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May 28th 2013

We study the existence of grain boundaries with angle in (0, π) in the Swift-Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a
center manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ODEs in normal form. We show persistence of the leading order
approximation using transversality induced by wavenumber selection. Due to the qualitative
dynamical difference, we prove our theorem by two cases: the nonresonant case and the
resonant case.

The neural field equation (NFE) has been widely used to model the coarse-grained activity of large ensembles of cortical neurons. The resulting integro-differential equation describes the mean firing rate of neurons across a spatially continuous domain. Bifurcation analysis has proved effective in the study of the complex spatio-temporal patterns of activity that can arise in the homogeneous NFE (without input). However, taking visual areas as an example, the cortex is almost constantly under external stimulation and many analytical tools are not applicable in the presence of these inputs. Numerical continuation provides a means of detecting, classifying and tracking bifurcations under the variation model parameters. We demonstrate the effectiveness of this tool for studying the NFE with inputs in two case studies. In the first, the surface of the cortex is represented by the Euclidean plane and we investigate the spread of activity in the primary visual cortex for localised input. In the second, we consider an abstracted periodic feature space of motion direction and study alternations of direction-tuned responses for an ambiguous visual motion stimulus.

In this talk, I will present some results on hotspots of an urban crime model. I will show that the crime model gives rise to localised 1D states that undergo a process known as "homoclinic snaking" and and can form multi-pulses. We path-follow these localised states into a singular limit region where some more detail analysis can be done. We then discuss what happens to two-dimensional localised hotspots and relate the results to the crime application.

We will discuss the application of coarse-grained bifurcation analysis
to sociological agent-based models. We will start by reviewing basic concepts of coarse-grained bifurcation analysis, showing how standard path-following techniques can be adapted to stochastic systems. In sociological agent-based models, the microscopic variables are typically integers representing discrete states of the system, whereas macroscopic variables are often real fields, representing densities or ensemble averages. Coarse-grained bifurcation techniques for these types of models are severely affected by noise, leading to bad convergence properties of the Newton-Krylov solver. We will introduce a lifting strategy, based on a set of precomputed weights, which requires the solution of a minimisation problem at each Newton step. We will then present numerical experiments showing increased convergence properties of the stochastic root-finder, as well as a much more accurate estimation of Jacobian-vector products. We apply weighted lifting to a model of vendor lock-in, explaining global polarisations in terms of a coarse pitchfork bifurcation and comparing numerical results with an approximate mean-field description of the system. Finally, we will discuss the formation and stability of macroscopic fronts representing neighbouring factions of agents with different preferences.

**Multi-dimensional Localized Structures**

Lecture Video

Lecture Video

Wavespeed selection refers to the problem of determining the long time asymptotic speed of invasion of an unstable state by some secondary state. This tutorial will discuss wavespeed selection mechanisms for reaction-diffusion equations. Particular emphasis will be placed on the qualitative differences between wavespeed selection in systems of reaction diffusion equations and scalar problems. The tutorial will review the notions of the linear spreading speed, pushed and pulled fronts, marginal stability and the role of singularities of the pointwise Green's function. Examples will include a Lotka-Volterra competition model, the Keller-Segel model, and some examples of coupled Fisher-KPP equations.