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In this thesis we consider three-dimensional dynamical systems in the neighbourhood of a singular point with rank-one and rank-two resonant eigenvalues. We first introduce and generalize here a new technique extending previous work which was described by Aziz an Christopher (2012), where a second first integral of a 3D system can be found if the system has a Darboux-analytic first integral and an inverse Jacobi multiplier. We use this new technique to find two independent first integrals one of which contains logarithmic terms, allowing for non-zero resonant terms in the formal normal form of vector field. We also consider sufficient conditions for the existence of one analytic first integral for three dimensional vector fields around a singularity. Starting from the generalized Lotka-Volterra system with rank-one resonant eigenvalues, using the normal form method, we find an inverse Jacobi multiplier of the system under suitable conditions. Moreover, these conditions are sufficient conditions for the existence of one analytic first integral of the system. We apply this to demonstrate the sufficiency of the conditions in Aziz and Christopher (2014). In the case of two-dimensional systems, Christopher et al (2003) addressed the question of orbital normalizability, integrability, normalizability and linearizability of a complex differential system in the neighbourhood at a critical point. We here address the question of normalizability, orbital normalizability, and integrability of three-dimensional systems in the neighbourhood at the origin for rank-one resonance system. We consider the case when the eigenvalues of three-dimensional systems have rank-one resonance satisfying the condition the sum of eigenvalues is equal to zero a typical example, and we use a further change of coordinates to bring the formal normal form for three-dimensional systems into a reduced normal form which contains a finite number of resonant monomials. By using this technique, we can find two independent first integrals formally. The first one of these first integrals is of Darboux-analytic type, and other first integral contains logarithmic terms corresponding to non-zero resonant monomials of the original system. We introduce the monodromy map in three-dimensional vector fields by using these two independent first integrals to study a relationship between normalizability and integrability of systems. In the case of rank-one resonant eigenvalues, we get a monodromy map which is in normal form, and then in the same way as the case of vector fields, we use a further change of coordinates to reduce this map into a reduced map which contains only a finite number of resonant monomials. This thesis also examines briefly the case of rank-two resonant eigenvalues of three-dimensional systems. The normal form in this case contains an infinite number of resonant monomials, we were not able to find a reduced normal form with a finite number of resonant monomials. This situation is therefore much more complex than the rank-one case. Thus, we simplify the investigation by truncating the 3D system to a 3D homogeneous cubic system as a first step to understanding the general case. Even though we can find two independent first integrals, the second one involves the hypergeometric function, leading to some interesting topics for further investigation.

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.

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.

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 work presents an approach which provides the real-time estimation of the gas concentration in a plume using an unmanned aerial vehicle (UAV) equipped with concentration sensors. The plume is assumed to be generated by a moving aerial or ground source with unknown strength and location, and is modeled by the unsteady advection-diffusion equation with ambient winds and eddy diffusivities. The UAV dynamics is described using the point-mass model of a fixed-wing aircraft resulting in a sixth-order nonlinear dynamical system. The state (gas concentration) estimator takes the form of a Luenberger observer based on the advection-diffusion equation. The UAV in the approach is guided towards the region with the larger state-estimation error via an appropriate choice of a Lyapunov function thus coupling the UAV guidance with the performance of the gas concentration estimator. This coupled controls-CFD guidance scheme provides the desired Cartesian velocities for the UAV and based on these velocities a lower-level controller processes the control signals that are transmitted to the UAV. The finite-volume discretization of the estimator incorporates a second-order total variation diminishing (TVD) scheme for the advection term. For computational efficiency needed in real-time applications, a dynamic grid adaptation for the estimator with local grid-refinement centered at the UAV location is proposed. The approach is tested numerically for several source trajectories using existing specifications for the UAV considered. The estimated plumes are compared with simulated concentration data. The estimator performance is analyzed by the behavior of the RMS error of the concentration and the distance between the sensor and the source.

The Kari-Culik tilings are formed from a set of 13 Wang tiles that tile the plane only aperiodically. They are the smallest known set of Wang tiles to do so and are not as well understood as other examples of aperiodic Wang tiles. We show that a certain subset of the Kari-Culik tilings, namely those whose rows can be interpreted as Sturmian sequences (rotation sequences), is minimal with respect to the Z^2 action of translation. We give a characterization of this space as a skew product as well as explicit bounds on the waiting time between occurrences of m × n configurations. / Graduate / 0405

In this thesis we study the number of periodic points of congruential monomial dynamical system. By concept of index calculus we are able to calculate the number of solutions for congruential equations. We give formula for the number of r-periodic points over prime power. Then we discuss about calculating the total number of periodic points and cycles of length r for prime power.

The paper is devoted to asymptotic analysis of the Dirichlet problem for a second order partial differential equation containing a small parameter multiplying the highest order derivatives. It corresponds to a small perturbation of a dynamical system having a stationary solution in the domain. We focus on the case where the trajectories of the system go into the domain and the stationary solution is a proper node.

Herz–Schur multipliers of a locally compact group have a well developed theory coming from a large literature; they have proved very useful in the study of the reduced C∗-algebra of a locally compact group. There is also a rich connection to Schur multipliers,which have been studied since the early twentieth century, and have a large number of applications. We develop a theory of Herz–Schur multipliers of a C∗-dynamical system, extending the classical Herz–Schur multipliers, making Herz–Schur multiplier techniques available to study a much larger class of C∗-algebras. Furthermore, we will also introduce and study generalised Schur multipliers, and derive links between these two notions which extend the classical results describing Herz–Schur multipliers in terms of Schur multipliers. This theory will be developed in as much generality as possible, with reference to the classical motivation. After introducing all the necessary concepts we begin the investigation by defining generalised Schur multipliers. The main result is a dilation type characterisation of these multipliers; we also show how such multipliers can be represented using HilbertC∗-modules. Next we introduce and study generalised Herz–Schur multipliers, first extending a classical result involving the representation theory of SU(2), before studying how such functions are related to our generalised Schur multipliers. We give a characterisation of generalised Herz–Schur multipliers as a certain class of the generalised Schur multipliers, and obtain a description of precisely which Schur multipliers belong to this class. Finally, we consider some ways in which the generalised multipliers can arise; firstly, from the classical multipliers which provide our motivation, secondly, from the Haagerup tensor product of a C∗-algebra with itself, and finally from positivity considerations. We show that our theory behaves well with respect to positivity and give conditions under which our multipliers are automatically positive in a natural sense.