Construction and dynamics of knotted fields in soft matter systems

Knotted fields are physical fields containing knotted, linked, or otherwise topologically interesting structure. They occur in a wide variety of physical systems — fluids, superfluids, electromagnetism, optics and high energy physics to name a few. Far from being passive structures, the occurrence of knotting in a physical field often modifies its overall properties, rendering their study interesting from both a theoretical and practical point of view. In this thesis, we focus on knotted fields in ‘soft matter’ systems, systems which may be loosely characterised as those in which geometry plays a fundamental role, and which undergo substantial deformations in response to external forces, changes in temperature etc. Such systems are often experimentally accessible, making them natural testbeds for exploring knotted fields in all their guises. After providing an introduction to knotted fields with a focus on soft matter in the first chapter, in the second we introduce a method of explicitly constructing such fields for any knotted curve based on Maxwell’s solid angle construction. We discuss its theory, emphasising a fundamental homotopy formula as unifying methods for computing the solid angle, as well as describing a naturally induced curve framing, which we show is related to the writhe of the curve before using it to characterise the local structure in the neighbourhood of the knot. We then discuss its practical implementation, giving examples of its use and providing C code. In subsequent chapters we use this methodology to initialise simulations in our study of knotted fields in two soft matter systems: excitable media and twist-bend nematics. In excitable media we provide a systematic survey of knot dynamics up to crossing number eight, finding generically unsteady behaviour driven by a wave-slapping mechanism. Nevertheless, we also find novel complex knotted structures and characterise their geometry and steady state motion, as well as greatly expanding upon previous evidence to demonstrate the ability of the dynamics to untangle geometries without reconnection. In twist-bend nematics we describe their fundamental geometry, that of bend. The zeros of bend are a set of lines with rich geometric and topological structure. We characterise their local structure, describe how they are canonically oriented and discuss a notion of their self-linking. We then describe their topological significance, showing that these zeros compute Skyrmion and Hopfion numbers, with accompanying simulations in twist-bend nematics

Nonlinear Dynamics, Synchronisation and Chaos in Coupled FHN Cardiac and Neural Cells

Physiological systems are amongst the most challenging systems to investigate from a mathematically based approach. The eld of mathematical biology is a relatively recent one when compared to physics. In this thesis I present an introduction to the physiological aspects needed to gain access to both cardiac and neural systems for a researcher trained in a mathematically based discipline. By using techniques from nonlinear dynamical systems theory I show a number of results that have implications for both neural and cardiac cells. Examining a reduced model of an excitable biological oscillator I show how rich the dynamical behaviour of such systems can be when coupled together. Quantifying the dynamics of coupled cells in terms of synchronisation measures is treated at length. Most notably it is shown that for cells that themselves cannot admit chaotic solutions, communication between cells be it through electrical coupling or synaptic like coupling, can lead to the emergence of chaotic behaviour. I also show that in the presence of emergent chaos one nds great variability in intervals of activity between the constituent cells. This implies that chaos in both cardiac and neural systems can be a direct result of interactions between the constituent cells rather than intrinsic to the cells themselves. Furthermore the ubiquity of chaotic solutions in the coupled systems may be a means of information production and signaling in neural systems

Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps

What input signals will lead to synchrony vs. desynchrony in a group of biological oscillators? This question connects with both classical dynamical systems analyses of entrainment and phase locking and with emerging studies of stimulation patterns for controlling neural network activity. Here, we focus on the response of a population of uncoupled, elliptically bursting neurons to a common pulsatile input. We extend a phase reduction from the literature to capture inputs of varied strength, leading to a circle map with discontinuities of various orders. In a combined analytical and numerical approach, we apply our results to both a normal form model for elliptic bursting and to a biophysically-based neuron model from the basal ganglia. We find that, depending on the period and amplitude of inputs, the response can either appear chaotic (with provably positive Lyaponov exponent for the associated circle maps), or periodic with a broad range of phase-locked periods. Throughout, we discuss the critical underlying mechanisms, including slow-passage effects through Hopf bifurcation, the role and origin of discontinuities, and the impact of noiseComment: 17 figures, 40 page

Periodic pulse solutions to slowly nonlinear reaction-diffusion systems

The presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which vanishes on the spectrum of the linearization about the pattern. For certain 'slowly linear' prototype models it has been shown, via geometric arguments, that the Evans function factorizes in accordance with the scale separation. This leads to asymptotic control over the spectrum through simpler, lower-dimensional eigenvalue problems. Recently, the geometric factorization procedure has been generalized to homoclinic pulse solutions in slowly nonlinear reaction-di ffusion systems. In this thesis we study periodic pulse solutions in the slowly nonlinear regime. This seems a straightforward extension. However, the geometric factorization method fails and due to translational invariance there is a curve of spectrum attached to the origin, whereas for homoclinic pulses there is only a simple eigenvalue residing at 0. We develop an alternative, analytic factorization method that works for periodic structures in the slowly nonlinear setting. We derive explicit formulas for the factors of the Evans function, which yields asymptotic spectral control. Moreover, we obtain a leading-order expression for the critical spectral curve attached to origin. Together these approximation results lead to explicit stability criteria.NDNS+NWO-clusterAnalysis and Stochastic

Lines in the sand : behaviour of self-organised vegetation patterns in dryland ecosystems

Vast, often populated, areas in dryland ecosystems face the dangers of desertification. Loosely speaking, desertification is the process in which a relatively dry region loses its vegetation - typically as an effect of climate change. As an important step in this process, the lack of resources forces the vegetation in these semi-arid areas to organise itself into large-scale spatial patterns. In this thesis, these patterns are studied using conceptual mathematical models, in which vegetation patterns present themselves as localised structures (for example pulses or fronts). These are analysed using mathematical techniques from (geometric singular) perturbation theory and via numerous numerical simulations. The study of these ecosystem models leads to new advances in both mathematics and ecology. NWO Mathematics of Planet EarthAnalysis and Stochastic