Narrow Results

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 identified 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.

Recent progress towards an understanding of fluctuational escape from chaotic attractors (CAs) is reviewed and discussed in the contexts of both continuous systems and maps. It is shown that, like the simpler case of escape from a regular attractor, a unique most probable escape path (MPEP) is followed from a CA to the boundary of its basin of attraction. This remains true even where the boundary structure is fractal. The importance of the boundary conditions on the attractor is emphasized. It seems that a generic feature of the escape path is that it passes via certain unstable periodic orbits. The problems still remaining to be solved are identified and considered.