When many people think of fractals, they think of the beautiful images created by Mandelbrot’s set or the intricate dragons of Julia’s set. However, these are just the artistic stars of the fractal community. The theory behind the fractals is not necessarily pretty, but is very important to many areas outside the world of mathematics. This paper takes a closer look at various types of fractals, the fractal dimensionality of surfaces and chaotic dynamical systems. Some of the history and introduction of creating fractals is discussed. The tools used to prevent a modified Koch’s curve from overlapping itself, finding the limit of a curves length and solving for a surfaces dimensional measurement are explored. Lastly, an investigation of the theories of chaos and how they bring order into what initially appears to be random and unpredictable is presented. The practical purposes and uses of fractals throughout are also discussed. / text

Advances in machine autonomy hold great promise in advancing technology, economic markets, and general societal well-being. For example, the progression of unmanned air systems (UAS) research has demonstrated the effectiveness and reliability of these autonomous systems in performing complex tasks. UAS have shown to not only outperformed human pilots in some tasks, but have also made novel applications not possible for human pilots practical. Nevertheless, human pilots are still favored when performing specific challenging tasks. For example, transportation of suspended (sometimes called slung or sling) loads requires highly skilled pilots and has only been performed by UAS in highly controlled environments. The presented work begins to bridge this autonomy gap by proposing a trajectory optimization framework for operations involving autonomous rotorcraft with suspended loads. The framework generates optimized vehicle trajectories that are used by existing guidance, navigation, and control systems and estimates the state of the non-instrumented load using a downward facing camera. Data collected from several simulation studies and a flight test demonstrates the proposed framework is able to produce effective guidance during autonomous suspended load operations. In addition, variational integrators are extensively studied in this dissertation. The derivation of a stochastic variational integrator is presented. It is shown that the presented stochastic variational integrator significantly improves the performance of the stochastic differential dynamical programming and the extended Kalman filter algorithms. A variational integrator for the propagation of polynomial chaos expansion coefficients is also presented. As a result, the expectation and variance of the trajectory of an uncertain system can be accurately predicted.

The theoretical basis for the Lyapunov exponents of continuous- and discrete-time dynamical systems is developed, with the inclusion of the statement and proof of the Multiplicative Ergodic Theorem of Oseledec. The numerical challenges and algorithms to approximate Lyapunov exponents and vectors are described, with multiple illustrative examples. A novel generalized impulsive collision rule is derived for particle systems interacting pairwise. This collision rule is constructed to address the question of whether or not the quantitative measures of chaos (e.g. Lyapunov exponents and Kolmogorov-Sinai entropy) can be reduced in these systems. Major results from previous studies of hard-disk systems, which interact via elastic collisions, are summarized and used as a framework for the study of the generalized collision rule. Numerical comparisons between the elastic and new generalized rules reveal many qualitatively different features between the two rules. Chaos reduction in the new rule through appropriate parameter choice is demonstrated, but not without affecting the structural properties of the Lyapunov spectra (e.g. symmetry and conjugate-pairing) and of the tangent space decomposition (e.g. hyperbolicity and domination of the Oseledec splitting). A novel measure of the degree of domination of the Oseledec splitting is developed for assessing the impact of fluctuations in the local Lyapunov exponents on the observation of coherent structures in perturbation vectors corresponding to slowly growing (or contracting) modes. The qualitatively different features observed between the dynamics of generalized and elastic collisions are discussed in the context of numerical simulations. Source code and complete descriptions for the simulation models used are provided.

In this paper a modification of the DCSK (Direct chaos shift keying) is considered. This modification, instead of using the reference signal part as in DCSK, transmits the information bit without it. The receiver then uses a unique key, known only to it and the sender, to decode the bit. The effectiveness of the system and two ways of its realization are tried, with this paper describing the experiments that are held. The effectiveness is defined by systems capability to transmit data. The theoretical experiments include testing five different keys and testing noise resistibility. The practical experiments include production of two models of the system, their comparison in functional and synthesis levels. The experiments are successful if the system model can be implemented in both ways that are tried, it is implemented in both ways, and the synthesis results are comparable.

A computational technique is introduced to reveal the complex intrinsic structure of homoclinic and heteroclinic bifurcations in a chaotic dynamical system. This technique is applied to several Lorenz-like systems with a saddle at the center, including the Lorenz system, the Shimizu-Morioka model, the homoclinic garden model, and the laser model. A multi-fractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities is explored on a bi-parametric plane of those dynamical systems. Also a great detail is explored in the Shimizu-Morioka model as an example. The technique is also applied to a re exion symmetric dynamical system with a saddle-focus at the center (Chua's circuits). The layout of the homoclinic bifurcations near the primary one in such a system is studied theoretically, and a scalability ratio is proved. Another part of the dissertation explores the intrinsic mechanisms of escape in a reciprocally inhibitory FitzHugh-Nagumo type threecell network, using the phase-lag technique. The escape network can produce phase-locked states such as pace-makers, traveling-waves, and peristaltic patterns with recurrently phaselag varying.

