This thesis presents a study of the centre bifurcation and chaotic behaviour of three dimensional Lotka-Volterra systems. In two dimensional systems, Christopher (2005) considered a simple computational approach to estimate the cyclicity bifurcating from the centre. We generalized the technique to estimate the cyclicity of the centre in three dimensional systems. A lower bounds is given for the cyclicity of a hopf point in the three dimensional Lotka-Volterra systems via centre bifurcations. Sufficient conditions for the existence of a centre are obtained via the Darboux method using inverse Jacobi multiplier functions. For a given centre, the cyclicity is bounded from below by considering the linear parts of the corresponding Liapunov quantities of the perturbed system. Although the number obtained is not new, the technique is fast and can easily be adapted to other systems. The same technique is applied to estimate the cyclicity of a three dimensional system with a plane of singularities. As a result, eight limit cycles are shown to bifurcate from the centre by considering the quadratic parts of the corresponding Liapunov quantities of the perturbed system. This thesis also examines the chaotic behaviour of three dimensional Lotka-Volterra systems. For studying the chaotic behaviour, a geometric method is used. We construct an example of a three dimensional Lotka-Volterra system with a saddle-focus critical point of Shilnikov type as well as a loop. A construction of the heteroclinic cycle that joins the critical point with two other critical points of type planar saddle and axial saddle is undertaken. Furthermore, the local behaviour of trajectories in a small neighbourhood of the critical points is investigated. The dynamics of the Poincare map around the heteroclinic cycle can exhibit chaos by demonstrating the existence of a horseshoe map. The proof uses a Shilnikov-type structure adapted to the geometry of these systems. For a good understanding of the global dynamics of the system, the behaviour at infinity is also examined. This helps us to draw the global phase portrait of the system. The last part of this thesis is devoted to a study of the zero-Hopf bifurcation of the three dimensional Lotka-Volterra systems. Explicit conditions for the existence of two first integrals for the system and a line of singularity with zero eigenvalue are given. We characteristic the parameters for which a zero-Hopf equilibrium point takes place at any points on the line. We prove that there are three 3-parameter families exhibiting such equilibria. First order of averaging theory is also applied but we show that it gives no information about the possible periodic orbits bifurcating from the zero-Hopf equilibria.

Tunneling is a central result of quantum mechanics. It allows quantum particles to enter regions which are inaccessible by classical dynamics. Consequences of the tunneling process are most relevant. For example it causes the alpha-decay of radioactive nuclei and it is argued that proton tunneling is decisive for the emergence of DNA mutations. The theoretical prediction of corresponding tunneling rates is explained in standard textbooks on quantum mechanics for regular systems. Typical physical systems such as atoms or molecules, however, also show chaotic motion. Here the calculation of tunneling rates is more demanding. In this text a selection of articles on the prediction of tunneling rates in systems which allow for regular and chaotic motion is summarized. The presented approach is then used to explain consequences of tunneling on the quantum spectrum, such as the universal power-law behavior of small energy spacings and the flooding of regular states.

