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C M van den Bleek

Publications and source records attributed to C M van den Bleek.

4 recordsLinked to original sources

Learning chaotic attractors by neural networks.

An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured time series. During training, the algorithm learns to short-term predict the time series. At the same time a criterion, developed by Diks, van Zwet, Takens, and de Goede (1996) is monitored that tests the hypothesis that the reconstructed attractors of model-generated and measured data are the same. Training is stopped when the prediction error is low and the model passes this test. Two other features of the algorithm are (1) the way the state of the system, consisting of delays from the time series, has its dimension reduced by weighted principal component analysis data reduction, and (2) the user-adjustable prediction horizon obtained by "error propagation"-partially propagating prediction errors to the next time step. The algorithm is first applied to data from an experimental-driven chaotic pendulum, of which two of the three state variables are known. This is a comprehensive example that shows how well the Diks test can distinguish between slightly different attractors. Second, the algorithm is applied to the same problem, but now one of the two known state variables is ignored. Finally, we present a model for the laser data from the Santa Fe time-series competition (set A). It is the first model for these data that is not only useful for short-term predictions but also generates time series with similar chaotic characteristics as the measured data.

Algorithms↗

Synchronizing chaos in an experimental chaotic pendulum using methods from linear control theory.

Linear feedback control, specifically model predictive control (MPC), was used successfully to synchronize an experimental chaotic pendulum both on unstable periodic and aperiodic orbits. MPC enables tuning of the controller to give an optimal controller performance. That is, both the fluctuations around the target trajectory and the necessary control actions are minimized using a least-squares solution of the linearized problem. It is thus shown that linear control methods can be applied to experimental chaotic systems, as long as an adequate model is available that can be linearized along the desired trajectory. This model is used as an observer, i.e., it is synchronized with the experimental pendulum to estimate the state of the experimental pendulum. In contrast with other chaos control procedures like the map-based Ott, Grebogi, and York method [Phys. Rev. Lett. 64, 1196 (1990)], the continuous type feedback control proposed by Pyragas [Phys. Lett. A 170, 421 (1992)], or the feedback control method recently proposed by Brown and Rulkov [Chaos 7 (3), 395 (1997)], the procedure outlined in this paper automatically results in a choice for the feedback gains that gives optimum performance, i.e., minimum fluctuations around the desired trajectory using minimum control actions.

Journal Article↗