This thesis presents a generalized approach for the fully digital design and implementation of chaos generators through the numerical solution of chaotic ordinary differential equations. In particular, implementations use the Euler approximation with a fixed-point twos complement number representation system for optimal hardware and performance. In general, digital design enables significant benefits in terms of power, area, throughput, reliability, repeatability and portability over analog implementations of chaos due to lower process, voltage and temperature sensitivities and easy compatibility with other digital systems such as microprocessors, digital signal processing units, communication systems and encryption systems. Furthermore, this thesis introduces the idea of implementing multidimensional chaotic systems rather than 1-D chaotic maps to enable wider throughputs and multiplier-free architectures that provide significant performance and area benefits. This work focuses efforts on the well-understood family of autonomous 3rd order "jerk" chaotic systems. The effect of implementation precision, internal delay cycles and external delay cycles on the chaotic response are assessed. Multiplexing of parameters is implemented to enable switching between chaotic and periodic modes of operation. Enhanced chaos generators that exploit long-term divergence in two identical systems of different precision are also explored. Digital design is shown to enable real-time controllability of 1D multiscroll systems and 4th order hyperchaotic systems, essentially creating non-autonomous chaos that has thus far been difficult to implement in the analog domain. Seven different systems are mathematically assessed for chaotic properties, implemented at the register transfer level in Verilog HDL and experimentally verified on a Xilinx Virtex 4 FPGA. The statistical properties of the output are rigorously studied using the NIST SP. 800-22 statistical testing suite. The output is adapted for pseudo random number generation by truncating statistically defective bits. Finally, a novel post-processing technique using the Fibonacci series is proposed and implemented with a non-autonomous driven hyperchaotic system to provide pseudo random number generators with high nonlinear complexity and controllable period length that enables full utilization of all branches of the chaotic output as statistically secure pseudo random output.

This diploma thesis compares chaos theory with Elliott wave theory in order to find out whether there is an agreement in the area of prediction. Such formulation of main problem is considered original, new and pioneering issue. By solving an indirect problem of deterministic chaos, existence of the chaos was not proved in a respective time series. The possibility to predict future development of this time series in a short-term period was considered impossible with respect of chaos theory results. Nevertheless, subsequent prediction that used Elliott wave theory showed to be precise. Finally, agreement of both theories was not confirmed. The diploma thesis proved that knowledge of Elliott wave theory and ability to interpret it correctly is a valuable means of prediction.

Die vorliegende Arbeit kombiniert Aspekte der nichtlinearen Dynamik mit denen der Unordnungsphysik. Die bekannte Standardabbildung wird mit einem räumlich ungeordneten aber periodischen Potential modifiziert. Transportexponenten sowohl für den Impuls als auch die kanonisch konjugierte Koordinate für das Standard- und das Zufallsmodell werden gegenübergestellt. Für das Zufallspotential ergibt sich verstärkter Transport. Gemittelte Transportexponenten des Zufallspotentials werden präsentiert und für verschiedene Systemausdehnungen verglichen. / The thesis at hand combines aspects of nonlinear dynamics with the physics of disorder. The standard map potential is replaced by a spatially quenched random periodic potential. Transport exponents for the standard and the random model are determined for the momentum as well as the canonically conjugate coordinate. Transport for the disordered potential is increased in comparison to the standard map. For the random case, quenched average transport exponents are presented. Finite-size effects are examined.

As a non-linear theory of space-time, general relativity deals with interesting dynamical systems which can be expected more prone to chaos than their Newtonian counter-parts. In this thesis, we study the dynamics of time- like geodesics in the static and axisymmetric field of a Schwarzschild black hole surrounded, in a concentric way, by a massive thin disc or ring. We reveal the rise (and/or decline) of geodesic chaos in dependence on parameters of the sys- tem (the disc/ring mass and position and the test-particle energy and angular momentum), (i) on Poincaré sections, (ii) on time series of position and their power spectra, (iii) by applying two simple yet powerful recurrence methods, and (iv) by computing Lyapunov exponents and two other related quantifiers of or- bital divergence. We mainly focus on "sticky" orbits whose different parts show different degrees of chaoticity and which offer the best possibility to test and compare different methods. We also add a treatment of classical but dissipative system, namely the evolution of a class of mechanical oscillators described by non-standard constitutive relations.

