of proper parameter update rules to the true values of the parameters of the modeled system. The theory on how to construct this parameter update rules is given along with simple demonstrations with the Lorenz and Rössler systems. Both the scenario when the output represents the full system of the state, and the case when it is a scalar time series representing a function of the system variables are considered. We demonstrate how to apply the method for estimating the topology of a network of chaotic oscillators. Finally, we illustrate its application to estimating parameters of spatially extended systems that possess translational symmetry with a toy atmospheric model.

Keywords

Lyapunov Exponent Lyapunov Function Chaotic System Lorenz System Error Dynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

The work is supported by the European Commission (ERC Grant #266722). The authors thank the editor for his invaluable support and insightful discussions. We thank Gregory Duane for providing the results in Fig. 5.

References

  1. 1.
    Abarbanel, H.D.I., Creveling, D.R., Jeanne, J.M.: Estimation of parameters in nonlinear systems using balanced synchronization. Phys. Rev. E 77, 016208 (2008)Google Scholar
  2. 2.
    Abarbanel, H.D.I., Creveling, D.R., Farsian, R., Kostuk, M.: Dynamical state and parameter estimation. SIAM J. Appl. Dyn. Syst. 8, 1341–1381 (2009)Google Scholar
  3. 3.
    Afraimovich, V.S., Verichev, N.N., Rabinovich, M.I.: Stochastic synchronization of oscillations in dissipative systems. Radiophys. Quant. Electron. 29, 795–803 (1986)Google Scholar
  4. 4.
    Baake, E., Baake, M., Bock, H.G., Briggs, K.M.: Fitting ordinary differential equations to chaotic data. Phys. Rev. A 45, 5524–5529 (1992)Google Scholar
  5. 5.
    Bryant, P.H.: Optimized synchronization of chaotic and hyperchaotic systems. Phys. Rev. E 82, 015201 (2010)Google Scholar
  6. 6.
    Chen, M., Kurths, J.: Chaos synchronization and parameter estimation from a scalar output signal. Phys. Rev. E 76, 027203 (2007)Google Scholar
  7. 7.
    Cuomo, K.M., Oppenheim, A.V.: Circuit implementation of synchronized chaos with applications to communications. Phys. Rev. Lett. 71, 65–68 (1993)Google Scholar
  8. 8.
    Dai, C., Chen, W., Li, L., Zhu, Y., Yang, Y.: Seeker optimization algorithm for parameter estimation of time-delay chaotic systems. Phys. Rev. E 83, 036203 (2011)Google Scholar
  9. 9.
    Duane, G.S., Yu, D., Kocarev, L.: Identical synchronization, with translation invariance, implies parameter estimation. Phys. Lett. A 371, 416–420 (2007)Google Scholar
  10. 10.
    Evensen, G.: Data Assimilation. The Ensemble Kalman Filter, 2nd edn. Springer, Berlin (2009)Google Scholar
  11. 11.
    Freitas, U.S., Macau, E.E., Grebogi, C.: Using geometric control and chaotic synchronization to estimate an unknown parameter. Phys. Rev. E 71, 047203 (2005)Google Scholar
  12. 12.
    Fujisaka, H., Yamada, T.: Stability theory of synchronized motion in coupled-oscillator systems II. Prog. Theor. Phys. 70, 1240–1248 (1983)Google Scholar
  13. 13.
    Fujisaka, H., Yamada, T.: Stability theory of synchronized motion in coupled-oscillator systems III. Prog. Theor. Phys. 72, 885–894 (1984)Google Scholar
  14. 14.
    Huang, D.: Synchronization-based estimation of all parameters of chaotic systems from time series. Phys. Rev. E 69, 067201 (2004)Google Scholar
  15. 15.
    Julier, S.J., Uhlmann, J.K.: Unscented filtering and nonlinear estimation. Proc. IEEE 92, 401–422 (2004)Google Scholar
  16. 16.
    Kocarev, L., Tasev, T., Parlitz, U.: Synchronizing spatiotemporal chaos of partial differential equations. Phys. Rev. Lett. 79, 51–54 (1997)Google Scholar
  17. 17.
    Konnur, R.: Equivalence of synchronization and control of chaotic systems. Phys. Rev. Lett. 77 2937–2940 (1996)Google Scholar
  18. 18.
