Monostable Systems Research Articles

Fault diagnosis of rolling bearing based on the three-dimensional coupled periodic potential-induced stochastic resonance

Abstract Conventional bistable and monostable stochastic resonance (SR) methods exhibit certain limitations in their capacity to enhance and extract incipient characteristics. Firstly, the inherent potential function structure, characterized by a singular stable-state paradigm, proves inadequate in accommodating the heterogeneous and multifaceted condition monitoring signals. Secondly, the interconnected dynamic characteristics of the mechanical signals remain unaccounted for. Furthermore, conventional SR methods persist in utilizing a fixed constant as the critical system parameter, thereby neglecting the synergistic interaction among monitoring signals, potential function structures, and scale factors. Owing to the rich dynamic characteristics of the three-dimensional multi-stable coupled periodic potential SR system, it demonstrates superior noise utilization compared to monostable and bistable systems. In view of this, the present formulates a three-dimensional spatial model employing a coupled periodic potential model with nonlinear coupling. Subsequently, a pioneering method for diagnosing rolling bearing faults is introduced, utilizing the framework of three-dimensional multi-stable coupled periodic potential-induced SR. Simulation and experimental results illustrate that this approach effectively enhances and extracts the subtle fault characteristics of rolling bearings, ensuring a clear distinction between the spectral peak at the bearing fault characteristic frequency and the spectral peak originating from the interference noise.

Time-sharing orbit jump and energy harvesting in nonlinear piezoelectric energy harvesters using a synchronous switch circuit

Nonlinearity has enabled energy harvesting to advance toward higher power output and broader bandwidth in monostable, bistable, and multistable systems. However, operating in the preferable high-energy orbit (HEO) rather than the low-energy orbit (LEO) for making such advancement has restricted their applications. Based on a monostable nonlinear system, this paper proposes a self-contained solution for time-sharing orbit jump and energy harvesting. The joint dynamics of an electromechanical assembly consisting of a nonlinear energy harvester and a switched-mode piezoelectric interface circuit for high-capability energy harvesting is studied. The proposed solution is carried out by utilizing a cutting-edge switched-mode bidirectional energy conversion circuit (BECC), which enables time-sharing dual functions of energy harvesting and vibration exciting. A theoretical model is established based on impedance analysis and multiple time scales method to analyze the stability, frequency response, and phase evolution of the autonomous and nonautonomous nonlinear energy harvesting systems. In particular, the detailed dynamics for the orbit jumps with the vibration exciting mode of BECC are studied. Experiments are performed to validate the full-hysteresis-range orbit jumps with the monostable nonlinear energy harvester. The harvested power after orbit jumps yields a nine-fold increase, compensating for the energy consumption under vibration exciting mode quickly. The proposed solution also refrains the system from extra mechanical or electrical energy sources for orbit jumps, which leads to the first self-contained solution for simultaneous energy harvesting and orbit jump in nonlinear piezoelectric energy harvesting. This work enhances the practical utility of nonlinear energy harvesting technologies toward engineering applications.

Transient response of energy harvesting systems with multi-well potential under Poisson white noise excitations

This paper studies the transient response of vibration-based energy harvesting (VEH) systems characterized by multi-well potential functions exposed to Poisson white noise excitations. With the recently developed radial basis function neural networks (RBFNN) approach, the generalized Fokker–Planck–Kolmogorov (GFPK) equation in three-dimensional state space for the transient probability density function (PDF) is solved. The trial solution is composed of a series of radial basis Gaussian functions, each multiplied by a time-varying weight coefficient. The method of Lagrange multiplier is introduced for the constraint of weight coefficients to normalized the trial solution of the PDF solution. As examples to validate the proposed solution approach, the mono-stable and tri-stable energy harvester systems are studied. Monte Carlo simulations (MCS) have been applied to check the analytical solution. Some theoretical trends of the harvested mean square voltage (MSV) as a function of system parameters are examined. The non-Gaussian excitation and time-varying nature of the system can significantly affect the MSV output. It is noted that this work provides researchers and engineers a tool to optimize energy harvesting systems operated in random environment.

