Chaos theory is well known as one of three revolutions in physical sciences in 20th-century, as one physicist called it: Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurable process; and chaos eliminates the Laplacian fantasy of deterministic predictability". Specially, when chaos synchronization was found in 1991, chaos theory becomes more and more attractive. Chaos has been widely applied to many scientific disciplines: mathematics, programming, microbiology, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics. One of most important engineering applications is secure communication because of the properties of random behaviours and sensitivity to initial conditions of chaos systems. Noise-like dynamical behaviours can be used to mask the original information in symmetric cryptography. Sensitivity to initial conditions and unpredictability make chaotic systems very suitable to construct one-way function in public-key cryptography. In chaos-based secure communication schemes, information signals are masked or modulated (encrypted) by chaotic signals at the transmitter and the resulting encrypted signals are sent to the corresponding receiver across a public channel (unsafe channel). Perfect chaos synchronization is usually expected to recover the original information signals. In other words, the recovery of the information signals requires the receiver's own copy of the chaotic signals which are synchronized with the transmitter ones. Thus, chaos synchronization is the key technique throughout this whole process. Due to the difficulties of generating and synchronizing chaotic systems and the limit of digital computer precision, there exist many challenges in chaos-based secure communication. In this thesis, we try to solve chaos generation and chaos synchronization problems. Starting from designing chaotic and hyperchaotic system by first-order delay differential equation, we present a family of novel cell attractors with multiple positive Lyapunov exponents. Compared with previously reported hyperchaos systems with complex mathematic structure (more than 3 dimensions), our system is relatively simple while its dynamical behaviours are very complicated. We present a systemic parameter control method to adjust the number of positive Lyapunov exponents, which is an index of chaos degree. Furthermore, we develop a delay feedback controller and apply it to Chen system to generate multi-scroll attractors. It can be generalized to Chua system, Lorenz system, Jerk equation, etc. Since chaos synchronization is the critical technique in chaos-based secure communication, we present corresponding impulsive synchronization criteria to guarantee that the receiver can generate the same chaotic signals at the receiver when time delay and uncertainty emerge in the transmission process. Aiming at the weakness of general impulsive synchronization scheme, i.e., there always exists an upper boundary to limit impulsive intervals during the synchronization process, we design a novel synchronization scheme, intermittent impulsive synchronization scheme (IISS). IISS can not only be flexibly applied to the scenario where the control window is restricted but also improve the security of chaos-based secure communication via reducing the control window width and decreasing the redundancy of synchronization signals. Finally, we propose chaos-based public-key cryptography algorithms which can be used to encrypt synchronization signals and guarantee their security across the public channel.

Fractals have long been recognized to be a characteristic feature arising from chaotic dynamics; be it in the form of strange attractors, of fractal boundaries around basins of attraction, or of fractal and multifractal distributions of asymptotic measures in open systems. In this thesis we study fractal and multifractal measure distributions in leaky Hamiltonian systems. Leaky systems are created by introducing a fully or partially transparent hole in an otherwise closed system, allowing trajectories to escape or lose some of their intensity. This dynamics results in intricate (multi)fractal distributions of the surviving trajectories. These systems are suitable models for experimental