The past few years have witnessed a growth in the study of the long-time behaviour of physical, biological and economic systems using measure-theoretic and probabilistic methods. In this dissertation we present a study of the evolution of dynamical systems that display various types of irregular behaviour for large times. Large systems, containing many elements, like e.g. bacteria populations or ensembles of gas particles, are very difficult to analyse and contain elements of uncertainty. Also, in general, it is not necessary to know the evolution of each bacteria or each gas particle. Therefore we replace the "pointwise" description of the evolution of the system with that of the evolution of suitable averages of the population like e.g. the gas or the bacteria spatial density. In particular cases, when the quantity in the evolution that we analyse has the probabilistic interpretation, say, the probability of finding the particle in certain state at certain time, we will be talking about the evolution of (probability) densities. We begin with the establishment of results for discrete time systems and this is later followed with analogous results for continuous time systems. We observe that in many cases the system has two important properties: at each step it is determined by a non-negative function (for example the spatial density or the probability density) and the overall quantity of the elements remains preserved. Because of these properties the most suitable framework to investigate such systems is the theory of Markov operators. We shall discuss three levels of "chaotic" behaviour that are known as ergodicity, mixing and exactness. They can be described as follows: ergodicity means that the only invariant sets are trivial, mixing means that for any set A the sequence of sets S-n(A) becomes, asymptotically, independent of any other set B, and exactness implies that if we start with any set of positive measure, then, after a long time the points of this set will spread and completely fill the state space. In this dissertation we describe an application of two operators related to the generating Markov operator to study and characterize the abovementioned properties of the evolution system. However, a system may also display regular behaviour. We refer to this as the asymptotic stability of the Markov operator generating this system and we provide some criteria characterizing this property. Finally, we demonstrate the use of the above theory by applying it to a system that is modeled by the linear Boltzmann equation. / Thesis (M.Sc.)-University of Natal, Durban, 2001.

Controlling a system with chaotic nature provides the ability to control and maintain orbits of different periods which extends the functionality of the system to be flexible. A system with diverse dynamical behaviours can be achieved. Trajectory flows of chaotic systems can be periodically stabilised using only small perturbations from the controlled parameter. The method of chaos control is the Ott-Grebogi-Yorke method. In non-chaotic systems large system parameters changes are required for performance changes. A sagittal plane biped model which is capable of exhibiting periodic and chaotic locomotion was researched and investigated. The locomotion was either periodic or chaotic depending on the design parameters. Nonlinear dynamic tools such as the Bifurcation Diagram, Lyapunov Exponent and Poincaré Map were used to differentiate parameters which generated periodic motion apart from chaotic ones. Numerical analytical tools such as the Closed Return and Linearization of the Poincaré Map were used to detect unstable periodic orbit in chaotic attractors. Chaos control of the model was achieved in simulations. The system dynamic is of the non-smooth continuous type. Differing from other investigated chaotic systems, the biped model has varying phase space dimensions which can range from 3 to 6 dimensions depending on the phase of walking. The design of the biped was such that its features were anthropomorphic with respect to locomotion. The model, consisting of only the lower body (hip to feet), was capable of walking passively or actively and was manufactured with optimal anthropometric parameters based on ground clearance (to avoid foot scuffing) and basin of attraction simulations. During experimentation, the biped successfully walked down an inclined ramp with minimal aid. Real time data acquisitions were performed to capture the results, and the experimental data of the walking trajectories were analysed and verified against simulations. It was verified that the constructed biped exhibits the same walking trend as the derived theoretical model. / Thesis (M.Sc.Eng.)-University of KwaZulu-Natal, Durban, 2012.

Thesis (Ph. D.)--Open University. BLDSC no. DX176663.