We have shown that both deterministic and stochastic dynamics of a spatially periodic underdamped system in
the presence of, even a rather weak, adiabatic ac-drive drastically differs from that in the absence of the drive
or in the presence of other kinds of driving. This suggests promising applications.
We consider by means of the optimal fluctuation method the initial stage of the evolution of the noise-induced escape through various types of boundaries, especially concentrating on two types of the boundary - the wall and the boundary of the basin of attraction. We show in both cases that, if the damping is small enough, then the escape flux evolution possesses a remarkable property: it is it stairs-like i.e. intervals of a nearly constant flux alternate with intervals of a sharply increasing flux. This property is related to the successive increase of the number of turning points in the most probable escape path as time increases. Our results are relevant both for the absorbing and transparent boundaries. The major results of the theory are verified in computer simulations.
Two specialized algorithms for the numerical integration of the
equations of motion of a Brownian walker obeying detailed balance are introduced. The algorithms become symplectic in the appropriate limits, and reproduce the equilibrium distributions to some higher order in the integration time step. Comparisons with other existing integration schemes are carried out both for static and dynamical quantities.
We present a heuristic theory describing the recently discovered [PRL 90, 17410 (2003)] drastic facilitation of the onset of global chaos in time periodically perturbed Hamiltonian systems possessing two or more separatrices: the minimal magnitude of the perturbation which provides chaos in the whole energy range between the separatrices possesses deep spikes as a function of the perturbation
frequency. The relevant scenario is the combination of the overlap in the phase space between resonances of the same order and their overlap in energy with chaotic layers associated with the separatrices. We discuss a few applications: the increase of the dc conductivity of the 2D electron gas in a magnetic superlattice, the decrease of the activation barrier in the problem of noise-induced escape, the facilitation of the stochastic web formation in time periodically perturbed oscillators.
We have found that the width of the chaotic layer in case of a low
frequency of perturbation is significantly larger than that predicted by the conventional heuristic criteria. The underlying
reason is a randomness of the sign of the energy change if the
motion occurs in the vicinity of the separatrix, namely in the
energy band of the order of the separatrix split. Moreover, in case when the separatrix is unbounded while the time-periodic perturbation is of a dipole type, we have found the dramatic widening of the chaotic layer as the perturbation frequency decreases. This occurs because the system in the slowly rocked Hamiltonian is accelerated during long periods of time and, therefore manages to gain large energy. The acceleration periods alternate with the braking ones so that the system returns to the vicinity of the separatrix where it may be trapped for some time in one of the regions of the phase space inside the separatrix loops. We have developed the explicit adiabatic theory which nicely agrees with simulations.
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.
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.
If a cosinusoidal (harmonic) force acts on a locking dynamic system, the system may be synchronized not only to this force but to its harmonics also. This effect refers to parametric phenomena and has been studied in many systems. If a stationary random process is, for example, a signal phase, the angular vibration of the ring laser, or the phase of a periodic potential, the additive external noise in the systems is green one when its spectral density is zero at zero frequency. In this work we suppose that the green noise is the time derivative of a Ornstein-Uhlenbeck process and the locking system is the ring laser. We use a Krylov-Bogoluibov averaging method to find an effective potential which describes the system response near the locking regions located at the frequencies of the high harmonics of the force. We show that the effective Shapiro steps are well apparent but narrower then in the case of zero noise. The step size is given by the function of the external noise intensity and the harmonic force amplitude. This result is compared with that of numerical simulation accomplished by the predictor-corrector algorithm. The coincidence is excellent even if the green noise is strong enough. We also made the numeric simulation for the case of white noise. This showed that the parametric synchronization regions become ill-defined even for a very small white noise intensity.
In the problem of the activation energy for a noise-induced transition over a finite given time in an arbitrary overdamped one-dimensional potential system, we find and classify all extremal paths and provide a simple algorithm to explicitly
select which is the most probable transition path (MPTP).
The activation energy is explicitly expressed in quadratures.
For the transition beyond the top of the barrier, the MPTP does not possess turning points and the activation energy is a monotonously decreasing function of the transition time. For transitions between points lying on one and the same slope of the potential well,
which may be relevant e.g. for the problem of the tails of the prehistory probability density, the situation is more complicated: the activation energy is a non-monotonous function of time and, most important, may possess bends corresponding to jump-wise switches in the topology of the MPTP; it can be proved also that the number of turning points in the MPTP is necessarily less than two. The prefactor is calculated numerically using the scheme suggested by Lehmann, Reimann and Hanggi, PRE 55, 419 (1998). The theory is compared with simulations.
The Chirikov resonance-overlap criterion predicts the onset of global chaos if nonlinear resonances overlap in energy, which is conventionally assumed to require a non-small magnitude of perturbation. We show that, for a time-periodic perturbation, the onset of global chaos may occur at unusually small magnitudes of perturbation if the unperturbed system possesses more than one separatrix. The relevant scenario is the combination of the overlap in the phase space between resonances of the same order and their overlap in energy with chaotic layers associated with separatrices of the unperturbed system. One of the most important manifestations of this effect is a drastic increase of the energy range involved into the unbounded chaotic transport in spatially periodic system driven by a rather weak time-periodic force, which results in turn in the drastic increase either of the dc conductivity, if the system carries an electric charge, or the escape rate, if the system is subject to noise. We develop the asymptotic theory and verify it in simulations. Various generalizations are delineated, in particular for the case of a time-independent perturbation.
We consider a phase-locked loop for the case of an external signal with a stationary fluctuating phase. The problem reduces to the problem of a Brownian particle in a periodic potential driven by “green” noises. We numerically simulate the case in which the random phase is the Ornstain-Uhlenbeck process. The rapid irreversible transition from stationary random motion (a locked state) to a nonstationary one at a high near-constant rate (a running state) is shown to be possible for the case of the massive particle. We found that transition moments change suddenly for small variations of external parameters. We call this phenomenon the “catastrophe”. The numerical results are compared with those obtained by the Krylov-Bogoliubov averaging method. The first approximation of the method is found to be sufficiently accurate if the states coexist and the direct and backward transitions occur frequently enough.