TY - JOUR

T1 - Optimal fluctuations and the control of chaos.

AU - Luchinsky, D. G.

AU - Khovanov, Igor A.

AU - Beri, S.

AU - Mannella, R.

AU - McClintock, Peter V.E.

N1 - Electronic version of an article published as International Journal of Bifurcation and Chaos, 12, (3), 2002, 583-604 10.1142/S0218127402004528 © copyright World Scientific Publishing Company http://www.worldscinet.com/ijbc/ijbc.shtml

PY - 2002/3

Y1 - 2002/3

N2 - The energy-optimal migration of a chaotic oscillator from one attractor to another coexisting attractor is investigated via an analogy between the Hamiltonian theory of fluctuations and Hamiltonian formulation of the control problem. We demonstrate both on physical grounds and rigorously that the Wentzel-Freidlin Hamiltonian arising in the analysis of fluctuations is equivalent to Pontryagin's Hamiltonian in the control problem with an additive linear unrestricted control. The deterministic optimal control function is identied with the optimal fluctuational force. Numerical and analogue experiments undertaken to verify these ideas demonstrate that, in the limit of small noise intensity, fluctuational escape from the chaotic attractor occurs via a unique (optimal) path corresponding to a unique (optimal) fluctuational force. Initial conditions on the chaotic attractor are identified. The solution of the boundary value control problem for the Pontryagin Hamiltonian is found numerically. It is shown that this solution is approximated very accurately by the optimal fluctuational force found using statistical analysis of the escape trajectories. A second series of numerical experiments on the deterministic system (i.e. in the absence of noise) show that a control function of precisely the same shape and magnitude is indeed able to instigate escape. It is demonstrated that this control function minimizes the cost functional and the corresponding energy is found to be smaller than that obtained with some earlier adaptive control algorithms.

AB - The energy-optimal migration of a chaotic oscillator from one attractor to another coexisting attractor is investigated via an analogy between the Hamiltonian theory of fluctuations and Hamiltonian formulation of the control problem. We demonstrate both on physical grounds and rigorously that the Wentzel-Freidlin Hamiltonian arising in the analysis of fluctuations is equivalent to Pontryagin's Hamiltonian in the control problem with an additive linear unrestricted control. The deterministic optimal control function is identied with the optimal fluctuational force. Numerical and analogue experiments undertaken to verify these ideas demonstrate that, in the limit of small noise intensity, fluctuational escape from the chaotic attractor occurs via a unique (optimal) path corresponding to a unique (optimal) fluctuational force. Initial conditions on the chaotic attractor are identified. The solution of the boundary value control problem for the Pontryagin Hamiltonian is found numerically. It is shown that this solution is approximated very accurately by the optimal fluctuational force found using statistical analysis of the escape trajectories. A second series of numerical experiments on the deterministic system (i.e. in the absence of noise) show that a control function of precisely the same shape and magnitude is indeed able to instigate escape. It is demonstrated that this control function minimizes the cost functional and the corresponding energy is found to be smaller than that obtained with some earlier adaptive control algorithms.

KW - Optimal control

KW - control of chaos

KW - nonlinear oscillator

KW - large fluctuations

KW - optimal path

KW - optimal force

KW - escape

KW - stochastic process

U2 - 10.1142/S0218127402004528

DO - 10.1142/S0218127402004528

M3 - Journal article

VL - 12

SP - 583

EP - 604

JO - International Journal of Bifurcation and Chaos

JF - International Journal of Bifurcation and Chaos

SN - 0218-1274

IS - 3

ER -