For parametrized families of dynamical systems, two major goals are classifying the systems up to topological conjugacy, and understanding the structure of the bifurcation locus. The family Fλ = z^n + λ/z^d gives a 1-parameter, n+d degree family of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial z^n. This work presents several results related to these goals for the family Fλ, particularly regarding a structure of "necklaces" in the λ parameter plane. This structure consists of infinitely many simple closed curves which surround the origin, and which contain postcritically finite parameters of two types: superstable parameters and escape time Sierpinski parameters. First, we derive a dynamical invariant to distinguish the conjugacy classes among the superstable parameters on a given necklace, and to count the number of conjugacy classes. Second, we prove the existence of a deeper fractal system of "subnecklaces," wherein the escape time Sierpinski parameters on the previously known necklaces are themselves surrounded by infinitely many necklaces.

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Rotating wave solutions to evolution equations have been shown to govern many important biological and chemical processes. Much of the rigorous mathematical investigations of rotating waves rely on the model exhibiting a continuous Euclidean symmetry, which is only present in an idealized situation. Here we investigate the existence of rotationally propagating solutions in a discrete spatial setting, in which typical symmetry methods cannot be applied, thus presenting an unique perspective on rotating waves. Our goal in this thesis is to demonstrate the existence and potential stability of rotating wave solutions to a spatially discretized infinite systems of coupled differential equations. This goal is achieved by considering so-called Lambda-Omega systems, which have frequently been used to model typical oscillatory dynamics. Our work is broken into three major components: 1. An infinite system of coupled phase equations is investigated and we demonstrate that under some mild assumptions the system exhibits a phase-locked rotating wave solution. The phase system is derived from a limiting case of the original Lambda-Omega system, and therefore solutions of the phase equation will be useful in finding rotating wave solutions to the full Lambda-Omega system. 2. We examine the stability of the rotating wave solution found in the coupled phase equations. This is achieved by providing a link with an underlying graph-theoretic geometry endowed by the spatially discretized system. We use results from random walks on infinite graphs to provide a general stability theorem for coupled phase equations. 3. We use the rotating wave solution of the phase equations to extend to a rotating wave solution of the full Lambda-Omega system. This result is achieved using a non-standard Implicit Function Theorem, since we show that typical implicit function arguments cannot be applied to our present situation.

To the reader <p> Since I worked on two separate projects towards my doctorate thesis, the arrangement of my thesis is rather unusual. The reader will find that my thesis is divided into four parts. Part 1 is dedicated to a very general introduction about the basic knowledge needed to guide you, the reader, through the rest of the thesis. Within this part, different sections focus on different fundamental aspects of Biophysics related to my work. In Part 2, I discuss my studies of the distribution and dynamics of the nuclear protein Ran in live interphase HeLa cells. This part contains a background section specific to this project, the materials and methods used for this study, experimental results, a discussion of our findings, and it ends with conclusions. Part 3 is dedicated to the study of the dynamical mechanisms responsible for the establishment of the Bed protein concentration gradient along the anterior-posterior axis in live Drosophila melanogaster embryos. Again, a specific background section is included in this part, followed by the materials and methods used to perform this research, results, discussions and finally I will summarize my results to conclude this work. The last part, part 4, is rather short and contains the summary of the overall results of my work on both nuclear proteins with some emphasis on the similarities and differences in their dynamical behavior.</p> / Thesis / Doctor of Philosophy (PhD)

We study the time evolution of light fields inside a double cavity which is comprised of two perfect end mirrors and a parametrically driven, partially transmissive central mirror in both a classical and a quantum mechanical framework. It is common practise in the field of optomechanics to take a Hamiltonian approach \cite{aspelmeyer2014cavity} ignoring non-linear coupling terms between the light field and the moving mechanical element. By contrast, we start from the Maxwell wave equation which is second order in time and find that a first order in time Schr\"{o}dinger-type wave equation (equivalent to neglecting the non-linear coupling) is a valid approximation for low enough mirror reflectivity and speed and for large light frequencies. We also study adiabatic dynamics for the Maxwell wave equation and find it differs from the more familiar adiabaticity in the Schr\"{o}dinger equation. Next, we numerically simulate the dynamical Casimir effect (DCE) in the double cavity with a sinusoidally driven central mirror following earlier numerical work on the perfect single cavity \cite{Ruser2006NumericalDCE,ruser2005vibrating,naylor2009dynamical}. Because our central mirror is partially transmissive it is physically more realistic and circumvents fundamental problems associated with having perfectly reflecting moving mirrors \cite{Moore1970DCESingleCavity,barton1993quantum}. The corresponding photon creation rates are drastically lower when compared to the perfectly reflective mirror case. Furthermore, if we make one of the cavities much longer than the other we can simulate the DCE for a single open cavity coupled to an environment without having to make the Markov approximation. The resultant asymmetric double cavity (ADC) model is valid for times short enough that only a negligible number of the photons that has leaked out of the open cavity has sloshed back in again. As for the symmetric case, one advantage of the ADC is that driven mirror is partially transmissive rather than perfectly reflecting. / Thesis / Doctor of Philosophy (PhD)

