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Automated adaptive recursive control of unstable orbits in high-dimensional chaotic systems

Abstract

We develop and demonstrate an automated control strategy using an adaptive learning algorithm that can control and track periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The control system is designed to operate in real time, taking time series measurements of a single variable as input and providing as output the control parameter value required to stabilize the desired unstable periodic orbit (UPO). The control scheme directs the system to the fixed point itself rather than a stable manifold and works when the unstable Lyapunov multipliers are relatively large (approx. 6). The learning and control algorithm uses a time delay embedding with the full state vector collected within one period of the controlled orbit. Control is achieved by small perturbations of a single control parameter once each cycle using a control algorithm with one recursive term. A simulation is used to study the application of the control algorithm to the hyperchaotic Rossler system. The simulation demonstrates both control of a highly unstable UPO and tracking the UPO as system parameters slowly drift over a wide range. The difficulties encountered in tracking with recursive control are discussed.

PACS numbers: 05.45.+b, 87.10.+e