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• W2012081888 abstract "In the one-dimensional classical analogs to Anderson localization, whether optical, acoustical or structural dynamic, theperiodic system has its periodicity disrupted by having one or more of its parameters randomly disordered. Suchrandomized systems can be modeled via an infinite product of random transfer matrices. In the case where the transfermatrices are 2x2, the upper (and positive) Lyapunov exponent of the random matrix product is identified as thelocalization factor (inverse localization length) for the disordered one-dimensional model. It is this localization factorwhich governs the confinement of energy transmission along the disordered system, and for which the localizationphenomenon has been of interest.The theorem of Furstenberg for infinite products of random matrices allows us to calculate this upper Lyapunovexponent. In Furstenberg's master formula we integrate with respect to the probability measure of the random matrices,but also with respect to the invariant probability measure of the direction of the vector propagated by the long chain ofrandom matrices. This invariant measure is difficult to find analytically, and, as a result, either an approximatingassumption is frequently made, or, less frequently, the invariant measure is determined numerically.Here we calculate the invariant measure numerically using a Monte Carlo bin counting technique and then numericallyintegrate Furstenberg's formula to arrive at the localization factor for both continuous and discrete disorder of the mass.This result is cross checked with the (modified) Wolf algorithm." @default.
• W2012081888 created "2016-06-24" @default.
• W2012081888 creator A5081014047 @default.
• W2012081888 date "2009-03-26" @default.
• W2012081888 modified "2023-09-23" @default.
• W2012081888 title "Localization and the invariant probability measure for a structural dynamic system" @default.
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• W2012081888 doi "https://doi.org/10.1117/12.817760" @default.
• W2012081888 hasPublicationYear "2009" @default.
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