1. Solution of the inverse problems in nonlinear and chaotic dynamics (project 2.2) Research group: Prof. S.Berczynski, Szczecin University of Technology Prof. Yu.A.Kravtsov, Maritime University of Szczecin Potential partners: to be found

2. Observed Time Series The inverse problem of nonlinear and chaotic dynamics: to restore differential equations of the system studied from the noisy observed time series. time

3. The inverse problems can be successfully resolved only for low dimensional systems: n=3,4,5,.. Unfortunately, plasma is high dimensional system, and generally plasma dynamics can not be reconstructed by the method of inversion. Nevertheless, it is worth to search for low dimensional uncertain subsystems in tokamak, say, in the control subsystem, which might be reconstructed by the method of inversion. What follows is a brief description of the inversion procedure with special attention to the noise influence on accuracy of dynamical system reconstruction.

4. Method for the inverse problem solution Three kind of the processes in the identification procedure: 1) x(t) –dynamic (physical) process; 2) y(t) –process, observed by the sensors; 3) z(t) = - model process, which serves as estimator for physical process

5. 1) Dynamical equations for x(t) Nonlinear „forces” „Dynamical” noise Parameters 2) Equation for observed process: Observational noise 3) Equation for model process z(t) : b= – estimator for parameters

6. Polinomial Approximation for Nonlinear Forces: - ”multy-index” - monomials Model for identification:

7. Least square criterion for identification: where - Sampling interval

8. S. Berczyński, Yu.A. Kravtsov. Reconstruction of chaotic Rössler system from complete and incomplete noisy empirical data. Submitted to International Journal of Bifurcation and Chaos. Example: Noise influence on reconstruction of the chaotic R ö ssler system on the basis of complete and incomplete empirical data

9. Rössler system Model for identification on the basis of complete empirical data y 1, y 2 i y 3 :

10. Example of reconstruction at weak noise:

11. Example of reconstruction at strong noise:

12. Degree of agreement between observed and restored proceses can be described by a squared correlation coefficient: - Number of samples - Average value of y k (t)

13. Critical noise for reconstruction on the basis of one, two and three measured variables R2R2 0.8

14. The main publications: Anosov O.L., Butkovskii O.Ya., Kravtsov Yu. A. Strategy and algorithms for dynamical forecasting. In: Predictability of Complex Dynamical Systems, Kravtsov Yu.A., Kadtke J.B., Eds. Springer Verlag, Berlin, Heidelberg, 1996, p.105. S.Berczynski, P.Gutowski, Yu.A.Kravtsov. Identification of models for dynamic and chaotic systems. In: Problems of the Modal Analysis, T.Uhl, Ed., AGH, Kraków, 2003, p.13-19. S. Berczyński, Yu.A. Kravtsov. Inverse problem of nonlinear dynamics: algorithms and applications for chaotic systems. 18 Symposium „Modeling in Mechanics”, Wisła, 9-13 Feb., 2004. S. Berczyński, Yu.A. Kravtsov. Reconstruction of chaotic Rössler system from complete and incomplete noisy empirical data. Submitted to International Journal of Bifurcation and Chaos, 2006.