Noise Influence on a Dispersive Optical Bistable System with Trilinear Interaction

The influence of fluctuations in the driving fields on an optical bistable multimode system inside a ring resonator with a second order nonlinear material is considered. Steady state solutions of the resulting Fokker-Planck equations are found for different approximations in which one or two of the modes can be eliminated adiabatically. The results of the stochastics are compared with the deterministic behaviour.     

Critical dynamics and fluctuations for a mean-field model of cooperative behavior

The main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior. The model under consideration is a dynamic model of a collection of anharmonic oscillators in a two-well potential together with an attractive mean-field interaction. The system is studied in the limit as the number of oscillators goes to infinity. The limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established. The main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity. It is established that these fluctuations are non-Gaussian and occur at a time scale slower than the noncritical fluctuations. The method used is based on the perturbation theory for Markov processes developed by Papanicolaou, Stroock, and Varadhan adapted to the context of probability-measure-valued processes.     

Nucleation theory for a model bistable chemical reaction

The theory of homogeneous nucleation is developed for a model nonlinear bistable chemical reaction driven far from equilibrium (trimolecular Schlögl model). The theory is restricted to the vicinity of the stable/unstable transition, where the nucleation barrier is small but nonvanishing. The nucleation rates are derived for two types of fluctuations: first, fluctuations due to a homogeneous external white noise source, and second, internal chemical fluctuations, described by a reastion-diffusion multivariate master equation. In the white noise case, a Landau-Ginzburg potential can be defined, and the standard nucleation formalism can be applied; this is not true for the internal case and a new result is used. The inhomogeneous chemical fluctuations, due to the coupling between the nonlinear reaction and diffusion, are shown to have an influence on the nucleation rate. Quantitative conditions are also given to evaluate the possibility of homogeneous nucleation in nonlinear chemical systems.     

Bistable Behaviour of N Two-Level Atoms Interacting with the Field inside a High-Q Cavity Containing a Nonlinear Kerr-Like Medium

Taking the intensity-dependent coupling between atoms and cavity mode into account, we investigate a system consisting of N homogeneously broadened two-level atoms interacting with the field inside a single-mode Fabry- Perot cavity containing a nonlinear Kerr-like medium. We derive the steady-state bistable behaviour of the system, and further analyse in details the influence of several critical parameters on the bistable behaviour.     

Fluctuation-induced transitions of a bistable driven polariton system in the presence of damping

Fluctuation-induced transitions from the lower energy state of a bistable nonlinear driven microcavity oscillator are analyzed beyond the Fluctuation–Dissipation theorem. The sources of noise are: both fluctuations of the external pumping and inevitable interaction with the exciton reservoir in the cavity. We show that finite polariton lifetime strongly modifies the phase portrait and influences the temporal parameters of the transition within the bistable regime. To investigate the transient dynamics of the driven polariton system, three different approaches are realized: numerical experiment (1), i.e. direct solution of the quasiclassical dynamic equation for polariton amplitude driven by an external pump, (2) solution of two dimensional Fokker–Planck-equation and (3) effective one dimensional Fokker–Planck-equation, obtained within a low-damping approximation. We show that the escape times obtained within the numerical experiment and the 2D-Fokker–Planck-equation coincide. In contrast, the one dimensional Fokker–Planck-approximation fails for large damping parameters due to strong deviation of the phase trajectories from those obtained within the low-damping approximation. The range of the ratio between damping and detuning for which the 1d approximation is valid, is shown to be smaller than 0.04. Finally, we determine the impact of the fluctuations on experiments illustrated by the narrowing of the hysteresis cycle.     

Effects of saturation on optical bistability with coupled surface plasmons

We present exact numerical results for a symmetric layered medium (prism/Ag film/nonlinear dielectric/Ag film/prism) where the middle dielectric slab is assumed to have a saturation-type nonlinearity. We show bistable behaviour in the power dependence of the reflectivity ofp-polarized light under the condition when coupled surface modes are excited in the structure. Moreover, we study the effect of saturation on the bistable behaviour to show that multivalued character is inhibited by saturation effects. The field distributions corresponding to the minimum reflectivity states of the nonlinear structure are also presented.     

Diffusion in a bistable potential: A comparative study of different methods of solution

The problem of diffusion in a bistable potential is studied by considering the associated nonlinear Langevin equation and its equivalent Fokker-Planck equation. Two numerically exact methods of solution, namely, the Monte Carlo solution of the nonlinear Langevin equation and the solution of the Fokker-Planck equation via the finite difference technique, are considered. The latter method has the advantage that it directly gives the evolution of the probability distribution function. Approximate analyses of the fluctuations using the system size expansion, the Gaussian decoupling procedure, and the scaling approach are also carried out. These investigations are performed on a representative problem for two specific cases: (1) evolution from intrinsically unstable states and (2) evolution from extensive regime. The fluctuations obtained using these approximate methods are compared with those obtained via the numerically exact methods. The study brings out the advantages and limitations of each of the methods considered.     

