Thesis (Ph. D.)--West Virginia University, 2000. / Title from document title page. Document formatted into pages; contains xi, 113 p. : ill. (some col.). Includes abstract. Includes bibliographical references.

Thesis (Ph.D.)--Aberdeen University, 2008. / Includes bibliographical references.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2006. / Title from title screen (sites viewed on August 11, 2006). PDF text of dissertation: xx, 160 p. : ill. ; 1.73Mb. UMI publication number: AAT 3208123. Includes bibliographical references. Also available in microfilm, microfiche and paper format.

Thesis (Ph. D.)(Industrial Engineering)--University of Pretoria, 2005. / Includes summary. Includes bibliographical references. Available on the Internet via the World Wide Web.

M.Ing. (Electrical And Electronic Engineering) / Please refer to full text to view abstract

2003 / Some important ideas froni classical control theory are introduced with the intention of applying them to chaotic dynamical systems, in particular the coupled logistic equations. The structure of this dissertation is such that a strong foundation in control theory is first established before introducing the coupled logistic map or the methods of control and targetting in chaotic systems. In chapter 1 some aspects of classical control theory are reviewed. Continuous- and discrete-time dynamical systems are introduced and the existence and uniquendss criteria for the continuous case are explored via Lipschitz continuity. The matrix form of an inhomogeneous linear differential equation is presented and several properties of the associated transition matrix are discussed. Several linear algebraic ideas, most notably the Cayley-Hamilton theorem, are employed to explore the important concepts of controllability and observability in linear systems. The stabilisability problem is thoroughly investigated. Finally, the neighbourhood properties of continuous nonlinear dynamical systems with reference to controllability, stability and noise are established. Chapter 2 places emphasis on canonical forms, pole assignments and state observers. The decomposition of a general system into distinct components is facilitated by the general structure theorem, which is proved. The pole placement problem is described and the correspondence between the stabilisability of a system and the placement of poles is noted by the use'of a socalled feedback matrix. Lastly, the notion of a state observer, with reference to some dynamic feedback law, is introduced. The dynamics of the coupled logistic equations are studied in chapter 3. The fixed points of the map are calculated and the subsequent dynamical consequences explored. Using methods introduced in earlier chapters, the stability of the map is investigated. Using the so-called variational equations, the Lyapunov exponents are computed and used to classify, the motion of the system for the parameter values r and a. This chapter concludes with a discussion of the basins of attraction and critical curves associated with the coupled logistic equations. It is in chapter 4 that the models for controlling chaos are instantiated. The famous Ott-Grebogi- Yorke (OGY) method for controlling chaos is explained and related to the pole placement problem, discussed previously. The theory is extended to study the control of periodic orbits with periods greater than one.

Although a closed quantum system lacks clear signatures of classical chaos, it has been shown numerically that correspondence between an open quantum system and open classical system can be established. This phenomenon is explored for the case of an unconditioned evolution, where a system interacts with its environment, but the environment does not extract any information. This has been dubbed the “weak” transition and stands in contrast to the “strong” version where information is extracted by the environment. The transition is numerically mapped for the classically chaotic Duffing oscillator. Closed quantum and classically chaotic systems fail to agree due to the presence of fine scale structure in the classical evolution and the abundance of nonlocal interference in the quantum evolution. We show how noise mitigates both of these effects by suppressing the foliation of the classical unstable manifold while simultaneously acting as a passive filter of nonlocal quantum interference. The predicted transition parameter values are tested numerically for the Duffing oscillator. Finally, we explore whether these mechanisms are responsible for the emergence of classical chaos. While they do modify closed system spectral arguments against chaotic behavior, they do not provide a signature of chaotic dynamics. This stands in contrast to the trajectory level chaos observed in the strong transition. Keywords: nonlinear dynamics, quantum-classical transition, theoretical physics

For one-dimensional uninmodal maps hλ(x) a binary tree which includes all the periodic windows in the chaotic regime is constructed. By associating each element in the tree with the superstable parameter value of the corresponding periodic interval we define a different unimodal map. After applying a certain renormalization procedure to this new unimodal map, we find the period doubling fixed point g(x) which depends on the details of the map hλ(x) and the scaling constant α. The thermodynamics and the scaling function of the resulting dynamical system are also discussed. In addition, the total measure of the periodic windows is calculated with results in basic agreement with those obtained previously by Farmer. Up to 13 levels of the tree have been included, and the convergence of the partial sums of the measure is shown explicitly. It is conjectured that the asymptotic behavior of the partial sum of the measure as the number of levels goes to 00 is universal for the class of maps that have the same order of maximum. A new scaling law has been observed, i.e., the product of the length of a periodic interval characterized by sequence Q and the scaling constant of Q is found to be approximately 1. We also study two three-dimensional volume-preserving quadratic maps. There is no period doubling bifurcation in either case. We have also developed an algorithm to construct the symbolic alphabet for some given superstable symbolic sequences for one-dimensional unimodal maps. Using this symbolic alphabet and the approach of cycle expansion the topological entropy can be easily computed. Furthermore, the scaling properties of the measure of constant topological entropy are studied. Our results support the conjectures that for the maps with the same order of maximum, the asymptotic behavior of the partial sum of the measure as the level of the binary goes to infinity is universal and the corresponding 'fatness' exponent is universal. Numerical computations and analysis are also carried out for the clipped Bernoulli shift. / Ph. D.