High-energy orbit sliding mode control for nonlinear energy harvesting

Vibration energy harvesting has extensive application prospects in many significant occasions, such as mechanical structure health monitoring, vehicle tire pressure monitoring, IoT devices and human health monitoring. The nonlinearity is an effective method to improve the energy harvesting efficiency where there are low- and high-energy orbits in the multi-solution region of the system. The harvested power will be increased significantly when the system is guided from the low-energy orbit to the high-energy orbit. The sliding mode control is regarded as an easy, robust and adaptive method for orbit jump, but the implementation of this nonlinear control method has not been discussed. This paper proposes a high-energy sliding mode control method through rotatable magnets actuated by micro-motor. The electromechanical model of mono-stable and bi-stable systems with the identified nonlinear restoring force is established to design a sliding mode control algorithm for enhancing the energy harvesting performance. Simulation and experiment results demonstrate that the rotatable magnets with sliding mode control have a positive influence on reaching the high-energy orbit for both mono-stable and bi-stable systems within the multi-solution region. Moreover, the rotatable magnets method with a sliding mode control actuates the small magnets in the system for a short time with little theoretical consumption of energy. This research has provided a potential practical application of sliding mode control for high-energy orbit jump of the nonlinear energy harvesting.

Stochastic resonance in cascaded monostable systems with double feedback and its application in rolling bearing fault feature extraction

Stochastic resonance (SR) has been widely concerned and studied in the field of mechanical fault detection due to the capability of weak signal detection. However, research focused on how to improve the performance of SR system and enable it to process large-parameter signals in engineering practice better. Therefore, from the perspective of feedback control and multi-system synergy, this paper presents a double-feedback cascaded monostable SR (CMSR) system and analyzes the SR phenomenon in the system by using the signal-to-noise ratio as the measurement index. It investigates specifically influence of the coefficients on the system performance. Introducing feedback control and serial structure to a monostable system, consisting of a symmetric quartic potential function, not only alters the response characteristics of the monostable system but also significantly improve the system performance. Compared with the classical bistable SR system with low-pass filtering characteristics, the proposed system can exhibit low-pass filtering or band-pass filtering behavior through different combinations of feedback coefficients, which can better match the frequency characteristic of target signal. Finally, the proposed double-feedback CMSR system is used for extraction of the weak signal features in rolling bearing fault detection, and its superiority and feasibility are verified by comparison with other SR systems.

Bistable traveling waves in impulsive reaction-advection-diffusion models

In this paper, we study an impulsive reaction-advection-diffusion model given by a reaction-advection-diffusion equation with the Allee effect in high-dimensional space composed with a discrete-time map. We establish the existence, uniqueness up to translation, and global stability of bistable traveling waves for impulsive reaction-advection-diffusion models. The methods involve the monotone semiflow approach, the upper and lower solutions, and spreading speeds of monostable systems. Numerical simulations verify the correctness of our conclusions.

Unified mechanism of inverse stochastic resonance for monostability and bistability in Hindmarsh-Rose neuron.

Noise is ubiquitous and has been verified to play constructive roles in various systems, among which the inverse stochastic resonance (ISR) has aroused much attention in contrast to positive effects such as stochastic resonance. The ISR has been observed in both bistable and monostable systems for which the mechanisms are revealed as noise-induced biased switching and noise-enhanced stability, respectively. In this paper, we investigate the ISR phenomenon in the monostable and bistable Hindmarsh-Rose neurons within a unified framework of large deviation theory. The critical noise strengths for both cases can be obtained by matching the timescales between noise-induced boundary crossing and the limit cycle. Furthermore, different stages of ISR are revealed by the bursting frequency distribution, where the gradual increase of the peak bursting frequency can also be explained within the same framework. The perspective and results in this paper may shed some light on the understanding of the noise-induced complex phenomena in stochastic dynamical systems.