More than ever, organisations are recognising that they need to build and sustain healthy relationships with stakeholders in order to survive, grow and be successful. When an organisation is threatened by environmental changes – such as a crisis or competition as a result of information technology developments, increased customer demands, new legislations, even the threat of AIDS – the need for better communication increases. Successful organisations use the potential of communication management, not only to ease the transformation process, but also to improve their relationships with key stakeholders and the environment, and uphold their reputation. In this sense, communication practitioners are playing managerial, ethical and strategic roles during times of instability because change complexities involve having to deal with stakeholders’ trust and commitment. This thesis attempts to clarify the growing importance of communication management, particularly the role of relationship management. Proposed here is a different way of thinking about change communication strategies and building healthy relationships when organisations and their stakeholders have to make or adjust to change. Existing literature shows that most organisations tend to take a planned approach to change which is structured, consists of specific goals and objectives, and tightly controlled by management. Management sees its role within this paradigm as reducing conflict, creating order, controlling chaos and simplifying all the complexities created by the environment. Possible outcomes are predicted and alternatives for action are planned. Structures determine the information needed, and perceptions are managed by feeding the ‘right’ information or withholding information that might give rise to disorder and chaos. An alternative paradigm is the postmodern perspective, drawing specifically from chaos and complexity theories. These ‘emergent’ approaches to management consider organisations as living and holistic systems, more organic and ecological, seeking less control and more freedom. Organisations that operate like living systems are open, flexible, creative, caring and willing to adjust their strategies to adapt to the environment. While strategic planning is still considered important within the positivistic paradigms of management, it is nevertheless moving from the basic premise of control and prediction to scenario planning and processes of open communication, facilitation and networking. The emphasis is on relationship building via the full participation of the stakeholders involved. This two-way, symmetrical approach is also considered the more ethical. When an organisation is confronted by a problem, and if the constraints on communication are low, the organisation’s publics (employees/stakeholders) tend to feel connected to the problem and want to do something about it. Their need to actively seek information about the problem opens up many communication potentialities, including a willingness to change their attitudes, beliefs and behaviours. The result is a culture of shared responsibility, participation in decision-making, open and honest communication, which leads to a positive working climate and higher productivity. So an important assumption that can be made here is that an empowered public will strive for a positive relationship with the organisation. It can further be derived that a positive relationship between an organisation and its publics, particularly its internal stakeholders (employees), will lead to greater communication and a greater willingness to change. These assumptions were tested in this study. The research questions were: (1) What is the connection between the communication management strategy followed during change in organisations and the relationship and behavioural effects on internal stakeholders (employees)? (2) What are the effects of the communication management strategy followed during high change on relationships and behaviours with internal stakeholders (employees)? The methodology was an experimental approach which allows for the manipulation of independent variables and measurement of influences thereof on dependent variables. The independent variables were the communication and change strategies followed in organisations; the dependent variables were the relationship characteristics (trust, control mutuality, commitment, and satisfaction). The use of scenarios was most effective in this experimental context because different scenarios can project different outcomes. The researcher, therefore, had the opportunity to analyse the effects of the change strategies, as well as the changes that would take place. Data collection from 9 different organisations resulted in, more or less, 10–40 employees from different levels of management and non-management of each organisation. Each respondent evaluated 2 different scenarios, which resulted in 372 evaluations in total (186 for each scenario). The scenarios addressed changes in general and were about various issues. Basic descriptive statistics, as well as hypotheses testing using MANOVA (to test for meaningful differences between groups), were conducted. Other data analyses included testing for validity and reliability, analysis of variance and the Scheffe’s Test for significance of correlations. The findings showed significant proof of the internal validity of the experimental design used, indicating that: (a) the experimental manipulation (the two different change management strategies) had a definite effect on the relationship that internal stakeholders would have with organisations, and that except for the size of the organisation and the educational level of the employees, no other variables had an influence; (b) strong correlations exist between the strategy followed during change and the resulting projected relationships with internal stakeholders of the organisation; (c) high participation during high change led to significantly more positive overall relationships between an organisation and its internal stakeholders, as compared to low participation with a planned approach. In brief, the findings support a strong participative, two-way public relations strategy to be followed when organisations go through major change processes. The significance of these findings calls for a new paradigm in strategic communication and relationship management. Change cannot be solely based on plans and projections, but rather on understanding the complexities of situations and weighing different options available. Well-developed organisational change, therefore, is a strategically managed process that takes into consideration all the possibilities of change in the environment. Traditional studies and models of change management have either ignored the importance of strategic communication as a contribution to successful change, or saw communication as only a tool in the first stages of ransformation. However, to facilitate successful strategic change management is to recognise communication management as contributing significantly to guiding the complete transformation process in building important relationships internally and externally. Alternative emergent approaches to change recognise that change and, more specifically, transformation should be viewed as a continuous process linking to the complexities of the changing market, nature of work environments, new management approaches, organisational boundaries and relationships. Chaos and complexity theories, in particular, stress the importance of interconnectivity between subsystems of societies and organisations. Central to these theories is the observation that relationships built on open communication have the potential of producing something greater for an organisation. The complex and dynamic nature of the environment, structural alteration, and the need for employee flexibility are all recognised. Another important insight is the view that organisations can create visions and perform strategic planning around scenarios that guide actions. Another way of adapting to change is to influence back on change, that is, steer change through relationship building and participative decision-making. To become a true learning organisation requires the building of knowledge architecture with a strong supporting technical infrastructure. The main function of the communication manager, therefore, is to establish networks and structures for the collection and dissemination of information, and ultimately, the translation to knowledge. Communication leaders can connect teams and workgroups by driving communication and building trust. Involving staff in change management decision-making stimulates debate and criticisms, thus creates opportunities for innovation and revolutionary change. These are some of the main preconditions for sustainable change, and all can be achieved through sound communication management and the building of relationships with stakeholders. / Thesis (PhD)--University of Pretoria, 2006. / Communication Management / PhD / Unrestricted