setups such as optical microcavities or microwave resonators. In this thesis we perform an improved investigation of the fractality in these systems using the concept of effective dimensions. They are defined as the dimensions far from the usually considered asymptotics of infinite evolution time $t$, infinite sample size $S$, and infinite resolution (infinitesimal box-size $varepsilon$). Yet, as we show, effective dimensions can be considered as intrinsic to the dynamics of the system. We present a detailed discussion of the behaviour of the numerically observed dimension $D_mathrm{obs}(S,t,varepsilon)$. We show that the three parameters can be expressed in terms of limiting length scales that define the parameter ranges in which $D_mathrm{obs}(S,t,varepsilon)$ is an effective dimension of the system. We provide dynamical and statistical arguments for the dependence of these scales on $S$, $t$, and $varepsilon$ in strongly chaotic systems and show that the knowledge of the scales allows us to define meaningful effective dimensions. We apply our results to three main fields. In the context of numerical algorithms to calculate dimensions, we show that our findings help to numerically find the range of box sizes leading to accurate results. We further show that they allow us to minimize the computational cost by providing estimates of the required sample-size and iteration time needed. A second application field of our results is systems exhibiting non-trivial dependencies of the effective dimension $D_mathrm{eff}$ on $t$ and $varepsilon$. We numerically explore this in weakly chaotic leaky systems. There, our findings provide insight into the dynamics of the systems, since deviations from our predictions based on strongly chaotic systems at a given parameter range are a sign that the stickiness inherent to such systems needs to be taken into account in that range. Lastly, we show that in quantum analogues of chaotic maps with a partial leak, a related effective dimension can be used to explain the numerically observed deviation from the predictions provided by the fractal Weyl law for systems with fully absorbing leaks. Here, we provide an analytical description of the expected scaling based on the classical dynamics of the system and compare it with numerical results obtained in the studied quantum maps. / Es ist seit langem bekannt, dass Fraktale eine charakteristische Begleiterscheinung chaotischer Dynamik sind. Sie treten in Form von seltsamen Attraktoren, von fraktalen Begrenzungen der Einzugsbereiche von Attraktoren oder von fraktalen und multifraktalen Verteilungen asymptotischer Maße in offenen Systemen auf. In dieser Arbeit betrachten wir fraktal und multifraktal verteilte Maße in geöffneten hamiltonschen Systemen. Geöffnete Systeme werden dadurch erzeugt, dass man ein völlig oder teilweise transparentes Loch im Phasenraum definiert, durch das Trajektorien entkommen können oder in dem sie einen Teil ihrer Intensität verlieren. Die Dynamik in solchen Systemen erzeugt komplexe (multi)fraktale Verteilungen der verbleibenden Trajektorien, beziehungsweise ihrer Intensitäten. Diese Systeme sind zur Modellierung experimenteller Aufbauten, wie zum Beispiel optischer Mikrokavitäten oder Mikrowellenresonatoren, geeignet. In dieser Arbeit führen wir eine verbesserte Untersuchung der Fraktalität in derartigen Systemen durch, die auf dem Konzept der effektiven Dimensionen beruht. Diese sind als die Dimensionen definiert, die weit weg von den üblicherweise betrachteten Limites unendlicher Iterationszeit $t$, unendlicher Stichprobengröße $S$ und unendlicher Auflösung, also infinitesimaler Boxgröße $varepsilon$ auftreten. Dennoch können effektive Dimensionen, wie wir zeigen, als der Dynamik des Systems inhärent angesehen werden. Wir führen eine detaillierte Diskussion der numerisch beobachteten Dimension $D_mathrm{obs}(S,t,varepsilon)$ durch und zeigen, dass die drei Parameter $S$, $t$ und $varepsilon$ in Form grenzwertiger Längenskalen ausgedrückt werden können, die die Parameterbereiche definieren, in denen $D_mathrm{obs}(S,t,varepsilon)$ den Wert einer effektiven Dimension des Systems annimmt. Wir beschreiben das Verhalten dieser Längenskalen in