Observation of stochastic resonance near a subcritical bifurcation

A hysteretic Subcritical period-doubling bifurcation is observed in the nonlinear strain dynamics of a magnetostrictive oscillator. The dynamic strain response of the magnetostrictive oscillator was observed with a high-resolution fiber optic interferometer. The effects of low-frequency modulation and band-limited stochastic fluctuations on such a bifurcation are investigated. Power spectral density measurements show that for an optimal value of externally injected noise the signal-to-noise ratio of a low-frequency modulation signal is enhanced by greater than 14 dB, thus indicating the first experimental observation of stochastic resonance near a bistable period-doubling bifurcation.     

Finding thermoacoustic limit cycles for a ducted Burke-Schumann flame

This paper examines nonlinear thermoacoustic oscillations of a ducted Burke-Schumann diffusion flame. The nonlinear dynamics of the thermoacoustic system are studied using two distinct approaches. In the first approach, a continuation analysis is performed to find limit cycle amplitudes over a range of operating conditions. The strength of this approach is that one can characterize the coupled system’s nonlinear behaviour over a large parameter space with relative ease. It is not able to give physical insight into that behaviour, however. The second approach uses a Flame Describing Function (FDF) to characterize the flame’s response to harmonic velocity fluctuations over a range of forcing frequencies and forcing amplitudes, from which limit cycle amplitudes can be found. A strength of the FDF approach is that it reveals the physical mechanisms responsible for the behaviour observed. However, the calculation of the FDF is time consuming, and it must be recalculated if the flame’s operating conditions change. With the strengths and shortcomings of the two approaches in mind, this paper advocates combining the two to provide the dynamics over a large parameter space and, furthermore, physical insight into that behaviour at judiciously-chosen points in the parameter space. Further physical insight concerning the flame’s near-linear response at all forcing amplitudes is given by studying the forced flame in the time domain. It is shown that, for this flame model, the limit cycles arise because of the flame’s nonlinear behaviour when it is close to the inlet.     

Fluctuations in optical bistable systems

The bistable behaviour of an ensemble of two level atoms driven by a coherent external light field is discussed in the adiabatic limit when the relaxation of the atoms is fast compared to the relaxation of the field. The influence of the different noise sources is discussed separately in a one-dimensional approximation by neglecting the phase fluctuations. In the case where the fluctuations of the driving field play the dominant role, an exact solution of the two-dimensional stationary Fokker-Planck equation is derived describing amplitude as well as phase fluctuations.     

Output response identification in a multistable system for piezoelectric energy harvesting

In this paper we examine in detail the multiple responses of a novel vibrational energy harvester composed of a vertical bistable beam whose complex non-linear behavior is tuned via magnetic interaction. The beam was excited horizontally by a harmonic inertial force while mechanical vibrational energy is converted to electrical power through a piezoelectric element. The bistable laminate beam coupled to the piezoelectric transducer showed a variety of complex responses in terms of the beam displacement and harvested power output. The range of vibration patterns in this non-linear system included single-well oscillations and snap-through vibrations of periodic and chaotic character. Harvested power was found to be strongly dependent on the vibration pattern with nonlinearities providing a broadband response for energy harvesting. Wavelet analysis of measured voltage, displacement and velocity time histories indicated the presence of a variety of nonlinear periodic and also chaotic phenomena. To measure the complexity of response time series we applied phase portraits and determine stroboscopic points and multiscale entropy. It is demonstrated that by changing parameters such as the magnetic interaction, the characteristics of the bistable laminate harvester, such as the natural frequency, bandwidth, vibration response and peak power can be readily tailored for harvesting applications.     

Small fluctuations in a steady non-equilibrium state

An expression for the probability of fluctuations in a given probability distribution deduced in an earlier paper is used to derive a relation between autocorrelations of velocities and the dynamics of small fluctuations in a stable steady state. The steady state may lie in the region of nonlinear thermodynamics. The considerations are restricted to systems with a distinctly separated behaviour on short and long time scale and the corresponding use of Mori distributions.     

Growth and Nonlinear Saturation of Fluctuations in a Nonlinear Oscillator at the Threshold of a Bifurcation of Spontaneous Symmetry Breaking

We consider the phenomenon of prebifurcation noise amplification in a nonlinear oscillator at the threshold of a bifurcation of spontaneous symmetry breaking. The studies are based on the model of a nonlinear oscillator in which the potential relief transforms from monostable to symmetric bistable and the noise acting on the system is assumed Gaussian and short-correlated. The fluctuation variance as a function of the regime of the system and the rate at which the bifurcation threshold is reached are examined. Our analytical estimates are in good agreement with the results of numerical simulation for both the linear growth and the nonlinear saturation of fluctuations. It is noted that in the case of fast bifurcation transitions, a loop of noise-dependent hysteresis and breaking of probability symmetry of stable final states are observed in the nonlinear oscillator. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 48, No. 5, pp. 425–435, May 2005.     

