9406144 Knobloch This is a proposal to study the formation and stability properties of patterns occurring in physical systems. Because such patterns often exhibit a high degree of symmetry the techniques employed in the study are designed to take full advantage of such symmetries. In practical applications these symmetries are usually not exact. Therefore a particular emphasis of the proposed work is to study the effects of small symmetry-breaking ``imperfections'' in the system. The proposed work is thus designed to make symmetry-based techniques applicable to realistic systems. The proposal focuses on the effects of distant endwalls on patterns in the form of propagating waves, patterns of oscillations in acoustically excited spherical bubbles and the formation of three-dimensional patterns in morphogenesis. This is a proposal to study bifurcations and pattern formation in systems with symmetry. Such systems occur naturally in a variety of applications. The proposal focuses on pattern formation near onset. In this regime the process is described by amplitude equations. In symmetric systems these must respect the assumed symmetries. The resulting equations enable one to compute not only the possible patterns but also their relative stability. The proposal focuses on extending this approach to three-dimensional patterns (such as those created by the Turing instability or the modes of oscillation of an acoustically excited bubble). In addition in order to make the technique applicable to realistic systems in which the assumed symmetries are usually only approximate the proposal emphasizes the study of small externally imposed symmetry-breaking imperfections. Such imperfections may have important qualitative consequences and may be responsible for the introduction of chaotic dynamics into systems that would otherwise be nonchaotic. ***