stark chaotischen Systemen als Funktionen von $S$, $t$ und $varepsilon$ anhand statistischer Überlegungen und anhand von auf der Dynamik basierenden Aussagen. Weiterhin zeigen wir, dass das Wissen um diese Längenskalen die Definition aussagekräftiger effektiver Dimensionen ermöglicht. Wir wenden unsere Ergebnisse hauptsächlich in drei Bereichen an: Im Kontext numerischer Algorithmen zur Dimensionsberechnung zeigen wir, dass unsere Ergebnisse es erlauben, diejenigen $varepsilon$-Bereiche zu finden, die zu korrekten Ergebnissen führen. Weiterhin zeigen wir, dass sie es uns erlauben, den Rechenaufwand zu minimieren, indem sie uns eine Abschätzung der benötigten Stichprobengröße und Iterationszeit ermöglichen. Ein zweiter Anwendungsbereich sind Systeme, die sich durch eine nichttriviale Abhängigkeit von $D_mathrm{eff}$ von $t$ und $varepsilon$ auszeichnen. Hier ermöglichen unsere Ergebnisse ein besseres Verständnis der Systeme, da Abweichungen von den Vorhersagen basierend auf der Annahme von starker Chaotizität ein Anzeichen dafür sind, dass im entsprechenden Parameterbereich die Eigenschaft dieser Systeme, dass Bereiche in ihrem Phasenraum Trajektorien für eine begrenzte Zeit einfangen können, relevant ist. Zuletzt zeigen wir, dass in quantenmechanischen Analoga chaotischer Abbildungen mit partiellen Öffnungen eine verwandte effektive Dimension genutzt werden kann, um die numerisch beobachteten Abweichungen vom fraktalen weyl'schen Gesetz für völlig transparente Öffnungen zu erklären. In diesem Zusammenhang zeigen wir eine analytische Beschreibung des erwarteten Skalierungsverhaltens auf, die auf der klassischen Dynamik des Systems basiert, und vergleichen sie mit numerischen Erkenntnissen, die wir über die Quantenabbildungen gewonnen haben.

Fractals have long been recognized to be a characteristic feature arising from chaotic dynamics; be it in the form of strange attractors, of fractal boundaries around basins of attraction, or of fractal and multifractal distributions of asymptotic measures in open systems. In this thesis we study fractal and multifractal measure distributions in leaky Hamiltonian systems. Leaky systems are created by introducing a fully or partially transparent hole in an otherwise closed system, allowing trajectories to escape or lose some of their intensity. This dynamics results in intricate (multi)fractal distributions of the surviving trajectories. These systems are suitable models for experimental setups such as optical microcavities or microwave resonators. In this thesis we perform an improved investigation of the fractality in these systems using the concept of effective dimensions. They are defined as the dimensions far from the usually considered asymptotics of infinite evolution time $t$, infinite sample size $S$, and infinite resolution (infinitesimal box-size $varepsilon$). Yet, as we show, effective dimensions can be considered as intrinsic to the dynamics of the system. We present a detailed discussion of the behaviour of the numerically observed dimension $D_mathrm{obs}(S,t,varepsilon)$. We show that the three parameters can be expressed in terms of limiting length scales that define the parameter ranges in which $D_mathrm{obs}(S,t,varepsilon)$ is an effective dimension of the system. We provide dynamical and statistical arguments for the dependence of these scales on $S$, $t$, and $varepsilon$ in strongly chaotic systems and show that the knowledge of the scales allows us to define meaningful effective dimensions. We apply our results to three main fields. In the context of numerical algorithms to calculate dimensions, we show that our findings help to numerically find the range of box sizes leading to accurate results. We further show that they allow us to minimize the computational cost by providing estimates of the required sample-size and iteration time needed. A second application field of our results is systems exhibiting non-trivial dependencies of the effective dimension $D_mathrm{eff}$ on $t$ and $varepsilon$. We numerically explore this in weakly chaotic leaky systems. There, our findings provide insight into the dynamics of the systems, since deviations from our predictions based on strongly chaotic