Theory and applications of symmetric bifurcation theory

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The research program I am pursuing consists in investigating the interaction between theory and applications of nonlinear dynamics with symmetry, in particular the use of bifurcation theory. Symmetry is often an assumption in mathematical modelling using differential equations. Its origin is sometimes intrinsic to the phenomena/equations under study or symmetry appears as part of simplifying assumptions on the model. I am planning to explore the interaction between symmetry and nonlinear dynamics in the context of three major application areas: pattern formation in animal aggregation models, symmetrically coupled devices (e.g lasers, gyroscopes) and symmetric periodic orbits in Hamiltonian systems. The goals of my research program will be achieved by using application areas as a springboard to develop new tools in nonlinear dynamics with symmetry, and by using existing and new theory to contribute significantly to the application area.***The phenomenon of animal aggregation of individuals of the same species are observed from cells to large mammals and is recently the focus of major mathematical modelling efforts to complement a large body of experimental and empirical data. However, the theoretical basis and methods for the study of several of the patterns emerging from the models follows at a much slower pace and I am committed in the coming years to make important contributions in this area. In particular, symmetry has been shown to be a major factor in the patterns observed numerically in some partial differential equation models and my research expertise is ideally suited to continue making significant contributions in this exciting area of knowledge. ***As the need for more powerful devices grows, networks have become popular alternatives to advance the fundamental limits of performance of an individual unit. Symmetrically coupling the devices is a convenient way to reduce the complexity of the system and favours the emergence of synchronization. The analysis of those models calls for the use of symmetric nonlinear dynamics and it is an important part of my research program to use my expertise in this domain to further the understanding of these coupled systems. Moreover, I expect that these application studies will lead to challenges which will stimulate the emergence of new theoretical results. ******The search for collisionless periodic orbits in the Newtonian N-body problem has been revived by the finding of new orbits (e.g. Hip-Hop, Figure-Eight). The use of time-reversing and spatial symmetries has been an important aspect of this renewed activity. The topic of stability and bifurcations of periodic orbits with symmetry in the N-body problem received less interest and many questions still lie open. Part of my research program seeks to characterize linear and nonlinear stability and explore bifurcation issues using symmetric, topological and numerical tools.  ********