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Num problema de espalhamento temos partículas incidentes sobre uma região de espalhamento que, depois de interagir por algum tempo nessa região, escapam para o infinito. Quando o espalhamento é caótico, a função de espalhamento (que é a relação entre uma variável antes do espalhamento e outra variável depois do espalhamento), apresenta singularidades sobre um conjunto de Cantor de condições iniciais. O espalhamento caótico pode ser dividido em dois tipos: espalhamento não-hiperbólico e hiperbólico. No espalhamento não-hiperbólico, o conjunto invariante contém órbitas estáveis. O decaimento das partículas que escapam do conjunto invariante é regido por uma lei de potência com relação ao tempo. No caso do espalhamento hiperbólico, a sela caótica é hiperbólica e todas as órbitas que a compõem são instáveis. O decaimento das partículas na região de espalhamento segue uma exponencial decrescente. Investigamos a transição do espalhamento não-hiperbólico para o hiperbólico quando ruído é adicionado à dinâmica do sistema. Isto porque prevíamos que o ruído reduzisse o efeito de aprisionamento (stickness) dos conjuntos de órbitas estáveis, provocando um decaimento exponencial. Introduzimos perturbações randômicas a fim de simular flutuações reais que ocorrem em sistemas físicos, como por exemplo, um vórtex que depende irregularmente do tempo no estudo de fluidos. Assim, usamos o conceito de mapas randômicos, que são mapas onde um ou mais parâmetros são variados aleatoriamente a cada iteração. Estudamos então, os efeitos provocados por perturbações randômicas em um sistema com espalhamento caótico não-hiperbólico. / In a scattering problem we have particles inciding on a scattering region and these particles, after spending some time in this region, escape towards infinity. When the scattering is chaotic, the scattering function (a function that relates an input variable with an output variable), is singular on a Cantor set of initial conditions. The chaotic scattering can be either non-hyperbolic or hyperbolic. In the non-hyperbolic scattering, the invariant set has stable orbits. This decay is governed by a power law in time. In the hyperbolic case, the chaotic saddle is hyperbolic and all the orbits are unstable. The decay of the particles is a decreasing exponential in the time. We investigate the transition from non-hyperbolic to hyperbolic scattering as noise is added to the system. One expects that noise will reduce the stickness of the regular regions, resulting in an exponential decay law, typical of hyperbolic systems. We apply random perturbations in order to simulate the real fluctuations that occur in physical systems, for example, an aperiodic vortex in a fluid flow. So, we work with random maps, where we change randomly one or more parameters on each iteration. We study thus, the effects of the random perturbations on a system having non-hyperbolic scattering.

Ph.D. / The work presented in this dissertation is concerned with the application of Chaos Theory to the field of Electrical and Electronic Engineering. A comprehensive study on electrical and electronic systems which exhibit chaotic behaviour, forms an integral part of this work. The objective of this survey is, firstly, to assess how widely chaos occurs in the field of electrical engineering. Secondly, the survey attempts to determine how successfully chaotic behaviour (in electrical systems) is identified and characterized. Finally, the survey aims to determine to what extent nonlinear phenomena and specifically chaos is applied to solve engineering problems. From this survey it is concluded that the study of chaos in electronics has reached a mature state. Currently, research focuses on controlling chaos, finding new applications of chaos as well as using chaos theory to gain a better understanding of the nonlinear world we live in. The other component of this dissertation consists of research done in the field of electronics. Contributions are made in controlling chaos and the analysis of chaotic systems both analytically and numerically.