The consequences of two key approximations of accretion-disc physics near black holes are studied in this thesis. First, the question of effective ``pseudo-Newtonian" potentials mimicking a black hole is investigated both through numerical simulations and analytical means, and second, the neglect of additional gravitating matter near accreted-upon black holes and its consequences are put to test. After some broader discussion of integrability, resonance and chaos, a general "pseudo-Newtonian" limit for geodesic motion is derived, and applied for the case of null geodesics near a glowing toroid and for time-like geodesics in the Kerr metric. Afterwards, a new Newtonian gravitational potential for non- singular toroids is proposed and its usefulness for the so-called Weyl space-times is discussed. Finally, a new pseudo-Newtonian potential is introduced and applied alongside already known potentials in models of free test particle motion in the field of a black hole with a disc or ring, in complete analogy with previous exact-relativistic studies, and the previous conclusion of chaos in disc/ring-hole models is confirmed. Overall, the pseudo-Newtonian framework is able to reproduce a number of key features of the original systems with notable differences arising only as a consequence of extremely strong or...

In dieser Arbeit wird für Strukturen, die im makroskopischen aufgrund unterschiedlicher Materialeigenschaften oder komplexer Geometrien eine hohe Netzfeinheit für Finite-Elemente-Berechnungen benötigen, eine neue Möglichkeit zur Berechnung effektiver Materialkennwerte vorgestellt. Durch einen modalen Ansatz, bei dem, je nach Struktur analytisch oder numerisch, mit Hilfe der modalen Kennwerte die Formänderungsenergie eines repräsentativen Volumens der Originalstruktur mit der Formänderungsenergie eines äquivalenten homogen Vergleichsvolumens verglichen wird, können effektive Materialkennwerte ermittelt und daran anschließend eine Finite-Elemente-Berechnung mit einem im Vergleich zum Originalmodell sehr viel gröberen Netz durchgeführt werden, was eine enorme Zeiteinsparung mit sich bringt. Weiterhin enthält die vorgestellte Methode die Möglichkeit, unsichere Eingabeparameter wie Geometrieabmessungen oder Materialkennwerte mit Hilfe der polynomialen Chaos Expansion zu approximieren, um Möglichkeiten zur Aussage bzgl. der daraus resultierenden Verteilungen modaler Kenngrößen auf eine schnelle und effektive Weise zu gewinnen.

The structure of wavefunctions of quantum systems strongly depends on the underlying classical dynamics. In this text a selection of articles on eigenfunctions in systems with fully chaotic dynamics and systems with a mixed phase space is summarized. Of particular interest are statistical properties like amplitude distribution and spatial autocorrelation function and the implication of eigenfunction structures on transport properties. For systems with a mixed phase space the separation into regular and chaotic states does not always hold away from the semiclassical limit, such that chaotic states may completely penetrate into the region of the regular island. The consequences of this flooding are discussed and universal aspects highlighted.

This study is about the conditions that may serve as prerequisites for the development of self-regulated learning (SRL). In the context of educational psychology, SRL is not only about the formal aspects of managing one's learning, but also the motivational and affective processes that drive learning, as well as the social and political arena which provide the context for learning. In this study, I will propose that personal, social and political factors can combine in complex ways to produce a learning situation which cannot be addressed adequately without addressing its inherent complexity. I examine some current issues in cognition and cognitive intervention and begin by drawing attention to some problematic aspects concerning children's thinking from an educational and psychological point of view. I review the sociopolitical context in which the conceptualisation and implementation of outcomes based education (OBE) and Curriculum 2005 takes place in South Africa and I also discuss some issues pertaining to the study of cognition from a more psychological point of view. I also address issues of complexity by examining how the meaning of the word has changed in response to changing paradigms in science and psychology and suggest that complexity theory is a metaphor that best fits current knowledge about cognition and problem-solving. Since this study is about the accommodation of complexity in cognitive intervention, an important feature of this study concerns a specific characteristic of complex systems, namely chaos. Chaos allows self-organisation in a complex system and is also the main reason why change in a chaotic system is non-linear and unpredictable. It is generally believed that complex systems need to be studied in an unrestricted context if one is to observe those features that lend the system its chaotic character. In the context of the present study, complexity and chaos are hypothesised to be necessary prerequisites for the development of children as self-regulated learners because they form the mechanisms by which cognitive change becomes possible. The research was carried out in two phases. In Phase One of the research, classroom observations were made and the Mediational Behaviour Observation Scale (MBOS) was especially designed for this purpose. Phase Two of the research was carried out in an intervention context by means of a design experiment. Verbatim transcriptions were made of the interaction between the researcher and the learners in nine group sessions which formed part of the design experiment. To enhance the reliability and validity of the data, re-coding and intra-coding consistencies were calculated before the data were analysed. The re-coding consistencies ensured that the subsequent analysis of patterns would enable reliable conclusions to be drawn, whereas the intra-coding consistencies helped to refine the MBOS by indicating which categories may have been flawed, poorly described or impure. As such, the examination of the intra-code consistencies could perhaps be likened to factor analysis which resulted in some codes being merged and others being rejected. These data were used to construct a revised and shortened version of the MBOS. Some of the more important results of the data-analysis on the design experiment indicated that when complexity and chaos are encouraged in cognitive intervention, some of the mediator behaviours that are most likely to be observed are (i) guidance of the way in which learners execute tasks, (ii) attempts to engage learners in group discussions, (iii) modelling or requiring learners to explore tasks systematically, (iv) positive interactions such as acknowledging responses or praising learners, (v) modelling analytical thinking and (vi) probing of learners' responses. / Dissertation (PhD(Educational Psychology))--University of Pretoria, 2003. / Educational Psychology / unrestricted

