Global Bifurcations in Symmetric Systems (Applied Nonlinear Dynamics)

Global bifurcations are of interest in many physical or biological applications since they describe phenomena such as solitary waves, pulses or general localised patterns. Moreover, they are often responsible for creating chaotic dynamics in dissipative differential equations. Much of the complicated behaviour found in chaotic systems can be explained by constructing maps that are valid near such a bifurcation. In many cases the underlying system exhibits symmetries, whose presence makes the analysis more difficult, and introduces the possibility of new types of phenomena: synchronisation, cycling chaos, and blow-out bifurcations. PhD projects would study these phenomena using modern numerical and analytical methods.

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.

Keywords: Applied Mathematics, Nonlinear dynamics, global bifurcations, chaos