Synchronization in chaotic systems: Coupling of chaotic maps, data assimilation and weather forecasting
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The theme of this thesis is the synchronization of coupled chaotic systems. Background and introductory material are presented in Chapter 1. In Chapter 2, we study the transition to coherence of ensembles of globally coupled chaotic maps allowing for ensembles of non-identical maps and for noise. The transition coupling strength is determined from a transfer function of the perturbation evolution. Analytical results are presented and tested using numerical experiments. One of our examples suggests that the validity of the perturbation theory approach can be problematic for an ensemble of noiseless identical `nonhyperbolic' maps, but can be restored by noise and/or parameter spread. The problem of estimating the state of a large evolving spatiotemporally chaotic system from noisy observations and a model of the system dynamics is studied in Chapters 3 - 5. This problem, refered to as `data assimilation', can be thought of as a synchorization problem where one attempts to synchronize the model state to the system state by using incoming data to correct synchronization error. In Chapter 3, using a simple data assimilation technique, we show the possible occurrence of temporally and spatially localized bursts in the estimation error. We discuss the similarity of these bursts to those occurring at the `bubbling transition' in the synchronization of low dimensional chaotic systems. In general, the model used for state estimation is imperfect and does not exactly represent the system dynamics. In Chapter 4 we modify an ensemble Kalman filter scheme to incorporate the effect of model bias for large chaotic systems based on augmentation of the system state by the bias estimates, and we consider different ways of parameterizing the model bias. The experimental results highlight the critical role played by the selection of a good parameterization model for representing the form of the possible bias in the model. In Chapter 5 we further test the method developed in Chapter 4 via numerical experiments employing previously developed codes for global weather forecasting. The results suggest that our method can be effective for obtaining improved forecasting results when using an ensemble Kalman filter scheme.