PhD Thesis: Removing observational uncertainty from orbits of nonlinear dynamical systems


The main goal of the research is to develop general, novel techniques and algorithms to retrieve information about orbits (trajectories) of nonlinear dynamical systems lost in the process of data ackquisition due to observational uncertainty. Observational uncertainty is a result of measurements (or observations) which do not perturb the system being measured. Imprecise measurements, additive noise, sampling of continuous variables, and/or incomplete knowledge are examples of observational uncertainty. Its presence causes the states of a system (i.e., points in some appropriate state space) to be replaced by uncertainty sets surrounding the exact "true" states. Given the uncertainty sets, the problem investigated in the thesis is to reconstruct the original states of the system with improved precision.

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 {xi} is replaced by a blurred orbit -- a sequence of uncertainty sets {Ai} surrounding the states, i.e., xi in Ai. The task is to recover the original orbit {xi} with a better accuracy (i.e., with smaller uncertainty sets) knowing the blurred orbit. The probability distribution of the states within the uncertainty sets may or may not be known. The amount of possible improvement will generally depend on the properties of the system and on the accuracy with which the model of the system is known . When the uncertainty in data is due to a noise whose statistical characteristics can be estimated, filtering or statistical techniques can be employed. In the last few years, several different noise-reduction techniques based on the nonlinear character of the dynamics producing the data have appeared. The main goal of these techniques is to decrease the noise level in time series. It is either assumed that the laws governing the dynamics are known, or that the laws are also derived from the noisy data. Since the number of variables in the observed time series is often lower than the number of degrees of freedom needed to fully describe the dynamical system, the dynamical laws need to be reconstructed from the time series. This is usually achieved by first determining the embedding dimension and then approximating the dynamics by a nonlinear mapping.

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 Ai are kept in the form of parallelepipeds, and the dynamics is linearized on each uncertainty set. By mapping the sets backward and forward to a fixed time i and intersecting the sets, the size of the uncertainty set Ai can be significantly decreased. The methodology is somewhat similar in nature to the approach suggested by Sidorowich and Farmer. The difference is that no a priori probability distribution for the noise is assumed. In addition, much more specific results can be obtained for the case of discretized orbits. While the technique works best when the model of the system is known, the technique appears to be robust with respect to the dynamical model.


Clone of an accurate measuring device

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.

Noise reduction in time series

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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