Intensive Care Units (ICUs) host patients in critical condition who are being monitored by sensors which measure their vital signs. These vital signs carry information about a patient’s physiology and can have a very rich structure at fine resolution levels. The task of analysing these biosignals for the purposes of monitoring a patient’s physiology is referred to as physiological condition monitoring. Physiological condition monitoring of patients in ICUs is of critical importance as their health is subject to a number of events of interest. For the purposes of this thesis, the overall task of physiological condition monitoring is decomposed into the sub-tasks of modelling a patient’s physiology a) under the effect of physiological or artifactual events and b) under the effect of drug administration. The first sub-task is concerned with modelling artifact (such as the taking of blood samples, suction events etc.), and physiological episodes (such as bradycardia), while the second sub-task is focussed on modelling the effect of drug administration on a patient’s physiology. The first contribution of this thesis is the formulation, development and validation of the Discriminative Switching Linear Dynamical System (DSLDS) for the first sub-task. The DSLDS is a discriminative model which identifies the state-of-health of a patient given their observed vital signs using a discriminative probabilistic classifier, and then infers their underlying physiological values conditioned on this status. It is demonstrated on two real-world datasets that the DSLDS is able to outperform an alternative, generative approach in most cases of interest, and that an a-mixture of the two models achieves higher performance than either of the two models separately. The second contribution of this thesis is the formulation, development and validation of the Input-Output Non-Linear Dynamical System (IO-NLDS) for the second sub-task. The IO-NLDS is a non-linear dynamical system for modelling the effect of drug infusions on the vital signs of patients. More specifically, in this thesis the focus is on modelling the effect of the widely used anaesthetic drug Propofol on a patient’s monitored depth of anaesthesia and haemodynamics. A comparison of the IO-NLDS with a model derived from the Pharmacokinetics/Pharmacodynamics (PK/PD) literature on a real-world dataset shows that significant improvements in predictive performance can be provided without requiring the incorporation of expert physiological knowledge.

In this dissertation, we developed predictive models for legged and limbless locomotion on dry, homogeneous granular media. The vertical plane Resistive Force Theory (RFT) for frictional granular fluids accurately predicted the reaction forces on intruders (with complex geometries) translating and rotating at low speeds ( < 0.5 m/s). Using RFT and multibody simulation, we predicted the forward moving speed of legged robots. During the locomotion of lightweight robots and animals where instantaneous limb penetration speed can reach values greater than ~0.5 m/s, a Discrete Element Method (DEM) simulation was developed to capture the limb-ground interaction. We demonstrated that hydrodynamic-like forces generated by accelerated particles can balance the robot weight and inertia, and promote the rapid movement on granular media. Forces from the environment can not only determine locomotion dynamics, but also affect the locomotion strategy. We studied and simulated the limbless locomotion of snakes in a heterogeneous environment and demonstrated how touch sensing was used to adjust the movement pattern. In heterogeneous environments, the long-term locomotion dynamics is also poorly understood. We presented a theory for transport and diffusion in such settings.

Synchronization is an important phenomenon which plays a central role in the function or dysfunction of a wide spectrum of biological and technological networks. Despite the vast literature on network synchronization, the majority of research activities have been focused on oscillators connected through one network. However, in many realistic biological and engineering systems the units can be coupled via multiple, independent networks. This thesis contributes toward the rigorous understanding of the emergence of stable synchronization in dynamical networks with mixed coupling. A mixed network is composed of subgraphs connecting a subnetwork of oscillators via one of the individual oscillator's variables. An illustrative example is a network of Lorenz systems with mixed couplings where some of the oscillators are coupled through the x-variable, some through the y-variable and some through both. This thesis presents a new general synchronization method called the Mixed Connection Graph method, which removes a long-standing obstacle in studying synchronization in mixed dynamical networks of different nature. This method links the stability theory, including the Lyapunov function approach with graph theoretical quantities. The application of the method to specific networks reveals surprising, counterintuitive effects, not seen in networks with one connection graph.

This Thesis proposes three main objectives: (i) to provide the concept of a new generalized non-standard synthesis model that would provide the framework for incorporating other non-standard synthesis approaches; (ii) to explore dynamic sound modeling through the application of new non-standard synthesis techniques and procedures; and (iii) to experiment with dynamic sound synthesis for the creation of novel sound objects. In order to achieve these objectives, this Thesis introduces a new paradigm for non-standard synthesis that is based in the algorithmic assemblage of minute wave segments to form sound waveforms. This paradigm is called Extended Waveform Segment Synthesis (EWSS) and incorporates a hierarchy of algorithmic models for the generation of microsound structures. The concepts of EWSS are illustrated with the development and presentation of a novel non-standard synthesis system, the Dynamic Waveform Segment Synthesis (DWSS). DWSS features and combines a variety of algorithmic models for direct synthesis generation: list generation and permutation, tendency masks, trigonometric functions, stochastic functions, chaotic functions and grammars. The core mechanism of DWSS is based in an extended application of Cellular Automata. The potential of the synthetic capabilities of DWSS is explored in a series of Case Studies where a number of sound object were generated revealing (i) the capabilities of the system to generate sound morphologies belonging to other non-standard synthesis approaches and, (ii) the capabilities of the system of generating novel sound objects with dynamic morphologies. The introduction of EWSS and DWSS is preceded by an extensive and critical overview on the concepts of microsound synthesis, algorithmic composition, the two cultures of computer music, the heretical approach in composition, non- standard synthesis and sonic emergence along with the thorough examination of algorithmic models and their application in sound synthesis and electroacoustic composition. This Thesis also proposes (i) a new definition for “algorithmic composition”, (ii) the term “totalistic algorithmic composition”, and (iii) four discrete aspects of non-standard synthesis.

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