Pattern Formation in Two and Three Dimensions

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9703684 Knobloch This project will involve the study of bifurcations and pattern formation in systems with symmetry. Such systems arise naturally in a variety of applications. The proposal focuses on the study of the consequences of weak system symmetry--breaking perturbations for pattern formation in two and three dimensions. In doing so it seeks to identify those aspects of equivariant dynamics that are robust under symmetry-breaking "imperfections". The approach promises to extend the applicability of equivariant bifurcation theory to more realistic situations and is an important step towards a theory of pattern formation in two and three dimensions. Applications to a number of continuum systems are envisaged, including pattern formation by the Faraday and Turing instabilities, and acoustically driven shape oscillations of drops and bubbles. This project will involve the study of the spontaneous formation of structure in systems with symmetry. Such systems occur naturally in a variety of applications, and may reflect the geometry of the system or a convenient idealization. Symmetries are known to have a profound effect on the typical behavior of the system. Near onset the process of structure or pattern formation can be described by simplified equations called amplitude equations. The form of these equations is dictated by the symmetries of the system. Often the symmetries constrain these equations to such an extent that only one possibility exists. The analysis of the resulting equations then provides a complete understanding of the structure formation process, and can be used to predict the type of structure that will form as the parameters of the system are varied. Since in nature the assumed symmetries are usually an idealization the proposal seeks to identify those aspects of the structure formation process that are insensitive to small symmetry-breaking imperfections, i.e., it seeks to identify those aspects that are robust. The a im of the proposal is thus to extend the applicability of this type of theory to realistic problems in both two and three dimensions. As examples of this approach we shall study pattern formation in vibration-induced surface waves, structure formation in chemical and biological systems via the Turing instability, and the patterns on the surface of an acoustically excited drop or bubble. An understanding of these relatively simple systems is a prerequisite for extending the techniques to more complex systems.