This paper is based on the proposition that the choice of different product tactics is influenced by the nature of the firm’s external environment. It illustrates the type of product activities suggested for a complex and turbulent environment, when viewing the environment through a chaos and complexity theory lens. A qualitative, case method, using depth interviews,investigated the product activities in two companies to identify the product activities adopted in a more successful, versus a less successful, firm in a complex/turbulent environment. The results showed that the more successful company uses some destabilizing product activities but also partially uses stabilizing product activities. These findings are of benefit to marketers as they emphasize a new way to consider future product activities in their firms. Since businesses and markets are complex adaptive systems, using complexity theory to understand how to cope in complex, turbulent environments is necessary, but has not been widely researched, with even less emphasis on individual components of the marketing mix.

Thesis (MA)--University of Stellenbosch, 2004. / ENGLISH ABSTRACT: The question of what it means to say one knows something, or has knowledge of something, triggered an epistemological study after the nature of knowledge and its acquisition. There are many different ways in which one can go about acquiring knowledge, manydifferent frameworks that one can use to search after truth. Because most real systems about which one could desire knowledge (organic, social, economic etc.) are non-linear, an understanding of non-linear systems is important for the process of acquiring knowledge. Knowledge exhibits the characteristics of a dynamic, adaptive system, and as such could be approached via a dynamic theory of adaptive systems. Therefore, chaos theory and complexity theory are two theoretical (non-linear) frameworks that can facilitate the knowledge acquisition process. As a modernist instrument for acquiring knowledge, chaos theory provides one with deterministic rules that make mathematical understanding of non-linear phenomenaa bit easier, but it is limited in that it can only provide one with certain knowledge up until the (system's) next bifurcation (i.e. when chaos sets in). After this, it is near impossible to predict what a chaotic system will do. Complexity theory, as a postmodern tool for knowledge acquisition, gives one insight into the dynamic, self-organising nature of the non-linear systems around one. By analysing the global stability complex systems produce during punctuated equilibrium, one can learn much about how these systems adapt, evolve and survive. Complexity and chaos, therefore, together can provide one with a useful framework for understanding the nature and workings of non-linear systems. However, it should be remembered that every observer of knowledge does so out of his/her own personal framework of beliefs, circumstances and history, and that knowledge therefore can never be 100 percent objective. Knowledge and truth can never be entirely relative either, however, for this would mean that all knowledge (and thereby all opposing claims and statements) is equally correct or true. This is clearly not possible. What is possible, though, is the fulfilling and successful pursuit of knowledge for the sake of the journey of learning and understandi ng. / AFRIKAANSE OPSOMMING: Die vraag na wat dit eintlik beteken om te sê mens weet iets, of dra kennis van iets, het na 'n epistemologiese soeke na die wese van kennis en die verwerwing daarvan toe gelei. Daar is baie maniere waarop mens kennis kan verwerf, baie verskillende raamwerke wat mens kan gebruik om te soek na waarheid. Omdat die meeste wesenlike stelsels waarvan mens kennis sou wou verkry (organies, sosiaal, ekonomies ens.) nie-lineêr is, is 'n verstaan van nie-lineêre stelsels belangrik vir die kennisverwerwingsproses. Kennis vertoon die eienskappe van I n dinamiese, aanpassende stelsel, en kan dus via 'n dinamiese teorie van aanpassendestelsels benader word. Daarom is chaosteorie en kompleksiteitsteorie twee teoretiese (nie-lineêre) raamwerke wat die proses van kennisverwerwing kan vergemaklik. As I n modernistiese instrument vir kennisverwerwing, verskaf chaosteorie deterministiese reëls wat die wiskundige verstaan van nie-lineêre verskynsels bietjie vergemaklik, maar dit is beperk deurdat dit net sekere kennis tot op die (stelsel se) volgende splitsing (d.w.s. waar chaos begin) verskaf. Hierna, word dit naasonmoontlik om te voorspel wat I n chaotiese stelsel gaandoen. Kompleksiteitsteorie, as I n postmodernistiese gereedskap vir kennisverwerwing, gee mens insig in die dinamiese, selforganiserende aard van die nie-lineêre stelsels om mens. Deur die globale stabiliteit wat komplekse stelsels gedurende onderbreekte ewewig ("punctuated equi/ibrium"}toon te analiseer, kan mens baie leer van hoe hierdie stelsels aanpas, ontwikkel en oorleef. Kompleksiteit en chaos, saam, kan mens dus van a nuttige raamwerk vir die verstaan van die wese en werkinge van nie-lineêre stelsels, voorsien. Daar moet egter onthou word dat elke waarnemer van kennis dit doen uit sy/haar persoonlike raamwerk van oortuiginge, omstandighede en geskiedenis, en dat kennis dus nooit 100 persent objektief kan wees nie. Kennis en waarheid kan egter ook nooit heeltemaal relatief wees nie, want dit sou beteken dat alle kennis (en hiermee ook alle teenstrydige aansprake en stellings) gelyk korrek of waar is. Hierdie is duidelik onmoontlik. Wat wel moontlik is, is die vervullende en suksesvolle strewe na kennis ter wille van die reis van leer en verstaan.