| The work presented in this thesis studies the phenomena of synchronization and bursting in dynamical systems. The work has three major parts. In the first part the bubbling transition is studied. We build theoretical models to derive the properties of a dynamical system containing an invariant manifold, when the system is perturbed in the direction transverse to the invariant manifold. We present a complete study that includes consideration of all generic bifurcations that may lead to bubbling. In the second part a useful method of looking at weather model data is described. This method accurately extracts wave packet envelopes from atmospheric wind velocity data. We also describe possible practical applications of the method. The last part describes a new data assimilation technique for numerical weather prediction. In the context of numerical weather prediction, data assimilation is the method of determining the best guess of the current state of the atmosphere, which is then used to make the subsequent forecast. From the point of view of dynamical systems, data assimilation acts as coupling between the numerical weather model and the atmosphere. In a broad sense, we desire to achieve perfect synchronization between the atmosphere and the model, thus leading to good quality forecasts. Due to model imperfections, measurement errors, and sparseness of the atmospheric data, the synchronization is not perfect. We observed bursting in the RMS error of the best guess state that has a similar nature to the bursting in the synchronization studies of Chapter I. Although study of the bursts of RMS error in determining the best guess state from the bubbling transition point of view is beyond the scope of this thesis, some preliminary results hint that these bursts may be explained as bubbling. |