Mean motion resonances (MMRs) play an important role in the formation and evolution of planetary systems and have significantly influenced the orbital properties and distribution of planets and minor planets in the solar system and in. exoplanetary systems. Most previous theoretical analyses have focused on the low- to moderate-eccentricity regime, but with new discoveries of high-eccentricity resonant minor planets and even exoplanets, there is increasing motivation to examine MMRs in the high-eccentricity regime. Here we report on a study of the high-eccentricity regime of MMRs in the circular planar restricted three-body problem. Numerical analyses of the 2: 1 and the 3: 2 interior resonances are carried out for a wide range of planet-to-star mass ratio mu, and for a wide range of eccentricity of the test particle. The surface-of-section technique is used to study the phase space structure near resonances. We find that new stable libration zones appear at higher eccentricity at libration centers that are. shifted from those at low eccentricities. We provide physically intuitive explanations for these transitions in phase space, and we present novel results on the mass and eccentricity dependence of the resonance widths. Our results show that MMRs have sizable libration zones at high eccentricities, comparable to those at lower eccentricities.

This dissertation is about some of the most important philosophical aspects of chaos research, a famous recent mathematical area of research about deterministic yet unpredictable and irregular, or even random behaviour. It consists of three parts. First, as a basis for the dissertation, I examine notions of unpredictability in ergodic theory, and I ask what they tell us about the justification and formulation of mathematical definitions. The main account of the actual practice of justifying mathematical definitions is Lakatos's account on proof-generated definitions. By investigating notions of unpredictability in ergodic theory, I present two previously unidentified but common ways of justifying definitions. Furthermore, I criticise Lakatos's account as being limited: it does not acknowledge the interrelationships between the different kinds of justification, and it ignores the fact that various kinds of justification - not only proof-generation - are important. Second, unpredictability is a central theme in chaos research, and it is widely claimed that chaotic systems exhibit a kind of unpredictability which is specific to chaos. However, I argue that the existing answers to the question "What is the unpredictability specific to chaos?" are wrong. I then go on to propose a novel answer, viz. the unpredictability specific to chaos is that for predicting any event all sufficiently past events are approximately probabilistically irrelevant. Third, given that chaotic systems are strongly unpredictable, one is led to ask: are deterministic and indeterministic descriptions observationally equivalent, i.e., do they give the same predictions? I treat this question for measure-theoretic deterministic systems and stochastic processes, both of which are ubiquitous in science. I discuss and formalise the notion of observational equivalence. By proving results in ergodic theory, I first show that for many measure-preserving deterministic descriptions there is an observationally equivalent indeterministic description, and that for all indeterministic descriptions there is an observationally equivalent deterministic description. I go on to show that strongly chaotic systems are even observationally equivalent to some of the most random stochastic processes encountered in science. For instance, strongly chaotic systems give the same predictions at every observation level as Markov processes or semi-Markov processes. All this illustrates that even kinds of deterministic and indeterministic descriptions which, intuitively, seem to give very different predictions are observationally equivalent. Finally, I criticise the claims in the previous philosophical literature on observational equivalence.

