S. Beri

OPTIMAL FLUCTUATIONS AND THE CONTROL OF CHAOS

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 Pontryagins 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.

Fluctuational Transitions through a Fractal Basin Boundary.

Fluctuational transitions between two co-existing chaotic attractors, separated by a fractal basin boundary, are studied in a discrete dynamical system. It is shown that the mechanism for such transitions is determined by a hierarchy of homoclinic points. The most probable escape path from the chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.

Noise-induced shift of singularities in the pattern of optimal paths

We analyse the non-equilibrium distribution in dissipative dynamical systems at finite noise intensities. The effect of finite noise is described in terms of topological changes in the pattern of optimal paths. Theoretical predictions are in good agreement with the numerical results.

Fast Monte Carlo Simulations and Singularities in the Probability Distributions of Nonequilibrium Systems.

A numerical technique is introduced that reduces exponentially the time required for Monte Carlo simulations of nonequilibrium systems. Results for the quasistationary probability distribution in two model systems are compared with the asymptotically exact theory in the limit of extremely small noise intensity. Singularities of the nonequilibrium distributions are revealed by the simulations.

Energy-optimal steering of transitions through a fractal basin boundary

We study fluctuational transitions in a discrete dynamical system having two co-existing attractors in phase space, separated by a fractal basin boundary. It is shown that transitions occur via a unique accessible point on the boundary. The complicated structure of the paths inside the fractal boundary is determined by a hierarchy of homoclinic original saddles. By exploiting an analogy between the control problem and the concept of an optimal fluctuational path, we identify the optimal deterministic control function as being equivalent to the optimal fluctuational force obtained from a numerical analysis of the fluctuational transitions between two states.

Solution of the boundary value problem for nonlinear flows and maps

Fluctuational escape via an unstable limit cycle is investigated in stochastic flows and maps. A new topological method is suggested for analysis of the corresponding boundary value problems when the action functional has multiple local minima along the escape trajectories and the search for the global minimum is otherwise impossible. The method is applied to the analysis of the escape problem in the inverted Van der Pol oscillator and in the Henon map. An application of this technique to solution of the escape problem in chaotic maps with fractal boundaries, and in maps with chaotic saddles embedded within the basin of attraction, is discussed.

Dynamics importance sampling for the activation problem in nonequilibrium continuous systems and maps

A numerical approach based on dynamic importance sampling (DIMS) is introduced to investigate the activation problem in two-dimensional nonequilibrium systems. DIMS accelerates the simulations and allows the investigation to access noise intensities that were previously forbidden. A shift in the position of the escape path compared to a heteroclinic trajectory calculated in the limit of zero noise intensity is directly observed. A theory to account for such shifts is presented and shown to agree with the simulations for a wide range of noise intensities.

Polarization switches in vertical-cavity surface-emitting lasers

The polarization dynamics of a vertical cavity surface emitting laser is investigated as a nonlinear stochastic dynamical system. The polarization switches in the device are considered as activation processes in a two dimensional system with a saddle cycle; the optimal way of switching is determined as the solution of a boundary value problem. The theoretical results are in good agreement with the numerical simulations.

DYNAMICS IMPORTANCE SAMPLING FOR THE COLLECTION OF SWITCHING EVENTS IN VERTICAL-CAVITY SURFACE-EMITTING LASERS

A numerical approach based on dynamic importance sampling (DIMS) is applied to investigate polarization switches in vertical-cavity surface-emitting lasers. A polarization switch is described as an activation process in a two-dimensional nonequilibrium system. DIMS accelerates the simulations and allows access to noise intensities that were previously forbidden, revealing qualitative changes in the shape of the transition paths with noise intensity.

Nonequilibrium distribution at finite noise intensity

The non-equilibrium distribution in dissipative dynamical systems with unstable limit cycle is analyzed in the next-to-leading order of the small-noise approximation of the Fokker-Planck equation. The noise-induced variations of the non-equilibrium distribution are described in terms of topological changes in the pattern of optimal paths. It is predicted that singularities in the pattern of optimal paths are shifted and cross the basin boundary in the presence of finite noise. As a result the probability distribution oscillates at the basin boundary. Theoretical predictions are in good agreement with the results of numerical solution of the Fokker-Planck equation and Monte Carlo simulations.