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 ͑UPO͒. The control scheme directs the system to the fixed point itself rather than a stable manifold and works when the the unstable Lyapunov multipliers are relatively large (Ϸ6). 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 Rössler 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.

Chaotic transitions and low-frequency fluctuations in semiconductor lasers with optical feedback

“…In cases where the lifetime is extremely long, how to apply control to stabilize the laser? While there has been a tremendous amount of work on control of chaotic dynamics [24][25][26][27][28][29][30][31] and on control of chaotic diode resonators [32,33], the problem of controlling a semiconductor laser with optical feedback remains to be challenging due to the extremely high dimensionality of the system.…”

Bifurcation to High-Dimensional Chaos

“…The mechanism by which the system becomes high-dimensionally chaotic via this route suggests an effective way to control it: By stabilizing the driving chaotic subsystem around some periodic orbits, the whole system becomes periodic. There has been a growing interest in techniques to control [Auerbach et al, 1992;Hu & He, 1993;Petrov et al, 1994;Petrov et al, 1995;Johnson et al, 1995;Ding et al, 1996;Rhode et al, 1996] and to synchronize [Kocarev & Parlitz, 1996;Peng et al, 1996;Lai, 1997;Pecora et al, 1997] high-dimensional chaotic systems, which are highly nontrivial and challenging problems. A good understanding of how nonlinear systems develop high-dimensional chaos may help to yield insights into devising methods to manipulate such systems.…”

“…In this paper, motivated by numerical simulation of nonlinear dynamical systems in high dimensions and by the growing recent interest in controlling [Romeiras et al, 1992;Auerbach et al, 1992;Hu & He, 1993, Petrov et al, 1994Johnson et al, 1995;Ding et al, 1996;Rhode et al, 1996] and synchronizing [Kocarev & Parlitz, 1996;Peng et al, 1996;Lai, 1997;Pecora et al, 1997;Grebogi & Lai, 1997] these systems, we study the characteristics of bifurcations to high-dimensional chaos. While at present there is no formal definition of low-dimensional versus high-dimensional chaos, we take the notion that low-dimensional chaos is characterized by one positive Lyapunov exponent, and high-dimensional chaos by more than one such exponent.…”

Chaos control via neural networks using radial basis functions

“…In all versions and modifications of the OGY method [1][2][3][4][5][6][7][8][9] it is assumed that: 1. There is no a priori knowledge available about the system dynamics.…”

“…In particular, Ott, Grebogi and Yorke [1] have introduced a control scheme, according to which the chosen target (either an unstable equilibrium point or an unstable periodic orbit of the system) can be stabilised applying small variations to the control parameter. Several extensions and modifications of this local approximation method using more than one control parameters and applied to higher dimensional systems have been proposed in the subsequent years [2][3][4][5][6][7][8][9].…”