Monostable Potential Research Articles

Nonlinear Resonance in a Position-Dependent Mass-Duffing Oscillator System with Monostable Potentials Driven by an Amplitude-Modulated Signal

This study examines the phenomenon of vibrational resonance (VR) in a classical positiondependent mass (PDM) system characterized by three types of single-well potentials. These potentials are influenced by an amplitude-modulated (AM) signal with $\Omega\gg\omega$. Our analysis is limited to the following parametric choices: <br> (i) $\omega_0^2$, $\beta$, $m_0$, $\lambda>0$ (type-1 single-well), <br> (ii) $\omega_0^2>0$, $\beta <0$, $2< m_0 <3$, $1< \lambda <2$ (type-2 single-well), <br> (iii) $\omega_0^2>0$, $\beta <0$, $0< m_0 <2$, $0<\lambda<1$ (type-3 single-well). <br> The system presents an intriguing scenario in which the PDM function significantly contributes to the occurrence of VR. In addition to the analytical derivation of the equation for slow motions of the system based on the high-frequency signal’s parameters using the method of direct separation of motion, numerical evidence is presented for VR and its basic dynamical behaviors are investigated. Based on the findings presented in this paper, the weak low-frequency signal within the single-well PDM system can be either attenuated or amplified by manipulating PDM parameters, such as mass amplitude ($m_0$) and mass spatial nonlinearity $\lambda$. The outcomes of the analytical investigations are validated and further supported through numerical simulations.

Vibrational resonance in a bistable van der Pol–Mathieu–Duffing oscillator

In this paper, we study the nonlinear response of a bistable van der Pol–Mathieu–Duffing (VMD) oscillator under the influence of two periodic excitations of widely different frequencies. We have shown that by systematically modulating the strength of the high-frequency drive as well as the strength of the parametric oscillation, a symmetrically oscillating bistable potential can be converted to a symmetrically oscillating monostable potential. In addition to this effect, the strength of the fast drive modifies the damping as well, allowing us to define a threshold value of this strength at which a supercritical Hopf bifurcation occurs. All analytical results have shown to be numerically consistent.

Stochastic resonance in time-delayed exponential monostable system driven by weak periodic signals

Based on the exponential monostable potential, we study an exponential monostable system with time-delayed feedback driven by weak periodic signals and additive Gaussian white noises. The small delay approximation is used to deduce the steady-state probability distribution and the effective potential function is derived. The system parameters l and b, time delay τ, feedback strength β can change the shapes of the potential function. The mean first-passage time (MFPT) is calculated, which plays an extremely important role in the research of particles escape. And the signal-to-noise ratio (SNR) of the system can be obtained by using the adiabatic approximation theory. The phenomenon of stochastic resonance is investigated under different system parameters and time-delayed feedback. The amplitude of SNR can be changed by adjusting the system parameters. When the feedback strength β is positive (or negative), the time delay τ can promote (or suppress) the stochastic resonance phenomenon. The SNR versus the noise intensity D presents the stochastic resonance phenomenon. In addition, the SNR increases non-monotonically with the increasing feedback strength β and the parameter b. Also, the analysis and numerical simulation results of SNR are in good agreement with the formula simulation.

Random response evaluation of mono-stable and bi-stable Duffing systems with Dahl friction

The random responses of Duffing systems with dynamic friction are investigated, and a uniform procedure is established to derive the explicit expressions of stationary probability densities for mono-stable potential and bi-stable potential cases. The Dahl friction model is adopted to describe the dynamic friction behavior, and the friction force is first expressed as a time-domain integration of the velocity function by directly integrating the auxiliary differential equation with non-smooth term. By introducing the associated generalized harmonic transformations for mono-stable potential and bi-stable potential cases, the friction force is further simplified and only approximately depends on the present velocity, amplitude and phase. The friction force is then mandatorily separated as conservative and dissipative components through the quasi-linearization technique. At last, the mono-stable and bi-stable Duffing systems with dynamic friction are equivalent to approximate nonlinear systems with modified damping and modified stiffness. The stationary probability densities of the original friction systems are evaluated by those of the approximate nonlinear systems, which can be analytically expressed. The effectiveness and accuracy of the uniform procedure are verified by comparison with the Monte Carlo simulation for mono-stable potential (including hardening stiffness and softening stiffness) and bi-stable potential cases.