This Thesis presents an optimal framework with dynamic perceptibility and chaos compensation for the control of robot manipulators. The fundamental objective of this framework is to obtain a variety of control laws for implementing dynamic visual servoing systems. In addition, this Thesis presents different contributions like the concept of dynamic perceptibility that is used to avoid image and robot singularities, the framework itself, that implements a delayed feedback controller for chaos compensation, and the extension of the framework for space robotic systems. Most of the image-based visual servoing systems implemented to date are indirect visual controllers in which the control action is joint or end-effector velocities to be applied to the robot in order to achieve a given desired location with respect to an observed object. The direct control of the motors for each joint of the robot is performed by the internal controller of the robot, which translates these velocities into joint torques. This Thesis mainly addresses the direct image-based visual servoing systems for trajectory tracking. In this case, in order to follow a given trajectory previously specified in the image space, the control action is defined as a vector of joint torques. The framework detailed in the Thesis allows for obtaining different kind of control laws for direct image-based visual servoing systems. It also integrates the dynamic perceptibility concept into the framework for avoiding image and robot singularities. Furthermore, a delayed feedback controller is also integrated so the chaotic behavior of redundant systems is compensated and thus, obtaining a smoother and efficient movement of the system. As an extension of the framework, the dynamics of free-based space systems is considered when determining the control laws, being able to determine trajectories for systems that do not have the base attached to anything. All these different steps are described throughout the Thesis. This Thesis describes in detail all the calculations for developing the visual servoing framework and the integration of the described optimization techniques. Simulation and experimental results are shown for each step, developing the controllers in an FPGA for further optimization, since this architecture allows to reduce latency and can be easily adapted for controlling of any joint robot by simply modifying certain modules that are hardware dependents. This architecture is modular and can be adapted to possible changes that may occur as a consequence of the incorporation or modification of a control driver, or even changes in the configuration of the data acquisition system or its control. This implementation, however, is not a contribution of this Thesis, but is necessary to briefly describe the architecture to understand the framework’s potential. These are the main objectives of the Thesis, and two robots where used for experimental results. A commercial industrial seven-degrees-of-freedom robot: Mitsubishi PA10, and another three-degrees-of-freedom robot. This last one’s design and implementation has been developed in the research group where the Thesis is written.

Among Wolf-Rayet stars, those of subtype WN8 are the intrinsically most variable. We have explored the long-term photometric variability of the brightest known WN8 star, WR 40, through four contiguous months of time-resolved, single-passband optical photometry with the BRIght Target Explorer nanosatellite mission. The Fourier transform of the observed light curve reveals that the strong light variability exhibited by WR 40 is dominated by many randomly triggered, transient, low-frequency signals. We establish a model in which the whole wind consists of stochastic clumps following an outflow visibility promptly rising to peak brightness upon clump emergence from the optically thick pseudo-photosphere in the wind, followed by a gradual decay according to the right-half of a Gaussian. Free electrons in each clump scatter continuum light from the star. We explore a scenario where the clump size follows a power-law distribution, and another one with an ensemble of clumps of constant size. Both scenarios yield simulated light curves morphologically resembling the observed light curve remarkably well, indicating that one cannot uniquely constrain the details of clump size distribution with only a photometric light curve. Nevertheless, independent evidence favours a negative-index power law, as seen in many other astrophysical turbulent media.