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Rate equations for semiconductor lasers subjected to simultaneous near-resonant optical injection and microwave current modulation are examined by combined analytical-numerical bifurcation techniques. Simple qualitative criteria are given for a bistable response. These results compare well with experimental measurements.
The optimized expansion is used to formulate a systematic approximation scheme to the probability distribution of a stochastic system. The first-order approximation for the one-dimensional system driven by noise in an anharmonic potential is shown to agree well with the exact solution of the Fokker-Planck equation. Even for a bistable system the whole period of evolution to equilibrium is correctly described at various noise intensities.
We investigate dynamics and bifurcations of a single-mode semiconductor laser subject to phase-conjugate feedback near the locking region. The system is described by rate equations which are a three-dimensional system with a delay. With tools that go much beyond mere simulation, we find and follow steady states regardless of their stability and compute unstable manifolds of saddle points. Furthermore, we identify heteroclinic bifurcations, which turn out to be responsible for bistability and excitability at the locking boundaries.
It has been well established that spatially extended, bistable systems that are driven by an oscillating field exhibit a nonequilibrium dynamic phase transition (DPT). The DPT occurs when the field frequency is of the order of the inverse of an intrinsic lifetime associated with the transitions between the two stable states in a static field of the same magnitude as the amplitude of the oscillating field. The DPT is continuous and belongs to the same universality class as the equilibrium phase transition of the Ising model in zero field [G. Korniss et al., Phys. Rev. E 63, 016120 (2001); H. Fujisaka et al., Phys. Rev. E 63, 036109 (2001)]. However, it has previously been claimed that the DPT becomes discontinuous at temperatures below a tricritical point [M. Acharyya, Phys. Rev. E 59, 218 (1999)]. This claim was based on observations in dynamic Monte Carlo simulations of a multipeaked probability density for the dynamic order parameter and negative values of the fourth-order cumulant ratio. Both phenomena can be characteristic of discontinuous phase transitions. Here we use classical nucleation theory for the decay of metastable phases, together with data from large-scale dynamic Monte Carlo simulations of a two-dimensional kinetic Ising ferromagnet, to show that these observations in this case are merely finite-size effects. For sufficiently small systems and low temperatures, the continuous DPT is replaced, not by a discontinuous phase transition, but by a crossover to stochastic resonance. In the infinite-system limit, the stochastic-resonance regime vanishes, and the continuous DPT should persist for all nonzero temperatures.
We examine the large-network, low-loading behavior of an attractor neural network, the so-called bistable gradient network (BGN), and compare it with that of the Hopfield network (HN). We use analytical and numerical methods to characterize the attractor states of the network and their basins of attraction. The energy landscape of BGN is more complex than that of the HN and depends on the strength of the coupling among units. At weak coupling, the BGN acts as a highly selective associative memory; the input must be close to the one of the stored patterns in order to be recognized. A category of spurious attractors occurs which is not present in the HN. Stronger coupling results in a transition to a more Hopfield-like regime with large basins of attraction. The basins of attraction for spurious attractors are noticeably suppressed compared to the Hopfield case, even though the Hebbian synaptic structure is the same and there is no stochastic noise.
The phenomenon of stochastic resonance in a bistable nonlinear system is investigated when both the multiplicative noise and the coupling between additive and multiplicative noise are colored with different values of noise correlation time tau(1) and tau(2). Combining the functional analysis and unified colored noise approximation, the two different kinds of colored noise in the nonlinear system can be simplified. The signal-to-noise ratio is calculated when a weakly periodic signal is added to the system. It is found that there appears a transition between one peak and two peaks in the curve of the signal-to-noise ratio when either the noise correlation time tau(1) and tau(2) or the coupling strength lambda between additive and multiplicative noise is increased. The transition between one and two peaks depending on tau(1) and lambda is more complex than that depending on tau(2).
We consider an overdamped bistable oscillator subject to the action of a biharmonic force with very different frequencies, and study the response of the system when the parameters of the high-frequency force are varied. A resonantlike behavior is obtained when the amplitude or the frequency of this force is modified in an experiment performed by means of an analog circuit. This behavior, confirmed by numerical simulations, is explained on the basis of a theoretical approach.
A detailed theoretical analysis of the dynamics of a sinusoidally driven noisy asymmetric bistable system is presented. The results are valid for any two-state system, however, the specific case of the Duffing potential is considered in detail. The dynamics are considered in the weak noise limit, i.e., when the response of the system to the external periodic field is strongly nonlinear. The system asymmetry is created by a nonzero dc component of the external force, and manifests itself as an asymmetry between the mean switching times between the potential wells. We obtain explicit analytic expressions for the whole hierarchy of switching time distributions (including the residence time and return time distributions). We also obtain expressions for the average residence times and describe how they depend on asymmetry, together with an explicit expression for the difference between the residence times in the weak noise limit; the results are presented in the context of using the switching dynamics to detect weak dc target signals.