systems at a given parameter range are a sign that the stickiness inherent to such systems needs to be taken into account in that range. Lastly, we show that in quantum analogues of chaotic maps with a partial leak, a related effective dimension can be used to explain the numerically observed deviation from the predictions provided by the fractal Weyl law for systems with fully absorbing leaks. Here, we provide an analytical description of the expected scaling based on the classical dynamics of the system and compare it with numerical results obtained in the studied quantum maps. / Es ist seit langem bekannt, dass Fraktale eine charakteristische Begleiterscheinung chaotischer Dynamik sind. Sie treten in Form von seltsamen Attraktoren, von fraktalen Begrenzungen der Einzugsbereiche von Attraktoren oder von fraktalen und multifraktalen Verteilungen asymptotischer Maße in offenen Systemen auf. In dieser Arbeit betrachten wir fraktal und multifraktal verteilte Maße in geöffneten hamiltonschen Systemen. Geöffnete Systeme werden dadurch erzeugt, dass man ein völlig oder teilweise transparentes Loch im Phasenraum definiert, durch das Trajektorien entkommen können oder in dem sie einen Teil ihrer Intensität verlieren. Die Dynamik in solchen Systemen erzeugt komplexe (multi)fraktale Verteilungen der verbleibenden Trajektorien, beziehungsweise ihrer Intensitäten. Diese Systeme sind zur Modellierung experimenteller Aufbauten, wie zum Beispiel optischer Mikrokavitäten oder Mikrowellenresonatoren, geeignet. In dieser Arbeit führen wir eine verbesserte Untersuchung der Fraktalität in derartigen Systemen durch, die auf dem Konzept der effektiven Dimensionen beruht. Diese sind als die Dimensionen definiert, die weit weg von den üblicherweise betrachteten Limites unendlicher Iterationszeit $t$, unendlicher Stichprobengröße $S$ und unendlicher Auflösung, also infinitesimaler Boxgröße $varepsilon$ auftreten. Dennoch können effektive Dimensionen, wie wir zeigen, als der Dynamik des Systems inhärent angesehen werden. Wir führen eine detaillierte Diskussion der numerisch beobachteten Dimension $D_mathrm{obs}(S,t,varepsilon)$ durch und zeigen, dass die drei Parameter $S$, $t$ und $varepsilon$ in Form grenzwertiger Längenskalen ausgedrückt werden können, die die Parameterbereiche definieren, in denen $D_mathrm{obs}(S,t,varepsilon)$ den Wert einer effektiven Dimension des Systems annimmt. Wir beschreiben das Verhalten dieser Längenskalen in stark chaotischen Systemen als Funktionen von $S$, $t$ und $varepsilon$ anhand statistischer Überlegungen und anhand von auf der Dynamik basierenden Aussagen. Weiterhin zeigen wir, dass das Wissen um diese Längenskalen die Definition aussagekräftiger effektiver Dimensionen ermöglicht. Wir wenden unsere Ergebnisse hauptsächlich in drei Bereichen an: Im Kontext numerischer Algorithmen zur Dimensionsberechnung zeigen wir, dass unsere Ergebnisse es erlauben, diejenigen $varepsilon$-Bereiche zu finden, die zu korrekten Ergebnissen führen. Weiterhin zeigen wir, dass sie es uns erlauben, den Rechenaufwand zu minimieren, indem sie uns eine Abschätzung der benötigten Stichprobengröße und Iterationszeit ermöglichen. Ein zweiter Anwendungsbereich sind Systeme, die sich durch eine nichttriviale Abhängigkeit von $D_mathrm{eff}$ von $t$ und $varepsilon$ auszeichnen. Hier ermöglichen unsere Ergebnisse ein besseres Verständnis der Systeme, da Abweichungen von den Vorhersagen basierend auf der Annahme von starker Chaotizität ein Anzeichen dafür sind, dass im entsprechenden Parameterbereich die Eigenschaft dieser Systeme, dass Bereiche in ihrem Phasenraum Trajektorien für eine begrenzte Zeit einfangen können, relevant ist. Zuletzt zeigen wir, dass in quantenmechanischen Analoga chaotischer Abbildungen mit partiellen Öffnungen eine verwandte effektive Dimension genutzt werden kann, um die numerisch beobachteten Abweichungen vom fraktalen weyl'schen Gesetz für völlig transparente Öffnungen zu erklären. In diesem Zusammenhang zeigen wir eine analytische Beschreibung des erwarteten Skalierungsverhaltens auf, die auf der klassischen Dynamik des Systems basiert, und vergleichen sie mit numerischen Erkenntnissen, die wir über die Quantenabbildungen gewonnen haben.