Invasion entire solutions for two-species diffusive monostable competitive systems

We study the existence of entire solutions of two-species diffusive monostable Lotka–Volterra competitive systems. Our results cover both cases of monostable with stable coexistence equilibrium and monostable with competition–extinction equilibria. We prove the existence of entire solutions describing different types of invasion scenarios of an empty territory by two competitors as the time variable t→−∞. Depending on the mobile species, these invasion phenomena can occur either from one side or both sides of the x-axis, leading to the existence of different types of entire solutions. In particular, the existence of entire solution characterizing the invasion of an empty habitat from one side by a weaker competitor and from both sides of the same territory by a stronger competitor as the time variable t→−∞ are established. We construct delicate super–sub solutions and employ the comparison principle to prove our main results.

Response of a vibro-impact energy harvesting system with bilateral rigid stoppers under Gaussian white noise

In this paper, responses of the stochastic energy harvesting systems with bilateral rigid stoppers are studied, in which the Newton’s impact model is adopted to describe the impact force between the proof mass and stoppers. To the best knowledge of the authors, the dynamical characteristics of the vibro-impact energy harvesting system have been scarcely investigated by utilizing the stochastic averaging of energy envelope. First, such theoretical method is employed to study the mono-stable energy harvesting system with bilateral rigid impact. After obtaining the equivalent system and considering the energy loss due to impact in every quasi period, the probability density functions are theoretically derived, and the effectiveness of the theoretical method is validated by numerical simulations. Then, the time history of response for the stochastic bi-stable system with bilateral rigid stoppers is given to analyze the effects of the noise intensity and the impact distance on the system. In addition, effects of the coefficients on the mean square voltage in the mono-stable and bi-stable systems are discussed to help to enhance the energy harvesting performance, and some interesting phenomena are observed.

Robustness in spatially driven bistability in signaling systems

Biological systems are spatially organized. This microscopic heterogeneity has been shown to produce emergent complex behaviors such as bistability. Even though the connection between spatiality and dynamic response is essential to understand biological output, its robustness and extent has not been sufficiently explored. This work focuses on a previously described system which is composed of two monostable modules acting on different cellular compartments and sharing species through linear shuttling reactions. One of the two main purposes of this paper is to quantify the frequency of occurrence of bistability throughout the parameter space and to identify which parameters and in which value ranges control the emergence and the properties of bistability. We found that a very small fraction of the sampled parameter space produced a bistable response. Most importantly, shuttling parameters were among the most influential ones to control this property. The other goal of this paper is to simplify the same system as much as possible without losing compartment-induced bistability. This procedure provided a simplified model that still connects two monostable systems by a reduced set of linear shuttling reactions that circulates all the species around the two compartments. Bistable systems are one of the main building blocks of more complex behaviors such as oscillations, memory, and digitalization. Therefore, we expect that the proposed minimal system provides insight into how these behaviors can arise from compartmentalization.

The effect of initial values on extinction or persistence in degenerate diffusion competition systems.

When the asymptotic spreading for classical monostable Lotka-Volterra competition diffusion systems is concerned, extinction or persistence of the two competitive species is completely determined by the dynamics of the corresponding kinetic systems, while the size of initial values does not affect the final states. The purpose of this paper is to demonstrate the rich dynamics induced by the initial values in a class of degenerate competition diffusion systems with weak Allee effect. We present various extinction or persistence results by selecting different initial values although the corresponding kinetic system is fixed, which also implies the existence of balance between degenerate nonlinear reaction and diffusion. For example, even if the positive steady state of the corresponding kinetic system is globally asymptotically stable, we observe four different spreading-vanishing phenomena by selecting different initial values. In addition, the interspecific competition of one species may be harmful to the persistence of the other species by taking proper initial values. Our results show that the superior competitor in the sense of the corresponding kinetic system is not always unbeatable, it can be wiped out by the inferior competitor in the sense of the corresponding kinetic system depending on the size of initial habitats as well as the intensity of Allee effect.