The method of moments is applied to an underdamped bistable oscillator driven by Gaussian white noise and a weak periodic force for the observations of stochastic resonance and the resulting resonant structures are compared with those from Langevin simulation. The physical mechanisms of the stochastic resonance are explained based on the evolution of the intrawell frequency peak and the above-barrier frequency peak via the noise intensity and the fluctuation-dissipation theorem, and the three possible sources of stochastic resonance in the system are confirmed. Additionally, with the noise intensity fixed, the stochastic resonant structures are also observed by adjusting the nonlinear parameter.
We study the effects of correlations between additive and multiplicative noise on relaxation time in a bistable system driven by cross-correlated noise. Using the projection-operator method, we derived an analytic expression for the relaxation time T(c) of the system, which is the function of additive (alpha) and multiplicative (D) noise intensities, correlation intensity lambda of noise, and correlation time tau of noise. After introducing a noise intensity ratio and a dimensionless parameter R=D/alpha, and then performing numerical computations, we find the following: (i) For the case of R<1, the relaxation time T(c) increases as R increases. (ii) For the cases of R>/=1, there is a one-peak structure on the T(c)-R plot and the effects of cross-correlated noise on the relaxation time are very notable. (iii) For the case of R<1, T(c) almost does not change with both lambda and tau, and for the cases of R>/=1, T(c) decreases as lambda increases, however T(c) increases as tau increases. lambda and tau play opposite roles in T(c), i.e., lambda enhances the fluctuation decay of dynamical variable and tau slows down the fluctuation decay of dynamical variable.
Conventional stochastic resonance can be viewed as an amplitude effect, in which a small (subthreshold) input signal receives assistance from noise to trigger a stronger response from a nonlinear system. We demonstrate another mechanism of improvement by the noise, which is more of a temporal effect. An intrinsically slow system has difficulty to respond to a fast (suprathreshold) input, and the noise plays a constructive role by spurring the system for a more efficient response. The possibility of this form of stochastic resonance is established and studied here in a double-well bistable dynamic system, driven by a suprathreshold random binary signal, with the noise accelerating the switching between wells.
We study an extended system that without noise shows a monostable dynamics, but when submitted to an adequate multiplicative noise, an effective bistable dynamics arises. The stochastic resonance between the attractors of the noise-sustained dynamics is investigated theoretically in terms of a two-state approximation. The knowledge of the exact nonequilibrium potential allows us to obtain the output signal-to-noise ratio. Its maximum is predicted in the symmetric case for which both attractors have the same nonequilibrium potential value.
We study the overdamped motion of a particle in a bistable potential subject to the action of a bichromatic force and additive noise, within the context of the vibrational resonance phenomenon. Under appropriate conditions, we obtain analytical expressions for the relevant observables which quantifies this phenomenon. The theoretical results are compared with those obtained by the numerical solution of the stochastic differential equation which describes the dynamics of the system. The limits of validity of the theoretical approach are also discussed.
We study the correlation properties of noise-driven bistable systems with multiple time-delay feedbacks. For small noisy perturbation and feedback magnitude, we derive the autocorrelation function and the power spectrum based on the two-state model with transition rates depending on the earlier states of the system. A comparison between the single and double time delays reveals that the auto correlation functions exhibit exponential decay with small undulation for the double time delays, in contrast with the remarkable oscillatory behavior at small time lags for the single time delay.
We analyze theoretically and experimentally the residence time distribution of bistable systems in the presence of noise and time-delayed feedback. We explain various nonexponential features of the residence time distribution using a two-state model and obtain a quantitative agreement with an experiment based on a Schmitt trigger. The limitations of the two-state model are also analyzed theoretically and experimentally using a semiconductor laser with optoelectronic feedback.
The stabilization of the metastable 180 degrees twist state in a bistable chiral splay nematic liquid crystal device is studied. The stability of the metastable 180 degrees twist state is explained using the azimuthal anchoring strength, which is a function of the twist elastic constant and the cell gap. A stable 180 degrees twist state as well as the permanent retention time can be obtained by increasing the anchoring energy above a certain threshold.
We present a method to construct a modulation frequency response curve for bistable systems in the presence of noise. To this end, a small sinusoidal modulation is applied to the system such that it switches between its two stable states. The response curve we construct yields information on the nature of the physical mechanism underlying the switching process and is furthermore comparable to the standard response curves of linear systems. Our semianalytical approach, which only needs approximate Kramer rates, is in good agreement with numerical simulations. The concept is applicable to a wide range of systems.
We observe breathing front dynamics which select three distinct types of bistable patterns in the 2:1 resonance regime of the periodically forced oscillatory Belousov-Zhabotinsky reaction. We measure the curvature-driven shrinking of a circular domain R approximately t(1/2) at forcing frequencies below a specific value, and show that the fast time scale front oscillations (breathing) drive this slow time scale shrinking. Above a specific frequency, we observe fronts of higher curvature grow instead of shrink and labyrinth patterns form. Just below the transition frequency is a relatively narrow range of frequencies where the curvature-driven coarsening is balanced by a competing front interaction, which leads to a pattern of localized structures. The length scale of the localized structure and labyrinth patterns is set by the front interactions.