Stochastic resonance of charge carrier diffusion in a nonhomogeneous medium in the presence of an harmonic external potential

The dynamics of charge carriers (electrons) hopping through a nonhomogeneous medium in semiconductor layer is investigated by changing a thermal noise of strength D and an external harmonic potential V(x). The nonhomogeneous medium exhibits denser trap distribution around the center, which biases the electrons to therein concentrate. Applying also a monostable potential at the center further enhances the accumulation of electrons. However, by applying a nonhomogeneous hot temperature in the vicinity of the potential minimum forced the electrons to diffuse away from the center and redistribute around two points. Thermally activated rate of hopping and diffusion of electrons in a nonhomogeneous medium, as a function of model parameters, is also considered in the high barrier limit. Using two states approximation, I have also studied the stochastic resonance (SR) of the electrons dynamics in the presence of a time-varying signal. I found a strong spectral amplification η and lower temperature occurrence of its peak as compared to previous works [M. Asfaw, B. Aragie and M. Bekele, Eur. Phys. J. B 79, 371 (2011); B. Aragie, Y. B. Tateka and M. Bekele, Eur. Phys. J. B 87, 101 (2014)].

Influence of potential function asymmetries on the performance of nonlinear energy harvesters under white noise

To improve the broadband transduction capabilities of vibratory energy harvesters (VEHs) under random and non-stationary excitations, many researchers have resorted to purposefully introducing nonlinearities into the restoring force of the harvester. While performing this task, it is often very challenging to maintain a perfectly symmetric restoring force which yields a VEH with an asymmetric potential energy function. This paper investigates the influence of potential function asymmetries on the performance of nonlinear VEHs under white noise inputs. To that end, a quadratic nonlinearity is introduced into the restoring force and its influence on the mean output power of the harvester for mono- and bi-stable quartic potentials is investigated. It is shown that, for VEHs with a mono-stable quartic potential function, the mean output power increases with the degree of potential function asymmetry. On the other hand, for energy harvesters with a bi-stable quartic potential function, asymmetries in the restoring force appear to worsen performance especially for low to moderate noise intensities. When the noise intensity becomes sufficiently large, the influence of the potential function׳s asymmetry on the mean power diminishes. Results also reveal that a VEH with a symmetric bi-stable quartic potential function produces higher mean power levels than the one with the most asymmetric mono-stable potential. As such, it is concluded that a VEH with a symmetric bi-stable potential is most desirable to improve performance under white noise.

A theoretical investigation of the stochastic filtering effect

A first-order nonlinear radio filter with the monostable potential of the signal-noise interaction is investigated within the framework of the spectral approach. It is shown that, in the case when signal frequencies are not very high, a stationary solution to the equation describing the behavior of the filter can be sought for in the form of a Fourier series. The solution obtained in the study is used to explain the basic features of the stochastic signal filtering effect. As a result of this effect, the output signal-to-noise ratio exceeds its input value. Theoretical predictions are compared to experimental data.

Single and multiple vibrational resonance in a quintic oscillator with monostable potentials

