Experimental modeling of chaotic fields

Igor Grabec

Faculty of Mechanical Engineering
University of Ljubljana, Ljubljana, Slovenia

Physical description of natural laws is based on evolution equations of fields but their analytical formulation is often not possible for very complex chaotic phenomena [1]. In the lecture we show how a method of chaotic time series prediction can be generalized to statistical modeling of chaotic fields [2, 3]. For this purpose we assume that a record of the field is provided by an experiment and that the field amplitude $\phi({\bf s})$ at a point of observation ${\bf s}$ is related to amplitudes in a surrounding region. We represent the field values in the surrounding region by the vector ${\bf g}({\bf s})$, and describe the field evolution by the mapping equation

$$\phi ({\bf s})=G({\bf g}({\bf s})), \eqno{(1)}$$

in which the function $G$ is estimated statistically. For this purpose $N$ samples of the joint state vector $\{(\phi({\bf s}_i),{\bf g}({\bf s}_i))=(\phi_i,{\bf g}_i); i=1\ldots N\}$ are first extracted from the given record. As an optimal non-parametric estimator of the field at point ${\bf s}$ we employ the conditional average, which is expressed by

$$\hat \phi ({\bf s})={\rm E}[\phi({\bf s})\vert {\bf g}({\bf ... ...rac{1}{N}\sum_{n=1}^N B_n ({\bf g}({\bf s})) \phi_n \eqno{(2)}$$

Here $B_n({\bf g}({\bf s}))=w({\bf g}({\bf s})-{\bf g}_n)/\sum_{k=1}^N w({\bf g}({\bf s}-{\bf g}_k))$ describes a similarity between the given vector ${\bf g}({\bf s})$ and a sample ${\bf g}_n$, while $w$ denotes a kernel function, such as Gaussian. The vector ${\bf g}({\bf s})$ is considered as a given condition and is comprised from field values in the surrounding of point ${\bf s}$. During the calculation of the conditional average the surrounding can be arbitrary selected, which is an advantage of non-parametric estimator. To calculate a field distribution in some domain, the field must be first specified in a sub-domain. From given values the field distribution in the surrounding of sub-domain can be estimated by Eq.2. The estimated values are then considered as given ones and the procedure of field estimation is iteratively continued. In the lecture optimal statistical methods for selection of surrounding region of a point ${\bf s}$ and self-organized determination of number $N$ are explained. Various examples of estimated chaotic filed distributions, such as profiles of a rough surface [3, 4], charge density and electron temperature in turbulent ionization waves in plasma, etc, are demonstrated. The performance of the proposed statistical modeling is described by comparing correlation functions and spectra of experimentally recorded and estimated fields.

References
[1] Grabec I, 1986, Phys. Lett., A117(8), pp 384-386
[2] Grabec I, Sachse W, 1997, Synergetics of Measurement, Prediction and Control, (Springer-Verlag, Heidelberg)
[3] Grabec I, Mandelj S, 1997, Proc. Int. Conf. EANN'97 - Stockholm, Eds.: A. B. Bulsari, S. Kallio, (Åbo Akademis Tryckery, Turku, Finland), pp 357-360
[4] Mandelj S, Grabec I, Govekar E, 2000, CIRP J. of Manuf. Systems, 30(3), pp 281-287