Phase oscillators are a common starting point for the reduced description of many single neuron models that exhibit a strongly attracting limit cycle. The framework for analysing such models in response to weak perturbations is now particularly well advanced, and has allowed for the development of a theory of weakly connected neural networks. However, the strong-attraction assumption may well not be the natural one for many neural oscillator models. For example, the popular conductance based Morris-Lecar model is known to respond to periodic pulsatile stimulation in a chaotic fashion that cannot be adequately described with a phase reduction. In this paper, we generalise the phase description that allows one to track the evolution of distance from the cycle as well as phase on cycle. We use a classical technique from the theory of ordinary differential equations that makes use of a moving coordinate system to analyse periodic orbits. The subsequent phase-amplitude description is shown to be very well suited to understanding the response of the oscillator to external stimuli (which are not necessarily weak). We consider a number of examples of neural oscillator models, ranging from planar through to high dimensional models, to illustrate the effectiveness of this approach in providing an improvement over the standard phase-reduction technique. As an explicit application of this phase-amplitude framework, we consider in some detail the response of a generic planar model where the strong-attraction assumption does not hold, and examine the response of the system to periodic pulsatile forcing. In addition, we explore how the presence of dynamical shear can lead to a chaotic response.

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When two or more chaotic systems are coupled, they may exhibit synchronized chaotic oscillations. The synchronization of chaos is usually understood as the regime of chaotic oscillations in which the corresponding variables or coupled systems are equal to each other. This kind of synchronized chaos is most frequently observed in systems specifically designed to be able to produce this behaviour. In this thesis, one particular type of synchronization, called impulsive synchronization, is investigated and applied to low dimensional chaotic, hyperchaotic and spatiotemporal chaotic systems. This synchronization technique requires driving one chaotic system, called response system, by samples of the state variables of the other chaotic system, called drive system, at discrete moments. Equi-Lagrange stability and equi-attractivity in the large property of the synchronization error become our major concerns when discussing the dynamics of synchronization to guarantee the convergence of the error dynamics to zero. Sufficient conditions for equi-Lagrange stability and equi-attractivity in the large are obtained for the different types of chaos-generating systems used. The issue of robustness of synchronized chaotic oscillations with respect to parameter variations and time delay, is also addressed and investigated when dealing with impulsive synchronization of low dimensional chaotic and hyperchaotic systems. Due to the fact that it is impossible to design two identical chaotic systems and that transmission and sampling delays in impulsive synchronization are inevitable, robustness becomes a fundamental issue in the models considered. Therefore it is established, in this thesis, that under relatively large parameter perturbations and bounded delay, impulsive synchronization still shows very desired behaviour. In fact, criteria for robustness of this particular type of synchronization are derived for both cases, especially in the case of time delay, where sufficient conditions for the synchronization error to be equi-attractivity in the large, are derived and an upper bound on the delay terms is also obtained in terms of the other parameters of the systems involved. The theoretical results, described above, regarding impulsive synchronization, are reconfirmed numerically. This is done by analyzing the Lyapunov exponents of the error dynamics and by showing the simulations of the different models discussed in each case. The application of the theory of synchronization, in general, and impulsive synchronization, in particular, to communication security, is also presented in this thesis. A new impulsive cryptosystem, called induced-message cryptosystem, is proposed and its properties are investigated. It was established that this cryptosystem does not require the transmission of the encrypted signal but instead the impulses will carry the information needed for synchronization and for retrieving the message signal. Thus the security of transmission is increased and the time-frame congestion problem, discussed in the literature, is also solved. Several other impulsive cryptosystems are also proposed to accommodate more solutions to several security issues and to illustrate the different properties of impulsive synchronization. Finally, extending the applications of impulsive synchronization to employ spatiotemporal chaotic systems, generated by partial differential equations, is addressed. Several possible models implementing this approach are suggested in this thesis and few questions are raised towards possible future research work in this area.

