Mathematical Sciences: Symmetry Breaking in Spatially-Extended Systems and in Globally-Coupled Oscillator Arrays

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9410115 Silber The investigator will develop equivariant bifurcation theory for applications to symmetry-breaking bifurcations in hydrodynamic systems and in globally coupled oscillator arrays. The first part of the research focuses on translation symmetry breaking in hydrodynamic systems that are extended in more than one space dimension. Both the situation where translation symmetry is broken spontaneously by an attracting solution of the governing equations, and the situation where it is broken externally by perturbing the equations will be considered. In the former case, the manifestations of the underlying translation symmetry in the possible evolution of instabilities of the symmetry-broken state will be investigated. In the case of external symmetry breaking, the research focuses on the specific physical system of rotating Rayleigh-Benard convection; this is a system for which it has already been established that breaking translation symmetry can have a profound effect on certain instabilities. The second part of the research project consists of a group theoretic/dynamical systems analysis of the $S_n$-equivariant Hopf bifurcation problem. This bifurcation problem is pertinent to the dynamics of n globally-coupled identical limit cycle oscillators. This research is motivated by recent analytic and numerical studies of series arrays of Josephson junctions. Global coupling represents an important limiting case for the coupling of oscillators; this aspect of the research project should contribute to our understanding of oscillator arrays ubiquitous in the modeling of physical, chemical, and biological systems. ***