We analyze the occurrence of vibrational resonance in a damped quintic oscillator with three cases of single well of the potential V(x)=1/2omega(0)(2)x(2)+1/4betax(4)+1/6gammax(6) driven by both low-frequency force f cos omegat and high-frequency force g cos Omegat with Omega >> omega. We restrict our analysis to the parametric choices (i) omega(0)(2), beta, gamma > 0 (single well), (ii) omega(0)(2), gamma > 0, beta < 0, beta(2) < 4omega(0)(2)gamma (single well), and (iii) omega(0)(2) > 0, beta arbitrary, gamma < 0 (double-hump single well). From the approximate theoretical expression of response amplitude Q at the low-frequency omega we determine the values of omega and g (denoted as omega(VR) and g(VR)) at which vibrational resonance occurs. We show that for fixed values of the parameters of the system when omega is varied either resonance does not occur or it occurs only once. When the amplitude g is varied for the case of the potential with the parametric choice (i) at most one resonance occur while for the other two choices (ii) and (iii) multiple resonance occur. Further, g(VR) is found to be independent of the damping strength d while omega(VR) depends on d. The theoretical predictions are found to be in good agreement with the numerical result. We illustrate that the vibrational resonance can be characterized in terms of width of the orbit also.

LÉVY FLIGHT SUPERDIFFUSION: AN INTRODUCTION

After a short excursion from the discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. Further development on this idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As a particular case we obtain the fractional Fokker–Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally, we discuss the results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained by different approaches.

Dynamical bimodality in equilibrium monostable systems

General features of the stochastic dynamics of classical systems approaching a thermodynamic equilibrium Gibbs state are studied via the numerical analysis of time-dependent solutions of the Fokker-Planck equation for an overdamped particle in various monostable potentials. A large class of initial states can dynamically bifurcate during its time evolution into bimodal transient states, which in turn wear off when approaching the long-time regime. Suitable quantifiers characterizing this transient dynamical bimodality, such as its lifetime, the positions of maxima, and the time-dependent well depth of the probability distribution, are analyzed. Some potential applications are pointed out that make use of this interesting principle which is based on an appropriately chosen initial preparation procedure.

Relaxation rate and stochastic resonance of a single-mode nonlinear optical syst em

The variational method is applied to a single-mode stochastic nonlinear optical system and the dependence of its relaxation rate and spectral amplification factor on the noise intensity based on the linear response theory is investigated.The investigation shows that when the biased input is vanishing,the relaxation rate has different dependence in monostable and multistable cases.The investigation also shows that the stochastic resonance occurs in one-dimensional symmetric monostable potential,and the possible condition for it is pointed out.

Effect of asymmetry on stochastic resonance and stochastic resonance induced by multiplicative noise and by mean-field coupling.

In the paper, we investigate the effect of asymmetry of the potential on stochastic resonance (SR) for a model with an asymmetric bistable potential and driven by additive noise, the signal-to-noise ratio (SNR) for a model with a monostable potential and driven by additive and multiplicative noises, and the SNR for a mean-field coupled model with infinite globally coupling oscillators driven by additive noises. It is shown that for the first model,the asymmetry of the potential can weaken the phenomenon of SR; for the second and third models, a SR induced by multiplicative noise and a different one caused by mean-field coupling are found.

Transient bimodality of nonequilibrium states in monostable systems with noise

We discuss a new type of monostable potential profiles for which the phenomenon of transient bimodality occurs in dynamic systems. It is shown that transient bimodality occurs if the potential profile describing the dynamic system has a linear part. A new method for calculating the lifetime of the transient bimodality is proposed. We show that the lifetime of transient bimodality for the considered potential profiles increases in proportion to the fluctuation intensity.

Probability distributions and escape rates for systems driven by quasimonochromatic noise.

The motion of an overdamped particle in a bistable potential U(x), driven by quasimonochromatic noise (high-frequency, narrow-band noise), has been investigated by means of electronic analog simulation. The escape rate from one potential well to another was found to be exponentially small compared to the reciprocal mean first-passage time to the top of the potential barrier. The logarithm of the quasi- stationary probability distribution was observed to fall extremely sharply at a particular value of x, quite close to the equilibrium position. Theory describing the nonanalytic dependence of this logarithm on the bandwidth of the noise is presented and shown to be in good agreement with experiment. Data are also presented for a symmetric monostable potential. In a certain parameter range, the quasistationary distribution is demonstrated to be independent of the form of such a potential.