Title:

Identification and control of chaotic maps : a FrobeniusPerron operator approach

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 matrixbased approach is proposed in the thesis to solve the generalised inverse FrobeniusPerron 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 FrobeniusPerron matrix, and finally constructing the underlying piecewise linear semiMarkov map. The approach is subsequently extended to more general onedimensional nonlinear systems. Compared with the previous solutions to the inverse FrobeniusPerron 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 oneday 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 onedimensional 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 socalled 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 FrobeniusPerron matrix associated with the unknown chaotic map, is derived. Inspired from the proposed approach for solving the generalised inverse FrobeniusPerron problem, a novel twostep matrixbased method is developed to identify the FrobeniusPerron matrix which gives rise to the reconstruction of the unperturbed chaotic map. The asymptotic stability of the probability density functions of the onedimensional 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 onedimensional 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.
