9704589 Kuske The first objective of this project is understanding how small noise can effect qualitative changes in nonlinear behavior. For example, small noise can eliminate chaotic behavior in slow-fast dynamics. Recently the principal investigator has developed a new approach for studying these systems in terms of probability densities. These techniques will be adapted to investigate small noise effects in delay phenomena in excitable systems and chemical reactions. Comparison of these problems will provide a basis for characterizing a variety of noisy phenomena. The second objective is to discover conditions for spatially localized oscillations in systems of coupled oscillators. In particular, the principal investigator will consider bistability of localized and nonlocalized dynamics in laser arrays and structural dynamics. Numerical simulations and singular perturbation methods are combined to explore competing influences of the system parameters, initial conditions, and resonance. The third objective is to explore pattern dynamics in reaction-diffusion systems. Complementary analytical numerical methods are used to derive evolution equations and study the dynamics and stability of modulated patterns. These techniques will be applied in general reaction-diffusion problems and a specific application of burner-stabilized flames. They will also be applied to investigate modulated nonequilateral hexagonal patterns and transitions of two- dimensional patterns in spatially inhomogeneous systems. The first objective of this project is motivated by the dramatic effects of small fluctuations in systems with interacting components. In some examples of chaotic interactions, such as convection, waves in plasma, turbulence and laser dynamics, small noise causes nearly periodic behavior to replace large fluctuations and chaos. Noise can also change the transition from a flat state to an oscillatory state in the propagation of nerve impulses an d in laser intensities. When a time variation in a system parameter controls the transition, there can be a delay or lag in the transition. The introduction of small, rapid fluctuations in the control parameter reduces this delay, as observed experimentally. Realizing the first objective will facilitate the development of mathematical models which account for system noise and describe realistic physical behavior. The second goal is motivated by the effects of localized behavior in several examples of interacting oscillations. Oscillating quantities include chemical concentrations and electrical currents which act as timing or reaction mechanisms in biological and chemical systems. Localization of these oscillations can limit the propagation of an electrical impulse or chemical reaction. Coupled oscillators also model fluctuations in laser arrays, used for applications which require a high optical power output, such as space communications and high speed optical recording. The power of coupled lasers is influenced by variations in intensities and synchronization, which occur in localized oscillations about a steady intensity. Vibrations in repetitive engineering structures are also coupled oscillations, such as in large space structures and bladed disk and beam assemblies. Irregularities in engineering structures can cause vibrations to be spatially localized. Localization can result in increased local stresses and cracking or serve as a desirable damping of energy. Understanding the causes of localization allows the control of the phenomena by adjusting the physical system. The third objective of this project focuses on the dynamics of two dimensional patterns which occur away from equilibrium in many physical or chemical systems, including convection, chemical reactions, and sedimentation in rivers. Patterns occur as superpositions of periodic waves of components such as temperature, concentration or velocity. Spatial variation of geometry or physical control parameters can cause spatial intermittency of the patterns. For example, in a meandering river, rippled sediment deposits are found in certain regions and not in others. In reaction-diffusion systems, slight spatial variation of control species causes periodic variation of chemical concentration in limited regions of the reaction. Since the equations describing pattern evolution appear in a variety of applications, understanding these patterns in one application leads to the understanding of patterns in others.
