Localisation and Delocalisation (Applied Nonlinear Dynamics)

In recent years there has been increased interest in the localisation and delocalisation of solutions to partial differential equations. Crudely put, localisation occurs when the solution becomes trapped in one part of an extended domain leaving the rest of the domain free from disturbance. Localisation can be either self-regulating or the result of outside influences (or a mixture of the two) and can be modified by the degree of nonlinearity and the breaking of symmetry. The mathematics of localisation includes local and global bifurcation theory and the solution of PDEs by both asymptotic and numerical methods. Applications of the theory are important in chemistry, nonlinear optics, fluid dynamics, theory of thin films, as well as geophysical and astrophysical fluid dynamics.

Keywords: applied mathematics, nonlinear dynamics, bifurcation theory, localised patterns, homoclinic orbit, fluid dynamics

Nonlinear dynamics and its applications at Leeds has for many years enjoyed a reputation for a distinctive interdisciplinary approach. The Centre for Nonlinear Studies was established at Leeds in 1984 to enhance existing and foster new research collaborations between mathematicians, scientists and engineers throughout the university and beyond. Twenty five years later, the research group retains its character as an applications driven centre, applying dynamical systems theory to a range of natural phenomena. It has recently expanded with the appointment of several new members of staff, bringing the total to ten permanent members of staff working with five postdocs and postgraduate students.

Applied Nonlinear Dynamics is a vibrant research area lying at the heart of problems of fundamental and practical importance. It employs a wealth of mathematical techniques, from statistical to geometrical, from computational to algebraic, and from qualitative to analytical. The main concern is systems that change with time, where the presence of nonlinearities can produce hugely complicated behaviour. The range of activities in Applied Nonlinear Dynamics is extremely broad. Core areas of investigation include chaos, global bifurcation theory and the role of symmetry, localised solutions (both in spectral and physical space), coupled oscillators and synchronisation, ergodic theory and stochastic dynamics, and pattern formation in fluid mechanics and reaction-diffusion systems. Developments in the basic theory and techniques of Nonlinear Dynamics go hand-in-hand with investigations of particular applications, such as fluid dynamics experiments, dynamics on complex networks and mixing in microfluidics.