Synchronization and Signal Enhancement in Nonlinear and Stochastic Systems
Bennett, Matthew Raymond
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In the first part of this dissertation we explore the consequences of high frequency operation of Josephson junction arrays. At high frequencies these systems are no longer well modeled by Kirchhoffs laws, and new dynamical equations are derived directly from Maxwells equations. From these equations we derive a reduced set of averaged equations which greatly simplify the analysis of high frequency arrays. The averaged equations allow us to examine experimental strategies for obtaining higher power outputs from arrays. These strategies rely on resonant architectures that place the junctions near antinodes of a desired standing wave mode of the fluctuating current. Simple, heuristic rules are derived for the proper placement of junctions. The second part of the dissertation is devoted to stochastic resonance. A new theory is proposed to explain both two-state and excitable stochastic resonance. Previous theories explaining the two types of stochastic resonance yield similar results while using different analytic strategies. A constrained asymmetric rate model is derived that in one limit produces the proper result for the two-state system, while in another limit models the excitable system. The result that the constrained asymmetric rate model gives in the excitable limit is off by a factor of two, and this discrepancy is examined. Furthermore, we study the consequences of adding a colored noise source to the classic two-state model of stochastic resonance. We will find that when both white and colored noise sources are present, stochastic resonance will occur as a function of colored noise strength only if the correlation time of the colored noise source is small enough. Two theories are proposed to explain this phenomenon and both are examined in detail.