It is assumed that observational uncertainty for discrete-time dynamical systems (DDSs) manifests itself in the following way. An orbit of a DDS (a sequence of points {x

The reconstructing method proposed in the dissertation is based solely on geometric ideas. The primary focus of is on the uncertainty removal after the system's dynamics has been approximately found using one of the methods cited in the references above. The uncertainty sets A

The reconstructing analysis can be used to make a copy of a very accurate (and perhaps unique) measuring device, preserving the accuracy of measurements. It is described how to build a simple chaotic "clone" of the original device that will be capable of measuring physical quantities with a precision approximately equal to the accuracy of the original device.

The idea of using the sensitivity of chaotic systems for construction of highly accurate measuring devices has been previously suggested by Wiesenfeld, and by Böhme et al. Wiesenfeld describes a method which uses a period-doubling bifurcation for the detection of weak signals. The method is based on the sensitivity of a nonchaotic system to parameters rather than on the sensitivity to initial conditions. This creates problems with manufacturing such devices since a very precise tuning of electronic components is necessary. Böhme creates the concept of a chaotic bridge which can be used as an amplification sensor for weak signals. In his approach, the initial states of two (identical) chaotic circuits are set such that their difference is the quantity to be measured (amplified). The time evolution of both systems is then used for estimating the difference in the initial conditions. However, the accuracy as well as the practical implementation may be hindered by the requirement that the two circuits be identical. This difficulty is not present in our approach since only a single chaotic device is used.

When the only information about a dynamical system is in the form of an imprecise time series, and no known model (i.e., the governing difference/differential equations) is available, the dynamical laws have to be extracted from the data, before our reconstructing procedure can be applied. Then, the accuracy with which the dynamics is reconstructed limits on the degree of possible improvement that can be achieved using our reconstructing method. The noise-removing technique consists of two separate parts: dynamics reconstruction and uncertainty removal. There are techniques where the dynamics reconstruction and the noise removal are done at the same time in one step. The dynamics reconstruction phase is done in two stages. First, the minimal necessary number of degrees of freedom of a dynamical system capable of producing the time series is determined. Second, the time series is then embedded into a finite-dimensional space, and the data is fitted by a nonlinear mapping representing the dynamical laws. Having an approximation to the dynamics, our reconstructing technique can be used to decrease the amount of noise in the time series. The entire procedure may be iterated by replacing the original time series with the new one.

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