Computational Analysis of Large Dynamical Systems

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This work consists of three projects related to the computational analysis of dynamical systems with many degrees of freedom. The first concerns networks of coupled biological oscillators and excitable elements driven by fluctuating external stimuli. The goal is to elucidate, within a class of biologically-relevant architectures, the relation between a network's structure and the reproducibility, or reliability, of its response. The second project concerns the emergence of macroscopic transport processes when an open system is coupled to unequal heat reservoirs at its boundaries. This project focuses on a prototypical class of models that includes both deterministic and stochastic microscopic dynamics. The aim here is to gain insights into the properties of nonequilibrium steady states in a concrete class of model systems. The third project aims to develop efficient numerical algorithms for computing statistical averages, e.g., Lyapunov exponents, that are frequently used to characterize nonlinear dynamical systems. The algorithms to be developed are based on exploiting approximate prior knowledge of the quantity to be computed, e.g., approximate knowledge of the system's invariant measure. This will be done via coupling multiple simulations of the system in question, and to use such couplings to produce unbiased estimators with potentially significantly smaller variance. The efficacy of the algorithms in various biological and physical settings, including the projects outlined above, will be investigated.Dynamical systems with many strongly nonlinear degrees of freedom arise in many scientific and technological problems. Their analysis and simulation is often difficult because of the complexity of their interactions and their often chaotic dynamics. The projects comprising this work seek to understand such large dynamical systems in some specific settings, and to develop general, efficient numerical algorithms for computing relevant statistical properties of nonlinear dynamical systems. The expected outcome of this research may lead to deeper insights into a range of phenomena, including the ability of biological neural networks to encode information and the emergence of macroscopic energy and matter transport in spatially-extended systems with complex microscopic interactions. The algorithms to be developed are potentially applicable to other application domains, e.g., stochastic chemical kinetics. It is expected that the projects will lead to interdisciplinary collaborations, e.g., with biological scientists, and to opportunities for graduate student training.