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Title: Identification and control of chaotic maps : a Frobenius-Perron operator approach
Author: Nie, Xiaokai
ISNI:       0000 0004 5363 1922
Awarding Body: University of Sheffield
Current Institution: University of Sheffield
Date of Award: 2015
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Deterministic dynamical systems are usually examined in terms of individual point trajectories. However, there are some deterministic dynamical systems exhibiting complex and chaotic behaviour. In many practical situations it is impossible to measure the individual point trajectories generated by an unknown chaotic dynamical system, but the evolution of probability density functions generated by such a system can be observed. As an alternative to studying point trajectories, such systems can be characterised in terms of sequences of probability density functions. This thesis aims to develop new approaches for inferring models of onedimensional dynamical systems from observations of probability density functions and to derive new methodologies for designing control laws to manipulate the shape of invariant density function in a desired way. A novel matrix-based approach is proposed in the thesis to solve the generalised inverse Frobenius-Perron problem, that is, to recover an unknown chaotic map, based on temporal sequences of probability density function estimated from data generated by the underlying system. The aim is to identify a map that exhibits the same transient as well as the asymptotic dynamics as the underlying system that generated the data. The approach involves firstly identifying the Markov partition, then estimating the associated Frobenius-Perron matrix, and finally constructing the underlying piecewise linear semi-Markov map. The approach is subsequently extended to more general one-dimensional nonlinear systems. Compared with the previous solutions to the inverse Frobenius-Perron problem, this approach is able to uniquely construct the transformation over the identified partition. The method is applied to heterogeneous human embryonic stem cell populations for inferring its dynamical model that describes the dynamical evolution based on sequences of experimentally observed flow cytometric distributions of cell surface marker SSEA3. The model that delineates the transitions of SSEA3 expression over one-day interval, can predict the long term evolution of SSEA3 sorted cell fractions, particularly, how different cell fractions regenerate the invariant parent distribution, and can be used to investigate the equilibrium points which are believed to correspond to functionally relevant substates, as well as their transitions. A new inverse problem is further studied for one-dimensional chaotic dynamical systems subjected to additive bounded random perturbations. The problem is to infer the unperturbed chaotic map based on observed temporal sequences of probability density functions estimated from perturbed data, and the density function of the perturbations. This is the so-called inverse Foias problem. The evolution of probability density functions of the states is formulated in terms of the Foias operator. An approximate matrix representation of Foias operator corresponding to the perturbed dynamical system, which establishes the relationship with Frobenius-Perron matrix associated with the unknown chaotic map, is derived. Inspired from the proposed approach for solving the generalised inverse FrobeniusPerron problem, a novel two-step matrix-based method is developed to identify the Frobenius-Perron matrix which gives rise to the reconstruction of the unperturbed chaotic map. The asymptotic stability of the probability density functions of the one-dimensional dynamical systems subjected to additive random perturbations is proven for the first time. The new result establishes the existence as well as the uniqueness of invariant densities associated to such transformations. Finally, this thesis addresses the problem of controlling the invariant density function. Specifically, given a one-dimensional chaotic map, the purpose of controller design is to determine the optimal input density function so as to make the resulting invariant density function as close as possible to a desired distribution. The control algorithm is based on the relationship between the input density function and the invariant density function derived earlier on.
Supervisor: Coca, Daniel Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available