Modeling and control of evolving, noisy chaotic dynamical systems

We study the modeling and control of evolving dynamical systems. In particular we model the dynamics of an evolving noisy iterative map, we study the extraction of the control parameter of the same map using a sparse time series from it, and we also study the dynamics of an evolving electronic circuit whose control parameter changes in proportion to low-pass filtering of one of its dynamical variables. First, we study a noisy one-dimensional iterative map whose system parameter evolves randomly in time. We find that there is an optimal number of model parameters and that there is an optimal number of data points to be retained as input for the model. These optimal numbers depend on the rate of change of the system parameter of the iterative map being modeled and its level of noise. Second, we study modeling based on a sparsely sampled time series. Using the same iterative map, the logistic map with noise, we fix the system parameter at a constant value and iterate the map repeatedly to generate data. We compare three different methods of extracting the system parameter from a sparse set of the map's data. Two of these methods employ an ensemble of test trajectories in order to determine if statistical properties, such as the law of large numbers, facilitate the search for the system parameter. The third extraction method uses a single test trajectory. These three methods are applied to both periodic and chaotic dynamics. We find in the periodic regime that, at low noise levels, the three methods all yield an accurate estimate of the system parameter and that this estimate surprisingly shows little variation as the sparseness of the data is increased eightfold. For large noise levels in the periodic regime, the two statistical extraction methods yield better results than the single trajectory method, particularly for test trajectories whose periodic sequence has been shifted out of phase with respect to the experimental trajectory by the presence of noise. In the chaotic regime, on the other hand, the results of all three methods depend much less on the noise level. Their estimates at low noise levels are at least an order of magnitude worse than at a similar noise level in the periodic regime. Third, we study self-adjusting dynamical systems. We study the logistic map and a chaotic electronic circuit, the Chua oscillator. In both, the system parameter that controls the type of dynamics is adjusted by low-pass filtered feedback. We find that when these systems begin in a chaotic region of phase space, they self-adjust their own dynamics to the edge of chaos or a periodic window neighboring chaos. The self-adjusted parameter diffuses through the chaotic region, and ordinary diffusion formulas are found to apply. From the periodic window or the edge of chaos, the system can occasionally re-enter the chaotic region. In addition, we study a self-adjusting system which has both low-pass filtered feedback and linear feedback control applied to its system parameter. The objective of the linear feedback control component is to drive the parameter to a target value in the presence of the low-pass component which behaves as described earlier. We find that the system parameter stays close to the target parameter value if the dynamics is non-chaotic. In the chaotic regime, the system parameter follows the target value only if the linear feedback component is large in comparison to the low-pass component. If the linear feedback is not large, then the system self-adjusts to the edge of chaos.U of I Onlythesis/dissertatio