The process of psychological change is complex, mirroring the complexity of life (Mahoney, 1991). Such complexity is nonlinear. Essentially, people are nonlinear dynamical systems and are characterised by an ever-changing, ever-adaptive movement from one state of order to another. This movement is a sequential flux, a turbulent ebb and flow of forces and form. Psychological change, too, follows this chaotic process of change (Butz, 1997). This is in line with the ‘new science’ of complexity. A postmodern vision, this is an ecological worldview that sees the world in terms of wholeness, interconnectedness, context, and nonlinear process (Goerner, 1995a). Three fields characterised by and concerned with complexity and which embrace the postmodern, ecological worldview are constructivism, ecopsychology, and chaos theory. Constructivism holds that people are meaning-making individuals who construct their own versions of reality; people are proactive, self-organising, and ever adapting to higher levels of complexity (Masterpasqua & Perna, 1997; Neimeyer & Mahoney, 1995). Ecopsychology is a synthesis of psychology and ecology; it is inspired by a holistic version of reality and posits the mutual embeddedness of humans and nature, the systemic connectedness of all that exists, and the evolutionary flux of the universe (Goerner, 1995a; Metzner, 1999). Chaos theory is the face of complexity; it is concerned with nonlinear dynamic systems as they evolve over time and the patterns and processes underlying such change (Cambel, 1993; Kellert, 1993). Although individually powerful and relevant for psychology, these fields are highly fragmented and often impractical. Much potential lies in their integration. Against this background, two goals were pursued in this study: 1) primarily, to simply and clearly demonstrate the concepts and application of chaos theory in a therapeutic situation; 2) to integrate the fields of constructivism, ecopsychology, and chaos theory relevant to the main goal of the study. Constructivism served as a grounding epistemology and, within this, ecopsychology served as a context within which chaos theory was utilised as a therapeutic applicatory model. The grounding epistemology, integration, and intervention are premised on the notions that: a) nature and humans are mutually and crucially embedded in each other; b) nature is characterised by nonlinear dynamical systems and the chaotic process of change, and thus humans (ie: dynamical systems) are also necessarily subject to such natural laws and principles; c) humans are proactive and may utilise the principles of chaos theory – notably self-organisation – to consciously initiate their own chaotic process of psychological change. The fields of constructivism, ecopsychology, and chaos theory are characterised by new and innovative forms of research and design methods. Such a pioneering spirit underpinned this study. The emphasis was on simplicity and pragmatic utility, using down-to-earth methods geared to producing practical and relevant data for use in therapy. A prime consideration was to ground the study in real-life. An empirical, descriptive field study was thus used, utilising an intensive single-case quantitative (time-series) design for data collection and a qualitative analysis. The intervention was aimed at initiating and facilitating psychological change, and was conducted with three participants. A nature-based metaphor and related guided imagery were used as a structure for the intervention. The intervention was conducted over three months. Participants completed self-report scales four times daily for the duration of the intervention, yielding time-series data. Analysis was by means of interpretation of three-dimensional geometric phase portraits and time-series graphs. Interpretations were used heuristically, triangulating them with clinical observations and verbal feedback from participants. Results showed that each of the three participants changed psychologically in different ways in the intervention, with certain aspects of chaos theory more applicable to one or the other. Considered together, the data pertaining to the three participants were clearly related to the principles of chaotic change. It was concluded that the concepts of chaos theory were shown to be relevant for therapy and that their application could be demonstrated simply and clearly. Chaos theory holds much potential as an applicatory model in psychology and would be well served by the use of more simple and pragmatic research methods. The use of triangulation in chaos theory analysis was found to be a particularly powerful methodology. The integration of constructivism, ecopsychology, and chaos theory proved to be a powerful framework for therapy and holds much potential for future development as a framework for broader psychological investigation and application. Much future research could be pursued from where this study leaves off. More studies focusing on simple and clear applications of chaos theory in therapy could be undertaken. Practical studies conducted in real-life therapeutic situations using innovative methodology would be particularly useful. A more comprehensive integration of constructivism, ecopsychology, and chaos theory could be undertaken. This could be a rich synthesis, going beyond unification of the core fundamentals to consider more widely related aspects of therapy and psychology. / Professor Gertie Pretorius

Femtosecond (fs) laser micromachining has been a useful technique either to modify and remove materials or to change the properties of a material, and can be applied to transparent and absorptive substances. Recently high-power fs laser radiation has drawn intensive attention for the induction of refractive index change to fabricate micro-structures in dielectric materials. This thesis studies the optical properties of a novel fiber random grating fabricated by fs laser micromachining technique and extends its applications from optical sensing to random fiber lasers and optical random bit generations. The thesis mainly consists of three parts. In the first part, the physical mechanism behind the fs laser micromachining technique and the fabrication of the fiber random grating are introduced. By employing a wavelength-division spectral cross-correlation algorithm, a novel multi-parameter fiber-optic sensor based on the fiber random grating is proposed and demonstrated to realize simultaneous measurements of temperature, axial strain, and surrounding refractive index. In the second part, Brillouin random fiber laser (BRFL) and Erbium-doped fiber random laser (EDFRL) are introduced, respectively. Firstly, we propose a novel Brillouin random fiber laser with a narrow linewidth of ~860 Hz based on the bi-directionally pumped stimulated Brillouin scattering (SBS) in a 10-km-long optical fiber. A random fiber Fabry-Perot (FP) resonator is built up through the pump depletion effects of SBS at both ends of the fiber. The novel laser is successfully applied for linewidth characterization beyond 860 Hz of light source under test. Secondly, the random grating-based FP resonator is introduced to build up a novel BRFL with narrow-linewidth of ~45.8Hz and reduced lasing threshold. The intensity and frequency noises of the proposed random laser are effectively suppressed due to the reduced resonating modes and mode competition. Finally, the fiber random grating is used as random distributed feedback in an EDFRL to achieve both static (temperature, strain) and dynamic (ultrasound) parameter sensing. Multiple lasing lines with high signal-to-noise ratio (SNR) up to 40dB are achieved, which gives an access for a high-fidelity multiple-static-parameter sensing application. By monitoring the wavelength shifts of each peak, temperature and strain have been simultaneously measured with small errors. The fiber random grating in the EDFRL is also able to sense the ultrasound waves. By achieving single mode lasing with the EDFRL, ultrasound waves with frequencies from 20kHz to 0.8MHz could be detected with higher sensitivity and SNR improvement up to 20dB compared with conventional piezoelectric acoustic sensors. In the third part, we demonstrate that a semiconductor laser perturbed by the distributed feedback from a fiber random grating can emit light chaotically without the time delay signature (TDS). A theoretical model is developed by modifying the Lang-Kobayashi model to numerically explore the chaotic dynamics of the laser diode subjected to the random distributed feedback. It is predicted that the random distributed feedback is superior to the single reflection feedback in suppressing the TDS. In experiments, The TDS with the maximum suppression is achieved with a value of 0.0088, which is the smallest to date.

