| The concept of stochastic resonance has been introduced previously to describe a curious phenomenon in bistable systems subject to both periodic and random forcing: an increase in the input noise can result in an improvement in the output signal-to-noise ratio. The first half of this thesis is a detailed theoretical and numerical study of stochastic resonance, based on a rate equation approach. The main result is an equation for the output signal-to-noise ratio as a function of the rate at which noise induces hopping between the two states. The manner in which the input noise strength determines this hopping rate depends on the precise nature of the bistable system. For this reason, the theory is applied to two classes of bistable systems, the double well (continuous) system and the two-state (discrete) system. The theory is tested in detail against digital simulations, and an experiment, in which stochastic resonance was observed in a ring laser, is presented.; In the second part of this thesis, an approach to the modeling of time series data from chaotic, nonlinear systems is introduced. With the goal of determining equations of motion which are in a sense optimal, a very general, iterative algorithm is presented and explained in detail. A wide variety of possible implementations are available, and some of the relative advantages and disadvantages of these are discussed. Analysis of time series data by this method may be used as an alternative approach to estimation of Lyapunov exponents, metric entropy, and dimension. Results with simulations and experiments indicate the promise of this approach, but also point out the difficulty of this task, even for systems that may seem quite simple. The algorithm utilizes concepts from nonlinear dynamics--phase space, attractors, chaos, and the importance of nonlinear models--and procedures from traditional statistical modeling. |
