Mathematical Sciences: "Chaos with Multiple Positive Lyapunov Exponents

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DMS-9423843 Yorke This proposal studies higher dimensional chaos. Lyapunov exponents measure the average rates of linear expansion or contraction of a dynamical system in different directions; a system with N positive Lyapunov exponents is expanding in N linearly independent directions, thus an attractor of the system should be at least N-dimensional. However, attractors and Lyapunov exponents can be very sensitive to parameter changes, leading to "windows" in parameter space for which the system exhibits lower-dimensional behavior. Also, the fluctuations over time of the number of positive finite-time Lyapunov exponents in some systems leads to important questions about the shadowing properties of numerical trajectories. We will study these and other fundamental aspects of dynamics systems with multiple positive Lyapunov exponents. Higher dimensional chaos can occur in models of many engineering devices and in meteorological models and climate models. This research will tell us if mathematical models can suddenly get trapped in artificial windows of regular behavior. These windows would be artificial in that any realistic level of noise would disrupt the regular behavior. Our research will also shed light on the reliability of numerical trajectories for higher dimensional models. This study is critical to realistic modeling.