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Nonlinear Dynamics has, for
many years, been one of the great strengths of Applied Mathematics at Leeds,
with a distinctive interdisciplinary approach facilitated by the Centre for
Nonlinear Studies and, at various times, involving colleagues in Chemistry,
several Engineering departments, Geography and Physiology. The Nonlinear
Dynamics group is engaged in research in applying bifurcation theory to a
number of natural phenomena, particularly those motivated by astrophysics and
chemical systems.
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| Perturbation Methods And Algebraic Manipulators (Mark Kelmanson) | ||
| A large number of realistically motivated problems in science and engineering are tractable only to approximate solution methods. Many nonlinear problems readily yield to asymptotic analysis and perturbation methods which, by their very construction, may be highly complicated from an algebraic point of view. As such, multi-purpose `pseudo-exact' (in the sense that the error is well understood) solvers for ODEs and PDEs are sought, particularly using both matched-asymptotics and multiple-scales. Many problems have already been conquered by automating this process using algebraic manipulators such as `Maple': for example, in recent work, new light has recently been shed on the classic problem of nonlinear surface waves in viscous coating flow on a rotating cylinder. Prof Kelmanson is also interested in developing automated asymptotic solvers for examining the effects of rounding errors in the numerical solution of both linear and nonlinear integral equations. | ||
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| Stochastics (Grant Lythe) |
RandomIdeas More details |
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| Stochastic phenomena are found everywhere: the growth of a bacterial population, the path of a particle in Brownian motion, the number of electrons in a cosmic ray shower, the price of a stock as a function of time. They are controlled by laws that are probabilistic rather than deterministic. (The word `stochastic' comes from the Greek word στωχωσ , meaning chance.) Solving a stochastic differential equation is like solving an ordinary differential equation: exact analytical solutions are seldom available, but paths can be generated in a matter of seconds on a computer and analysed using tools from pure and applied mathematics. On the other hand, stochastic partial differential equations describe spatially extended systems with randomness, such as vortices in a fluctuating environment. The theory of such equations is still in its infancy, but the rapid increase in the computing power available to scientists has permitted their increasing use. | ||
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| Spatio-temporal Structures In Chemical Systems (John Merkin) | ||
| Spatially distributed chemical systems can give rise to a wide range of complex structures which arise from the interaction of diffusion and chemical reaction. The basic forms are travelling waves, either as reaction fronts or pulses, and patterns (steady, spatially non-homogeneous responses). These can interact to generate periodic, aperiodic and chaotic behaviour and, in the presence of additional transport effects such as fluid flow or electric fields, further bifurcations are possible, for example between absolute and convective instabilities and to flow-driven patterns. The projects presently being considered are: (i) the effects of chaotic mixing on chemical reactions, (ii) the development of spatio-temporal structures within flow reactors and by applied electric fields, (iii) buoyant instabilities of reaction fronts, (iv) the initiation, inhibition and quenching of combustion reactions by endothermic processes. | ||
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| Pattern Formation (Alastair Rucklidge) | More details | |
| 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 equivariant bifurcation theory, but there are still numerous experimentally observed patterns that cannot yet be explained within this framework. Examples include quasipatterns and spatially modulated two-dimensional patterns. PhD projects would examine these kinds of patterns with an aim to developing new theory. | ||
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| Chaotic Dynamics With Symmetry (Alastair Rucklidge) | More details BBC interview |
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| Global bifurcations are often responsible for creating chaotic dynamics in dissipative differential equations. Much of the complicated behaviour exhibited by chaotic systems can be explained by constructing maps (usually one-dimensional) that are valid near such a bifurcation. The presence of symmetry makes the analysis more difficult, and introduces the possibility of new types of phenomena: synchronisation, cycling chaos, and blow-out bifurcations. New research would study these phenomena in cases where symmetry requires the use of higher-dimensional maps to describe the dynamics. | ||
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