We consider a quantum system whose unperturbed form consists of a self-adjoint Δ-operator on a 2-dimensional compact Riemannian manifold, which may or may not have a boundary. Then as a perturbation, we add a delta potential/point scatterer at some select point ρ. The perturbed self-adjoint operator is constructed rigorously by means of self-adjoint extension theory. We also consider a corresponding classical dynamical system on the cotangent/cosphere bundle, consisting of geodesic flow on the manifold, with specular reflection if there is a boundary. Chapter 2 describes the mathematics of the unperturbed and perturbed quantum systems, as well as outlining the classical dynamical system. Included in the discussion on the delta-perturbed quantum system is consideration concerning the strength of the delta potential. It is reckoned that the delta potential effectively has negative infinitesimal strength. Chapter 3 continues on with investigations from [KMW10], concerned with perturbed eigenfunctions that approximate to a linear combination of only two "surrounding" unperturbed eigenfunctions. In Thm. 4.4 of [KMW10], conditions are derived under which a sequence of perturbed eigenfunctions exhibits this behaviour in the limit. The approximating pair linear combinations belong to a class of quasimodes constructed within [KMW10]. The aim of Chapter 3 in this thesis is to improve on the result in [KMW10]. In Chapter 3, preliminary results are first derived constituting a broad consideration of the question of when a perturbed eigenfunction subsequence approaches linear combinations of only two surrounding unperturbed eigenfunctions. Afterwards, the central result of this Chapter, namely Thm. 3.4.1, is derived, which serves as an improved version of Thm. 4.4 in [KMW10]. The conditions of this theorem are shown to be weaker than those in [KMW10]. At the same time though, the conclusion does not require the approximating pair linear combinations to be quasimodes contained in the domain of the perturbed operator. Cor. 3.5.2 allows for a transparent comparison between the results of this Chapter and [KMW10]. Chapter 4 deals with the construction of non-singular rank-one perturbations for which the eigenvalues and eigenfunctions approximate those of the delta-perturbed operator. This is approached by means of direct analysis of the construction and formulae for the rank-one-perturbed eigenvalues and eigenfunctions, by comparison that of the delta-perturbed eigenvalues and eigenfunctions. Successful results are derived to this end, the central result being Thm. 4.4.19. This provides conditions on a sequence of non-singular rank-one perturbations, under which all eigenvalues and eigenbasis members within an interval converge to those of the delta-perturbed operator. Comparisons have also been drawn with previous literature such as [Zor80], [AK00] and [GN12]. These deal with rank-one perturbations approaching the delta potential within the setting of a whole Euclidean space Rⁿ, for example by strong resolvent convergence, and by limiting behaviour of generalised eigenfunctions associated with energies at every Eℓ(0,∞). Furthermore in Chapter 4, the suggestion from Chapter 2 that the delta potential has negative infinitessimal strength is further supported, due to the coefficients of the approximating rank-one perturbations being negative and tending to zero. This phenomenon is also in agreement with formulae from [Zor80], [AK00] and [GN12]. Chapter 5 first reviews the correspondence between certain classical dynamics and equidistribution in position space of almost all unperturbed quantum eigenfunctions, as demonstrated for example in [MR12]. Equidistribution in position space of almost all perturbed eigenfunctions, in the case of the 2D rectangular flat torus, is also reviewed. This result comes from [RU12], which is only stated in terms of the "new" perturbed eigenfunctions, which would only be a subset of the full perturbed eigenbasis. Nevertheless, in this Chapter it is explained how it follows that this position space equidistribution result also applies to a full-density subsequence of the full perturbed eigenbasis. Finally three methods of approach are discussed for attempting to derive this position space equidistribution result in the case of a more general delta-perturbed system whose classical dynamics satisfies the particular key property.

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The first line of defence against wireless attacks in Radio Frequency Identi cation (RFID) systems is authentication of tags and readers. RFID tags are very constrained in terms of power, memory and size of circuit. Therefore, RFID tags are not capable of performing sophisticated cryptographic operations. In this dissertation, we have designed light-weight authentication schemes to securely identify the RFID tags to readers and vice versa. The authentication schemes require simple binary operations and can be readily implemented in resource-constrained Radio Frequency Identi cation (RFID) tags. We provide a formal proof of security based on the di culty of solving the Syndrome Decoding (SD) problem. Authentication veri es the unique identity of an RFID tag making it possible to track a tag across multiple readers. We further protect the identity of RFID tags by a light-weight privacy protecting identifi cation scheme based on the di culty of the Learning Parity with Noise (LPN) complexity assumption. To protect RFID tags authentication against the relay attacks, we have designed a resistance scheme in the analog realm that does not have the practicality issues of existing solutions. Our scheme is based on the chaos-suppression theory and it is robust to inconsistencies, such as noise and parameters mismatch. Furthermore, our solutions are based on asymmetric-key algorithms that better facilitate the distribution of cryptographic keys in large systems. We have provided a secure broadcast encryption protocol to effi ciently distribute cryptographic keys throughout the system with minimal communication overheads. The security of the proposed protocol is formally proven in the adaptive adversary model, which simulates the attacker in the real world.

One of the major tasks in electronic circuit design is the ability to predict the performance of general circuits in the presence of uncertainty in key design parameters. In the mathematical literature, such a task is referred to as uncertainty quantification. Uncertainty about the key design parameters arises mainly from the difficulty of controlling the physical or geometrical features of the underlying design, especially at the nanometer level. With the constant trend to scale down the process feature size, uncertainty quantification becomes crucial in shortening the design time. To achieve the uncertainty quantification, this thesis presents a new approach based on the concept of generalized Polynomial Chaos (gPC) to perform variability analysis of general nonlinear circuits. The proposed approach is built upon a decoupling formulation of the Galerkin projection (GP) technique, where the large matrix is transformed into a block-diagonal whose diagonal blocks can be factorized independently. The proposed methodology provides a general framework for decoupling the GP formulation based on a general system of orthogonal polynomials. Moreover, it provides a new insight into the error level that is caused by the decoupling procedure, enabling an assessment of the performance of a wide variety of orthogonal polynomials. For example, it is shown that, for the same order, the Chebyshev polynomials outperforms other commonly used gPC polynomials.