The sigma-delta modulator is a popular signal amplitude quantization error (or noise) shaper used in oversampling analogue-to-digital and digital-to-analogue converter systems. The shaping of the noise frequency spectrum is performed by feeding back the quantization errors through a time delay element filter and feedback loop in the circuit, and by the addition of a possible stochastic dither signal at the quantizer. The aim in audio systems is to limit audible noise and distortions in the reconverted analogue signal. The formulation of the sigma-delta modulator as a discrete dynamical system provides a useful framework for the mathematical analysis of such a complex nonlinear system, as well as a unifying basis from which to consider other systems, from pseudorandom number generators to stochastic resonance processes, that yield equivalent formulations. The study of chaos and other complementary aspects of internal dynamical behaviour in previous research has left important issues unresolved. Advancement of this study is naturally facilitated by the dynamical systems approach. In this thesis, the general order feedback/feedforward sigma-delta modulator with multi-bit quantizer (no overload) and general input, is modelled and studied mathematically as a dynamical system. This study employs pertinent topological methods and relationships, which follow centrally from the symmetry of the circle map interpretation of the error state space dynamcis. The main approach taken is to reduce the nonlinear system into local or special case linear systems. Systems of sufficient structure are shown to often possess structured random, or random-like behaviour. An adaptation of Devaney's definition of chaos is applied to the model, and an extensive investigation of the conditions under which the associated chaos conditions hold or do not hold is carried out. This seeks, in part, to address the unresolved research issues. Chaos is shown to hold if all zeros of the noise transfer function lie outside the unit circle of radius two, provided the input is either periodic or persistently random (mod delta). When the filter satisfies a certain continuity condition, the conditions for chaos are extended, and more clear cut classifications emerge. Other specific chaos classifications are established. A study of the statistical properties of the error in dithered quantizers and sigma-delta modulators is pursued using the same state space model. A general treatment of the steady state error probability distribution is introduced, and results for predicting uniform steady state errors under various conditions are found. The uniformity results are applied to RPDF dithered systems to give conditions for a steady state error variance of delta squared over six. Numerical simulations support predictions of the analysis for the first-order case with constant input. An analysis of conditions on the model to obtain bounded internal stability or instability is conducted. The overall investigation of this thesis provides a theoretical approach upon which to orient future work, and initial steps of inquiry that can be advanced more extensively in the future.

This thesis is an exploration of Plato’s understanding of the power of disorder as it is presented in his cosmology, The Timaeus and in his predominantly religious work, Laws X. In the former work this causal force is presented as the disordering power responsible for the physical chaos prior to the generation of the universe, as well as for any residual disorder found within the cosmos after it has been ordered and is the antithesis of ‘nous’ or reason. In the latter work, however, Laws X, the causal force for disorder is now understood as a disordering power capable of endangering the soul, active long after the cosmos has been generated and itself, a ‘Soul’. What ultimately emerges is a dynamic theory of disorder and a metaphysics supporting that theory, weaving through, connecting across and separating apart these two works. In Part I, consisting of five chapters, I provide the Greek, an original translation and commentary on seven key passages from The Timaeus where Plato presents his ideas on disorder, both as an effect within the cosmos and as a causal power or force for disorder prior to its generation. In this regard, I look closely at Plato’s use of the Greek word &V&YK~ in its role as a disordering power, but which has also been commonly understood and translated as ‘necessity’. I contrast this with Plato’s understanding of the role which the ‘Demiurge’ or the ordering power of the cosmos has played, with its faculty of ~06s or ‘reason’ and its access to ideal ‘Forms’ or ‘ideas’ when ordering or generating the universe. In Part 11, consisting of four chapters, Laws X is similarly presented, providing the Greek, a translation (for the most part, that of A. E. Taylor) and commentary on eight key passages. Here I investigate Plato’s understanding of disorder as it pertains specifically to the ‘soul’ and of the soul’s relation to the disordering power(s) and to ‘evil’. In the final chapter a theory of disorder is proposed, in which an epistemology is outlined, an ontology is given and from which, it is argued, a metaphysics of disorder emerges.