Achieving high bit rate logical stochastic resonance in a bistable system by adjusting parameters

The phenomenon of logical stochastic resonance(LSR) in a nonlinear bistable system is demonstrated by numerical simulations and experiments. However, the bit rates of the logical signals are relatively low and not suitable for practical applications. First, we examine the responses of the bistable system with fixed parameters to different bit rate logic input signals, showing that an arbitrary high bit rate LSR in a bistable system cannot be achieved. Then, a normalized transform of the LSR bistable system is introduced through a kind of variable substitution. Based on the transform, it is found that LSR for arbitrary high bit rate logic signals in a bistable system can be achieved by adjusting the parameters of the system,setting bias value and amplifying the amplitudes of logic input signals and noise properly. Finally, the desired OR and AND logic outputs to high bit rate logic inputs in a bistable system are obtained by numerical simulations. The study might provide higher feasibility of LSR in practical engineering applications.     

左手材料与非线性光学材料界面处的内光学双稳态

从理论及数值上研究了左手材料与传统非线性材料界面处的电磁波性质。发现在界面处出现一种新的现象——光学双稳态。入射光强、入射角以及左手材料与非线性材料的磁导率比值强烈的影响光学双稳态的行为。比较了不同的电磁模式对双稳态行为的影响。结果表明在相同的参数下，TE模式比TM模式更容易获得光学双稳态。     

A nonlinear spring mechanism incorporating a bistable composite plate for vibration isolation

The High Static Low Dynamic Stiffness (HSLDS) concept is a design strategy for a nonlinear anti-vibration mount that seeks to increase isolation by lowering the natural frequency of the mount whilst maintaining the same static load bearing capacity. It has previously been proposed that an HSLDS mount could be implemented by connecting linear springs in parallel with the transverse flexure of a composite bistable plate — a plate that has two stable shapes between which it may snap. Using a bistable plate in this way will lead to lightweight and efficient designs of HSLDS mounts. This paper experimentally demonstrates the feasibility of this idea. Firstly, the quasi-static force–displacement curve of a mounted bistable plate is determined experimentally. Then the dynamic response of a nonlinear mass–spring system incorporating this plate is measured. Excellent agreement is obtained when compared to theoretical predictions based on the measured force–displacement curve, and the system shows a greater isolation region and a lower peak response to base excitation than the equivalent linear system.     

Nonlinear hydrodynamic and thermoacoustic oscillations of a bluff-body stabilised turbulent premixed flame

Turbulent premixed flames often experience thermoacoustic instabilities when the combustion heat release rate is in phase with acoustic pressure fluctuations. Linear methods often assume a priori that oscillations are periodic and occur at a dominant frequency with a fixed amplitude. Such assumptions are not made when using nonlinear analysis. When an oscillation is fully saturated, nonlinear analysis can serve as a useful avenue to reveal flame behaviour far more elaborate than period-one limit cycles, including quasi-periodicity and chaos in hydrodynamically or thermoacoustically self-excited system. In this paper, the behaviour of a bluff-body stabilised turbulent premixed propane/air flame in a model jet-engine afterburner configuration is investigated using computational fluid dynamics. For the frequencies of interest in this investigation, an unsteady Reynolds-averaged Navier–Stokes approach is found to be appropriate. Combustion is represented using a modified laminar flamelet approach with an algebraic closure for the flame surface density. The results are validated by comparison with existing experimental data and with large eddy simulation, and the observed self-excited oscillations in pressure and heat release are studied using methods derived from dynamical systems theory. A systematic analysis is carried out by increasing the equivalence ratio of the reactant stream supplied to the premixed flame. A strong variation in the global flame structure is observed. The flame exhibits a self-excited hydrodynamic oscillation at low equivalence ratios, becomes steady as the equivalence ratio is increased to intermediate values, and again exhibits a self-excited thermoacoustic oscillation at higher equivalence ratios. Rich nonlinear behaviour is observed and the investigation demonstrates that turbulent premixed flames can exhibit complex dynamical behaviour including quasiperiodicity, limit cycles and period-two limit cycles due to the interactions of various physical mechanisms. This has implications in selecting the operating conditions for such flames and for devising proper control strategies for the avoidance of thermoacoustic instability.     

多位数序列全加器的光电混合实现

本文报道用光不双稳开关构成多位数序列全加器的方法，并用非线性干涉滤光片双稳开关进行了原理性实验演示，完成了四位二进制数的全加运算。     

Nonlinear phenomena of left-handed nonlinear split-ring resonators

We theoretically analyze a two-dimensional periodic structure consisting of period nonlinear split-ring resonators (SRRs) with varactor diode. The diode is loaded into the slit of the SRRs. Then, we demonstrate nonlinear phenomena of left-handed nonlinear SRRs. This paper introduces nonlinear SRRs based on left-handed (LH) media and simulates self-tuning mechanisms, bistable effects in the microwave frequency range. In addition, with the increase of input power, the nonlinear response of the SRRs becomes multi-valued, paving a way for creating bistable tunable metamaterials.