Attaining the high-energy orbit of nonlinear energy harvesters by load perturbation

Abstract Energy harvesters, especially nonlinear systems with broad bandwidth, have made it possible for devices in IoT (Internet of Things) to extract energy from the ambient environment. However, due to the multistability feature of the nonlinear system, high and low energy orbits coexist in the system. It is critical to control the multistability so as to obtain high energy output. This paper proposes the load perturbation method to attain the high-energy orbit of nonlinear energy harvesters. Based on the electromechanical model, investigating the electrical load effects on the system shows that varying load results in different system states, which suggests that the load perturbation may trigger the transformation of system states. Therefore, a load perturbation by disconnecting the electrical load is employed to attain the high-energy orbit. Simulation and experiments reveal that the load perturbation can stimulate high-energy orbit oscillation for both monostable and bistable systems. The power output can be amplified for multifold times. The perturbation requires only one switch without additional mechanical structures. Furthermore, this method consumes little energy so that no external electrical energy source is needed. The load perturbation method opens opportunities for wide applications of nonlinear energy harvesters.

Levy noise-driven stochastic resonance in a coupled monostable system

Treating the signal-to-noise ratio (SNR) as the signal quality index, the stochastic resonance (SR) phenomenon in a coupled monostable system (CMS), which is nonlinearly constructed from two monostable systems with a feedback factor and driven by Levy noise, is investigated. First, the construction principle of the coupled system is analysed, and the mechanism of SR is explained from the perspective of microscopic particle motion. Second, the method of parameter adjustment is detailed, and the variations in the system output SNR with changes in different parameters are analysed. Finally, multichannel signals are applied to the coupled system. The research results show that (1) the system parameter intervals remain unchanged with variations in the characteristic noise parameter α when the system outputs the optimal SNR for single-channel signals; (2) in extracting single-channel, three-channel and five-channel signals, the optimal interval of parameter r does not change; and (3) by adjusting the value of the system parameter a, the differences among the power spectrum values of the different channel signals can be reduced. The SR induced by the CMS is greater than that for a monostable system.

Adjustable Nonlinear Mechanism System for Wideband Energy Harvesting in Rotational Circumstances

Nonlinear energy harvesters have already been exhibited to draw energy from ambient vibration owing to their particular dynamic characteristics, and are feasible to desirable responses for broadband excitations of bistable and monostable systems. This study proposes an energy harvester for rotational applications, in which a cantilever beam pasted piezoelectric film and magnets with the same polarity are comprised as a nonlinear vibrating system. As the rotationally angular velocity gradually increases, the tensile stress to the cantilever beam is also self-adjusted with the increscent centrifugal force, causing the potential barriers of bistable type become shallow, so that the cantilever beam has the ability to maintain the high energy orbit motion from bistable hardening type to monostable hardening behavior. From the implemented results, the valid bandwidth of angular frequency can be improved from 26 rad/s - 132 rad/s to 15 rad/s - 215 rad/s, under the case of the effect of centrifugal force on nonlinear vibrating behavior. It demonstrates that the centrifugal force can significantly promote the performance of nonlinear energy harvesters.

A Comparison of Deterministic and Stochastic Modeling Approaches for Biochemical Reaction Systems: On Fixed Points, Means, and Modes.

In the mathematical modeling of biochemical reactions, a convenient standard approach is to use ordinary differential equations (ODEs) that follow the law of mass action. However, this deterministic ansatz is based on simplifications; in particular, it neglects noise, which is inherent to biological processes. In contrast, the stochasticity of reactions is captured in detail by the discrete chemical master equation (CME). Therefore, the CME is frequently applied to mesoscopic systems, where copy numbers of involved components are small and random fluctuations are thus significant. Here, we compare those two common modeling approaches, aiming at identifying parallels and discrepancies between deterministic variables and possible stochastic counterparts like the mean or modes of the state space probability distribution. To that end, a mathematically flexible reaction scheme of autoregulatory gene expression is translated into the corresponding ODE and CME formulations. We show that in the thermodynamic limit, deterministic stable fixed points usually correspond well to the modes in the stationary probability distribution. However, this connection might be disrupted in small systems. The discrepancies are characterized and systematically traced back to the magnitude of the stoichiometric coefficients and to the presence of nonlinear reactions. These factors are found to synergistically promote large and highly asymmetric fluctuations. As a consequence, bistable but unimodal, and monostable but bimodal systems can emerge. This clearly challenges the role of ODE modeling in the description of cellular signaling and regulation, where some of the involved components usually occur in low copy numbers. Nevertheless, systems whose bimodality originates from deterministic bistability are found to sustain a more robust separation of the two states compared to bimodal, but monostable systems. In regulatory circuits that require precise coordination, ODE modeling is thus still expected to provide relevant indications on the underlying dynamics.

