Dynamics From Forced Symmetry-Breaking

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Systems with symmetries arise frequently in applications. Such systems exhibit behaviorthat is not typical of nonsymmetric systems. Even though this behavior may be more complexthan expected techniques from equivariant bifurcation theory permit one to analyze itsorigin and properties. However, in many applications the presence of symmetry is anidealization. It is important therefore to understand which aspects of the dynamics of thesymmetric system are robust with respect to small symmetry-breaking imperfections. In manycases the loss of both discrete and continuous symmetries introduces global bifurcationsinto the dynamics and hence the possibility of chaotic time-dependence. This projectproposes a study of such symmetry-breaking perturbations in several important classes ofsystems arising incontinuum mechanics (fluid flow, chemically reacting systems etc) withemphasis on identifying robust behavior both near threshold of a primary instability andfurther away from it. The former gives rise to low-dimensional systems with brokensymmetry via center manifold reduction, while the latter require a study of partialdifferential equations. Special emphasis is placed on understanding the effects ofbreaking both reflection symmetry and translation invariance (both separately andsimultaneously). Phenomena arising from the breaking of time-translation invariance due toparametric forcing and the loss of Galilean invariance will also be studied. The goal ofthe project is to gain a deep understanding of the limitations on the applicability ofidealized models and to provide a description of the type of behavior that is typical of''nearly-symmetric'' systems.The project aims at understanding the basic principles behind robust behavior. Forexample, understanding the effects of broken reflection symmetry is essential whenstudying the effects of a small inclination of an apparatus to the horizontal, and theeffects of through-flow or rotation on transitions from one state to another. Problems ofthis type arise in microscopic situations (thin films on horizontal or inclinedsubstrates) in geophysics (for example in oceanography) and everywhere in between.Similarly, understanding the trapping of wavelike disturbances by spatial inhomogemeitiesis of importance not only in catalysis where the inhomogeneities may be on atomic scale,and in the trapping of weather patterns by topography.