Chaotic systems have been studied for the past few decades because of its complex behaviour given simple governing ordinary differential equations. In the field of cryptology, several methods have been proposed for the use of chaos in cryptosystems. In this work, a method for harnessing the beneficial behaviour of chaos was proposed for use in RFID authentication and encryption. In order to make an accurate estimation of necessary hardware resources required, a complete hardware implementation was designed using a Xilinx Virtex 6 FPGA. The results showed that only 470 Xilinx Virtex slices were required, which is significantly less than other RFID authentication methods based on AES block cipher. The total number of clock cycles required per encryption of a 288-bit plaintext was 57 clock cycles. This efficiency level is many times higher than other AES methods for RFID application. Based on a carrier frequency of 13.56Mhz, which is the standard frequency of common encryption enabled passive RFID tags such as ISO-15693, a data throughput of 5.538Kb/s was achieved. As the strength of the proposed RFID authentication and encryption scheme is based on the problem of predicting chaotic systems, it was important to ensure that chaotic behaviour is maintained in this discretized version of Lorenz dynamical system. As a result, key boundaries and fourth order Runge Kutta approximation time step values that are unique for this new mean of chaos utilization were discovered. The result is a computationally efficient and cryptographically complex new RFID authentication scheme that can be readily adopted in current RFID standards such as ISO-14443 and ISO-15693. A proof of security by the analysis of time series data obtained from the hardware FPGA design is also presented. This is to ensure that my proposed method does not exhibit short periodic cycles, has an even probabilistic distribution and builds on the beneficial chaotic properties of the continuous version of Lorenz dynamical system.

No description available.

Sand is an appropriate material to embody and make visible the impulses which may pass through the moving hand and also permeate the universe. Its ability to absorb impressions means it can take on the full range of material states, solid, liquid or vapour, and allows it to speak with the rhythmic language the whole of nature itself uses. Developing a language of forms which more closely speaks this rhythmic language of nature requires an intimate knowledge of the processes of nature. Sand experiments undertaken show the formation a rich array of dazzling patterns. Chaos theory explains this self organizing capacity, and reveals the depth of interdependence of systems within nature. The aspect of self-similarity is also of central significance in appreciating how nature looks and function, and thus how it might be imaged. Drawings from nature were developed using sand as a parameter, starting with the single grain and translating the dynamism of moving sand into mark. Field trips involved looking for waves and repeated lines in nature, and observing how marks form in nature. Chaos theory provides a ground to bring together different spheres of knowledge ?? science, theology and art. It reveals the peculiarities of a material??s behaviour as being of critical importance in the mechanism of evolution. It also provides fresh insight into an incarnational Christian theological perspective, and the relational dynamic within the Trinity. The unity of far and near is also reflected in chaos theory in the self similarity of images. Romantic artists Turner and Van Gogh both engage in the search for a visual language of transcendence through nature using the use of the themes of chaos and order, with an emphasis on physicality and movement. Contemporary artists Goldsworthy, Blanchflower and Kirkeby ground their work in knowledge of material. Changes from solid and rigid to shifting and open show in the development of my work. The immediacy and dynamism of mark making in drawing and staining, ripping and sanding in painting gives process and materiality greater weight. The significance of relationality has reinforced the integrity of horizontal and vertical as expressed in nature and allowed for flexible repositioning of the image within a grid

The Floquet operator, defined as the time-evolution operator over one period, plays a central role in the work presented in this thesis on periodically perturbed quantum systems. Knowledge of the spectral nature of the Floquet operator gives us information on the dynamics of such systems. The work presented here on the spectrum of the Floquet operator gives further insight into the nature of chaos in quantum mechanics. After discussing the links between the spectrum, dynamics and chaos and pointing out an ambiguity in the physics literature, I present a number of new mathematical results on the existence of different types of spectra of the Floquet operator. I characterise the conditions for which the spectrum remains pure point and then, on relaxing these conditions, show the emergence of a continuous spectral component. The nature of the continuous spectrum is further analysed, and shown to be singularly continuous. Thus, the dynamics of these systems are a candidate for classification as chaotic. A conjecture on the emergence of a continuous spectral component is linked to a long standing number-theoretic conjecture on the estimation of finite exponential sums.

No description available.