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Prof Alastair Rucklidge
Applications accepted all year round
Competition Funded PhD Project (European/UK Students Only)
About the Project
Regular patterns, such as stripes, squares and hexagons, are ubiquitous in nature, and their formation and stability are governed by the intricate and complex interactions of symmetry and nonlinearity. Nonlinear interaction of waves in different directions can lead to the formation much more complicated and beautiful patterns: quasipatterns, spatio-temporal chaos and other forms of chaotic dynamics, depending on just how the waves interact. This project will involve using ideas from nonlinear dynamics: bifurcation theory, stability theory, three-wave interactions, chaos, symmetry and heteroclinic cycles, to understand the formation and stability of complex patterns such as quasipatterns, spatio-temporal chaos or turbulent spirals.
The distinct aspect of this project is that it will involve problems with two length scales, where waves of two different wavelengths can interact in many different ways. There will be emphasis on deep understanding of the underlying dynamics in the problem, using computational tools, bifurcation theory, asymptotic theory, weakly nonlinear theory, symbolic algebra, group theory, or whatever is needed. While the project will focus on solving a particular set of partial differential equations using asmptotic and numerical methods, one of the beauties of the nonlinear dynamics approach is that it can have wide applicability in different areas of mathematics, physics, chemistry or biology. The ideas that this project will explore have application to understanding patterns in fluid dynamics (the Faraday Wave experiment), soft matter physics (the formation of polymer quasicrystals) and chemistry (two-layer reaction-diffusion systems).
The distinct aspect of this project is that it will involve problems with two length scales, where waves of two different wavelengths can interact in many different ways. There will be emphasis on deep understanding of the underlying dynamics in the problem, using computational tools, bifurcation theory, asymptotic theory, weakly nonlinear theory, symbolic algebra, group theory, or whatever is needed. While the project will focus on solving a particular set of partial differential equations using asmptotic and numerical methods, one of the beauties of the nonlinear dynamics approach is that it can have wide applicability in different areas of mathematics, physics, chemistry or biology. The ideas that this project will explore have application to understanding patterns in fluid dynamics (the Faraday Wave experiment), soft matter physics (the formation of polymer quasicrystals) and chemistry (two-layer reaction-diffusion systems).
References
Three-dimensional Icosahedral Phase Field Quasicrystal, by P. Subramanian, A.J. Archer, E. Knobloch and A.M. Rucklidge. Physical Review Letters 117 (2016) 075501. doi:10.1103/PhysRevLett.117.075501
Project supervisors
Career overview
Professor Alastair M. Rucklidge holds a PhD from the University of Cambridge (Cantab), a SM from the Massachusetts Institute of Technology (MIT), and a BASc from the University of Toronto. He is currently a Professor in the School of Mathematics at the University of Leeds. His research interests focus on the formation and stability of regular patterns in nature, which are influenced by the interactions of symmetry, dynamics, and nonlinearity. Professor Rucklidge''s work explores how nonlinear interactions of waves lead to the emergence of quasipatterns in Faraday waves and quasicrystals in soft matter, as well as spatio-temporal chaos in reaction-diffusion systems. He investigates phenomena such as spirals in cyclic competition models and their stability, and he develops new models to understand mode interactions and localized patterns. His research encompasses quantitative explanations of fluid dynamics experiments and new ideas on heteroclinic bifurcations and networks. Professor Rucklidge is affiliated with research groups in Applied Mathematics and Applied Nonlinear Dynamics. He is a member of the London Mathematical Society, the Society for Industrial and Applied Mathematics, and the Royal Astronomical Society.
Research interests
Professor Rucklidge''s research focuses on the formation and stability of regular patterns in nature, governed by the intricate interactions of symmetry, dynamics, and nonlinearity. His work aims to understand how nonlinear interactions of waves in different directions lead to the formation of quasipatterns in Faraday waves and quasicrystals in soft matter, as well as spatio-temporal chaos in reaction-diffusion systems. He is particularly interested in the stability of spirals, such as those found in models of cyclic competition like Rock-Paper-Scissors. His research encompasses quantitative explanations of fluid dynamics experiments, the development of new models for understanding mode interactions and localised patterns, and novel ideas on heteroclinic bifurcations and networks.
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