# Browsing by Subject "Mappings (Mathematics)"

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Item Density evolution in systems with slow approach to equilibrium(2004) Nelson, Kevin Taylor; Driebe, Dean J.Show more This dissertation investigates the evolution of probability densities under the Frobenius-Perron operator U in chaotic iterated-map systems that are slow to reach equilibrium. It first concentrates on one-dimensional maps that are slow to reach equilibrium because they feature intermittent chaos due to the presence of a marginal fixed point. Using the method of shift states and coherent states under U, certain results are obtained concerning the spectrum of U in various functional spaces, using as the main example the cusp map f (x) = 1 - √ |1 - 2x |. Those results are applied to obtain corrections to the well-known leading 1/t form of the x-x auto correlation function C(t). The symbolic dynamics of one-dimensional maps are then investigated, with particular emphasis on the implications of the existence of intermittent chaos and with applications to topological conjugation. Next, the statistics of extreme values in one dimensional maps are investigated. Fn(x) is defined as the probability that a point chosen from an initial probability distribution and its first n- 1 iterates under a particular map are all less than x; the properties of Fn(x) are derived analytically for a wide variety of one-dimensional maps, and the on conclusions are confirmed numerically. Finally, higher-dimensional area-preserving maps are investigated. The technique of local spectral decomposition for U, in which approximate right and left eigenstates for U are constructed localized on unstable periodic points, is used to study density evolution and correlation of observables over time.Show more Item Design synthesis of multistable equilibrium systems(2004) King, Carey Wayne; Beaman, Joseph J.; Campbell, Matthew I.Show more Mechanical systems are often desired to have features that can adapt to changing environments. Ideally these systems have a minimum number of parts and consume as little power as possible. Unfortunately many adaptable systems either have a large number of heavy parts and/or continuous actuation of smart materials to provide the adaptive capabilities. For systems where both adaptability and power conservation are desired characteristics, adaptability can be limited by power consumption. Multistable equilibrium (MSE) systems aim to provide a type of adaptable system that can have multiple mechanical configurations, or states, that require no power to maintain each stable configuration. Power is only needed to move among the stable states, and a level of adaptability is maintained. The stable equilibrium configurations are defined by a system potential energy being at a minimum. The design of a MSE system is based around locally shaping a potential energy curve about desired equilibrium configurations, both stable and unstable, such that the basic design goals of position, linearized natural frequency, and transition energy can be specified for the MSE system. By mapping the performance space from the design space in tandem with stochastic numerical optimization methods, the designer determines if a certain system topology can be designed as a MSE system. Qualitative and quantitative mapping procedures enable the designer to decide whether or not the desired design lies near the center or periphery of a performance space. The performance space is defined by the desired design criteria (i.e. locations of the equilibria, natural frequency at the equilibria, etc.) that the designer deems important. If the desired design lies near the periphery of the performance space, a series of optimization trials is performed. This series shows the tendency of the problem to be solved as the desired MSE system characteristics are varied within the performance space from a location where the solution is known to exist to the true desired location where the solution is not guaranteed to exist. Upon analysis of the resulting optimization trends, the designer is able to determine whether or not a feasible limit in the system performance has been reached.Show more Item Renormalization and central limit theorem for critical dynamical systems with weak external random noise(2006) Díaz Espinosa, Oliver Rodolfo; Llave, Rafael de laShow more We study the effect on weak random noise on one dimensional critical dynamical systems, that is, maps with a renormalization theory. We establish a one dimensional central limit theorem for weak noises and obtain Berry– Esseen estimates for the rate of this convergence. We analyze in detail maps near the accumulation of period doubling and critical maps of the circle with golden mean rotation number. We derive scaling relations for several features of the effective noise after long times. These scaling relations are used to show that the central limit theorem for weak noise holds in both examples. We perform several numerical experiments that confirm our results and that suggest several conjectures.Show more