Pattern formation is the study of the spontaneous appearance of structure in nature and in the laboratory. Natural examples include sand ripples, geological structures such as the Giant's Causeway, cloud formations and animal coat markings. Laboratory examples span a diverse range of disciplines including fluid mechanics, granular media, chemistry (both at macroscopic and nanometre scales) and nonlinear optics. This broad range of motivating examples is mirrored in the similarly broad range of techniques that have been brought to bear on their analysis: dynamical systems theory, group representations, asymptotic analysis for differential equations and computational methods. Possibilities for progress rest on observations time and again of similar features in these many different experimental systems, pointing to universality that should be manifest in the underlying mathematics. Many pattern formation problems can be analysed using the theory of bifurcations with symmetry (equivariant bifurcation theory), but there are still numerous experimentally observed patterns that cannot yet be explained within this framework, often because of very deep mathematical issues. Examples include quasipatterns, spiral defect chaos and spatially modulated two-dimensional patterns. PhD projects would examine these kinds of patterns, using a combintation of numerical work, solving the partial differential equations for pattern-forming systems, and equivariant bifurcation theory or asymptotic analysis, with an overall aim of developing new theory.
Nonlinear dynamics and its applications at Leeds has for many years enjoyed 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.