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.

Domain nucleation and confinement in agent-controlled bistable systems.

We report a mechanism of pattern formation in growing bistable systems coupled indirectly. A modified Fujita etal. model is studied as an example of a reaction-diffusion system of nondiffusive activator and inhibitor molecules immersed in the medium of a fast diffusive agent. Here we show that, as the system grows, a new domain nucleates spontaneously in the area where the local level of the agent becomes critical. Newly nucleated domains are stable and the pattern formation is different from Turing's mechanism in monostable systems. Domains are spatially confined by the agent even if the activator and inhibitor molecules diffuse. With the spatial extension of the system, a larger domain may undergo a wave number instability, and the concentrations of active molecules within the neighboring elements of a domain can become sharply different. The mechanism reported in this work could be generic for pattern formation systems involving multistability, growth, and indirect coupling.

Energy harvesting in a nonlinear piezomagnetoelastic beam subjected to random excitation

This work addresses the influence of nonlinearities in energy harvesting from a piezomagnetoelastic structure subjected to random vibrations. Nonlinear equations of motion that describe the electromechanical system are given along with theoretical simulations. The numerical analysis presents a comparison between the voltage provided from a linear, nonlinear bistable and nonlinear monostable systems due to random vibration. Experimental performance of the generator exhibits qualitative agreement with the theory, showing an enhancement of piezoelectric power generation in a bistable system when it vibrates around both stable equilibrium points. A relationship between variations in the excitation and a bistable system response is established from numerical simulations, defining a region of enhanced power generation when compared to the linear and nonlinear monostable cases.

Validity conditions for moment closure approximations in stochastic chemical kinetics.

Approximations based on moment-closure (MA) are commonly used to obtain estimates of the mean molecule numbers and of the variance of fluctuations in the number of molecules of chemical systems. The advantage of this approach is that it can be far less computationally expensive than exact stochastic simulations of the chemical master equation. Here, we numerically study the conditions under which the MA equations yield results reflecting the true stochastic dynamics of the system. We show that for bistable and oscillatory chemical systems with deterministic initial conditions, the solution of the MA equations can be interpreted as a valid approximation to the true moments of the chemical master equation, only when the steady-state mean molecule numbers obtained from the chemical master equation fall within a certain finite range. The same validity criterion for monostable systems implies that the steady-state mean molecule numbers obtained from the chemical master equation must be above a certain threshold. For mean molecule numbers outside of this range of validity, the MA equations lead to either qualitatively wrong oscillatory dynamics or to unphysical predictions such as negative variances in the molecule numbers or multiple steady-state moments of the stationary distribution as the initial conditions are varied. Our results clarify the range of validity of the MA approach and show that pitfalls in the interpretation of the results can only be overcome through the systematic comparison of the solutions of the MA equations of a certain order with those of higher orders.

Front-Like Entire Solutions for Monostable Reaction-Diffusion Systems

This paper is concerned with front-like entire solutions for monostable reaction-diffusion systems with cooperative and non-cooperative nonlinearities. In the cooperative case, the existence and asymptotic behavior of spatially independent solutions (SIS) are first proved. Further, combining a SIS and traveling fronts with different wave speeds and propagation directions, the existence and various qualitative properties of entire solutions are established by using the comparison principle. In the non-cooperative case, we introduce two auxiliary cooperative systems and establish a comparison theorem for the Cauchy problems of the three systems, and then prove the existence of entire solutions via using the comparison theorem, the traveling fronts and SIS of the auxiliary systems. Our results are applied to some biological and epidemiological models. To the best of our knowledge, it is the first work to study the entire solutions of non-cooperative reaction-diffusion systems.