In this work, we construct numerical methods to solve a wide range of problems in finance. This includes the valuation under affine jump diffusion processes, chaotic and hyperchaotic systems, and pricing fractional cryptocurrency models. These problems are of extreme importance in the area of finance. With today’s rapid economic growth one has to get a reliable method to solve chaotic problems which are found in economic systems while allowing synchronization. Moreover, the internet of things is changing the appearance of money. In the last decade, a new form of financial assets known as cryptocurrencies or cryptoassets have emerged. These assets rely on a decentralized distributed ledger called the blockchain where transactions are settled in real time. Their transparency and simplicity have attracted the main stream economy players, i.e, banks, financial institutions and governments to name these only. Therefore it is very important to propose new mathematical models that help to understand their dynamics. In this thesis we propose a model based on fractional differential equations. Modeling these problems in most cases leads to solving systems of nonlinear ordinary or fractional differential equations. These equations are known for their stiffness, i.e., very sensitive to initial conditions generating chaos and of multiple fractional order. For these reason we design numerical methods involving Chebyshev polynomials. The work is done from the frequency space rather than the physical space as most spectral methods do. The method is tested for valuing assets under jump diffusion processes, chaotic and hyperchaotic finance systems, and also adapted for asset price valuation under fraction Cryptocurrency. In all cases the methods prove to be very accurate, reliable and practically easy for the financial manager. / Thesis (PhD)--University of Pretoria, 2021. / Mathematics and Applied Mathematics / PhD / Unrestricted

The emergence of the Internet of Things (IoT) brought about the widespread of Body Sensor Networks (BSN) that continuously monitor patients using a collection of tiny-powered and lightweight bio-sensors offering convenience to both physicians and patients in the modern health care environment. Unfortunately, the deployment of bio-sensors in public hacker-prone settings means that they are vulnerable to various security threats exposing the security and privacy of patient information. This thesis presents an authentication scheme for each of two applications of medical sensor networks. The first is an ECC based authentication scheme suitable for a hospital-like setting whereby the patient is hooked up to sensors connected to a medical device such as an ECG monitor while the doctor needs real-time access to continuous sensor readings. The second protocol is a Chebyshev chaotic map-based authentication scheme suitable for deployment on wearable sensors allowing readings from the lightweight sensors connected to patients to be sent and stored on a trusted server while the patient is on the move. We formally and informally proved the security of both schemes. We also simulated both of them on AVISPA to prove their resistance to active and passive attacks. Moreover, we analyzed their performance to show their competitiveness against similar schemes and their suitability for deployment in each of the intended scenarios.

The phenomenon of dynamically induced anomalous diffusion is both the classical and quantum kicked rotor is investigated in this dissertation. We discuss the capability of the quantum mechanical version of the system to reproduce for extended periods the corresponding classical chaotic behavior.

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Synchronization and estimation of unknown constant parameters for Lorenz-type transmitter are studied under the assumption that one of the three state variables is not transmitted and that transmitter parameters are not known apriori. An adaptive algorithm is proposed to estimate both the state and system parameters. Since Lorenz system shows the property of sensitivity to initial conditions and evolves in different mode with parameter variation, an equivalent system is introduced. The adaptive observer is designed based on this equivalent system without any requirement on initial conditions of the observer. It is shown by Lyapunov arguments and persistent excitation analysis that exponential stability of state and parameter estimation is guaranteed. Simulation results are included to demonstrate properties of the algorithm. In a practical communication system, the received signals presented at the receiver part differ from those which were transmitted due to the effects of noise. The proposed synchronization scheme is robust with regard to external bounded disturbance. When an additive white gaussian noise (AWGN) channel model is considered, estimates of state and parameter converge except for small errors. The results show promise in either coherent detection or the message decoding in telecommunication systems.