    Konnur, R.: Synchronization-based approach for estimating all model parameters of chaotic systems. Phys. Rev. E 67, 027204 (2003)Google Scholar
  19. 19.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)Google Scholar
  20. 20.
    Miguez, J., Marino, I.P.: Adaptive approximation method for joint parameter estimation and identical synchronization of chaotic systems. Phys. Rev. E 72, 057202 (2005)Google Scholar
  21. 21.
    Maybhate, A., Amritkar, R.E.: Use of synchronization and adaptive control in parameter estimation from a time series. Phys. Rev. E 59, 284–293 (1999)Google Scholar
  22. 22.
    Maybhate, A., Amritkar, R.E.: Dynamic algorithm for parameter estimation and its applications. Phys. Rev. E 61, 6461–6470 (2000)Google Scholar
  23. 23.
    Nijmeijer, H.: A dynamical control view on synchronization. Physica D 154, 219–228 (2001)Google Scholar
  24. 24.
    Nijmeijer, H., Mareels, I.M.Y.: An observer looks at synchronization. IEEE Trans. Circ. Syst. 44, 882–890 (1997)Google Scholar
  25. 25.
    Parlitz, U.: Estimating model parameters from time series by autosynchronization. Phys. Rev. Lett. 76, 1232–1235 (1996)Google Scholar
  26. 26.
    Parlitz, U., Junge, L., Kocarev, L.: Synchronization-based parameter estimation from time series. Phys. Rev. E 54, 6253–6259 (1997)Google Scholar
  27. 27.
    Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)Google Scholar
  28. 28.
    Pecora, L.M., Carroll, T.L., Johnson, G.A., Mar, D.J., Heagy, J.F.: Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos 7, 520–543 (1997)Google Scholar
  29. 29.
    Peng, H., Li, L., Yang, Y., Liu, L.: Parameter estimation of dynamical systems via a chaotic ant swarm. Phys. Rev. E 81 016297 (2010)Google Scholar
  30. 30.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge (2003)Google Scholar
  31. 31.
    Quinn, J.C., Bryant, P.H., Creveling, D.R., Klein, S.R., Abarbanel, H.D.I.: Parameter and state estimation of experimental chaotic systems using synchronization. Phys. Rev. E 80, 016201 (2009)Google Scholar
  32. 32.
    Schumann-Bischoff, J., Parlitz, U.: State and parameter estimation using unconstrained optimization. Phys. Rev. E 84 056214 (2011)Google Scholar
  33. 33.
    Timme, M.: Revealing networks connectivity from response dynamics. Phys. Rev. Lett. 98, 224101 (2007)Google Scholar
  34. 34.
    Toth, B.A., Kostuk, M., Meliza, C.D., Margoliash, D., Abarbanel, H.D.I., Creveling, D.R., Farsian, R.: Dynamical estimation of neuron and network properties I: variational methods. Biol. Cybern. 105, 217–237 (2010)Google Scholar
  35. 35.
    Wang, W.-X., Yang, R., Lai, Y.-C., Kovanis, V., Harison, M.A.F.: Time-series based prediction of complex oscillator networks via compressive sensing. Europhys. Lett. 94, 48006 (2011)Google Scholar
  36. 36.
    Yu, D., Parlitz, U.: Estimating parameters by autosynchronization with dynamics restriction. Phys. Rev. E. 77, 066221 (2008)Google Scholar
  37. 37.
    Yu, D., Parlitz, U.: Synchronization and control based parameter identification. In: Kocarev, L., Galias, Z., Lian, S. (eds.) Intelligent Computing Based on Chaos, pp. 227–249. Springer, Berlin (2009)Google Scholar
  38. 38.
    Yu, D., Parlitz, U.: Inferring network connectivity by delayed feedback control. PLoS One 6, e24333 (2011)Google Scholar
  39. 39.
    Yu, D., Righero, M., Kocarev, L.: Estimating topology of networks. Phys. Rev. Lett. 97, 188701 (2006)Google Scholar
  40. 40.
    Yu, W., Chen, G., Cao, J., L\(\ddot{u}\), J., Parlitz, U.: Parameter identification of dynamical systems from time series. Phys. Rev. E 75 067201 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Igor Trpevski
    • 1
    Email author
  • Daniel Trpevski
    • 1
  • Lasko Basnarkov
    • 2
  1. 1.Macedonian Academy for Sciences and ArtsSkopjeMacedonia
  2. 2.Faculty of Computer Science and EngineeringSkopjeMacedonia

Personalised recommendations