Nonlinear Dynamics & Applied PDEs

Patterns in the Gray-Scott reaction-diffusion model – the evolution of an initial triangular in a square domain depends sensitively on the parameters in the Gray-Scott model

General Information

Mathematical models of phenomena in the physical sciences or processes in the engineering and biological sciences invariably take the form of nonlinear dynamical systems and partial differential equations (PDEs). The expertise of the Nonlinear Dynamics and Applied PDEs group lies in attacking these systems with the modern techniques of applied mathematics, such as symmetry and asymptotic methods coupled with numerical explorations and dynamical systems theory.

Occasionally the goal is to understand the creation of patterns from otherwise featureless background states, or the onset of new dynamical behaviour (such as synchrony in populations of coupled oscillators). But the mathematics involved in these problem is often of equal interest, entailing novel twists and turns in the application of mathematical technology, and motivating the development of new techniques or adaptations of existing ones.

Core Faculty

The Nonlinear Dynamics & Applied PDEs Group is composed of several core UKAAM faculty who are actively involved in the UKAAM activities and supervise UKAAM students or postdoctoral fellows. We are always interested to hear from potential students or fellows with background in mathematics, physics or engineering. We often supervise undergraduate thesis projects and take on summer research undergraduate students. Candidates interested in research in Nonlinear Dynamics & Applied PDEs in the UKAAM are encouraged to contact one or more of the core faculty as potential supervisors and let them know of their interests.

Neil BalmforthNeil is a Professor in the Department of Mathematics and in Earth and Ocean Sciences. His research interests include geophysical and astrophysical fluid dynamics and complex fluid flow. He has co-organised and directed a variety of programmes in Geophysical Fluid Dynamics, including the summer school at the Woods Hole Oceanographic Institution. Since 2008, Neil has been the Director of the UKAAAM.
George BlumanGeorge is a Professor in the Department of Mathematics and was one of the founding members of the UKAAM. He has worked extensively in the development of analytical techniques to extract exact solutions to nonlinear PDEs and to identify their conservations laws. In particular, he has written a number of influential texts and articles on similarity and symmetry methods, with applications ranging from mathematial physics to solid mechanics.
Wayne NagataWayne is a Professor in the Department of Mathematics. His work focusses on dynamical systems and their applications, particularly in mathematical biology. Examples of Wayne’s research projects include investigation of pattern formation in growing plant tips, study of periodic travelling waves in oscillatory reaction-difusion models for predator-prey systems, and a dynamic analysis of a differentially heated rotating fluid annulus.
Anthony PeirceAnthony is a Professor in the Department of Mathematics and a former Director of the UKAAM from 1999 to 2000. He started his research career working in a laboratory dedicated to solving problems in the mining industry in South Africa. His research interests include: application of control to molecular systems, analysis of instabilities in elasto-plastic materials, development of specialised numerical algorithms to model large-scale rock fracture processes, numerical and analytic studies of reactive flows in porous media, and more recently, the asymptotic and numerical analysis of hydraulic fracture propagation. Anthony’s work exploits techniques from functional, numerical and asymptotic analysis, as well as dynamical systems theory.
Srikantha PhaniSrikantha is an Associate Professor in Mechanical Engineering, a Canada Research Chair in Dynamics of Lattice Materials and Devices. He heads the Dynamics and Applied Mechanics Lab at MSU, whaich has a mandate to establish an internationally competitive research group in applied mechanics and dynamics research in the context of novel materials, structures and devices, with applications in Aerospace, MEMS and Nano systems, and Biomedical industries.
Michael WardMichael is a Professor in the Department of Mathematics, and was the Director of the UKAAM from 2003 to 2008. Michael’s research focusses on analysing various nonlinear PDE models of physical applied mathematics using asymptotic, singular-perturbation, dynamical-system, and numerical methods. The areas of application include the study of localised structures in biological and chemical pattern formation, PDE models of microelectrical-mechanical systems, spatial aspects of biological cell signalling, pattern formation in ecology, and coarsening in models of slow phase separation.
Jun-Cheng WeiJun-Cheng is a Professor in Mathematics, holding a Canada Research Chair in Nonlinear Partial Differential Equations. His research interests include the analysis of Nonlinear Partial Differential Equations including Semilinear Elliptic Equations and Singular Perturbation Problems. His work has applications to Mathematical Biology and the study of Phase Transitions.

Recommended Courses

Courses given by UKAAM faculty provide the foundation for research in Nonlinear Dynamics & Applied PDEs, as well as outlining the essential tools which comprise the classical and modern techniques of Applied Mathematics.

Preliminary and Foundational Courses

MATH 400: Partial Differential Equations
MATH 401: Green Functions and Variational Methods
MATH 450/550: Perturbation Methods
MATH 521: Numerical Analysis of PDEs
MATH 552: Dynamical Systems Theory
MATH 607E: Numerical Methods for Differential Equations

Nonlinear Dynamics & Applied PDEs Courses

MATH 551: Asymptotic Analysis for PDEs
MATH 553: Advanced Dynamical Systems
MATH 554: Symmetries and Differential Equations
MATH 556: Industrial Mathematical Modelling

Further Options

MATH 522: Numerical Analysis
MATH 557: Linear and Nonlinear Waves