Bistable System Research Articles

Logical vibrational resonance in a symmetric bistable system: Numerical and experimental studies

In previous studies, double harmonic signals as external driving forces have been used for inducing logical vibrational resonance to facilitate the control of logic operations. It requires the bias parameter to break the symmetry of the bistable system. In hardware circuits, symmetric bistable system circuits are easy to build and even using few components. To address the problem of maintaining the symmetry of the system while inducing the logical vibrational resonance, we take advantage of the property that an anharmonic signal with a constant component in the Fourier series can replace the bias parameter to induce logical vibrational resonance. We add an anharmonic low-frequency signal to the symmetric bistable system as a slowly-varying signal. In addition, we add another high-frequency signal as a fast-varying signal. Through numerical simulation, we find the optimal logic output region of the system and the logic output region is explained by using bifurcation theory. Then, noise is added to the system to demonstrate its effect on the optimal logic output region. Finally, the simulation results are further verified by a hardware circuit. Both simulation and experimental results prove the occurrence of logical vibrational resonance and its necessary condition in the symmetric bistable system.

Uncertainty quantification of multi-scale resilience in networked systems with nonlinear dynamics using arbitrary polynomial chaos

Complex systems derive sophisticated behavioral dynamics by connecting individual component dynamics via a complex network. The resilience of complex systems is a critical ability to regain desirable behavior after perturbations. In the past years, our understanding of large-scale networked resilience is largely confined to proprietary agent-based simulations or topological analysis of graphs. However, we know the dynamics and topology both matter and the impact of model uncertainty of the system remains unsolved, especially on individual nodes. In order to quantify the effect of uncertainty on resilience across the network resolutions (from macro-scale network statistics to individual node dynamics), we employ an arbitrary polynomial chaos (aPC) expansion method to identify the probability of a node in losing its resilience and how the different model parameters contribute to this risk on a single node. We test this using both a generic networked bi-stable system and also established ecological and work force commuter network dynamics to demonstrate applicability. This framework will aid practitioners to both understand macro-scale behavior and make micro-scale interventions.

Reliability of the bi-stable piezoelectric energy harvester under random excitation

The bi-stable piezoelectric energy harvester can produce higher voltage and efficiency than traditional energy harvester. In this manuscript, the reliability of the bi-stable piezoelectric energy harvester excited by Gaussian white noise is investigated. The equivalent uncoupled equation of system is deduced on the basis of the generalized harmonic transformation for the bi-stable piezoelectric energy harvester original coupling system. The averaged stochastic differential equation for system energy of the equivalent uncoupled system is established using the stochastic averaging method. The conditional reliability function and the mean first-passage time are derived through the backward Kolmogorov equation and the generalized Pontryagin equation associated with the averaged stochastic differential equation. The influences of noise intensity, electromechanical coupling coefficient, and time constant ratio on the conditional reliability function and mean first-passage time are discussed. The analytical results are highly consistent with the numerical results that are obtained by the Monte Carlo numerical simulation.

The monotone traveling wave solution of a bistable three-species competition system via unconstrained neural networks.

In this paper, we approximate traveling wave solutions via artificial neural networks. Finding traveling wave solutions can be interpreted as a forward-inverse problem that solves a differential equation without knowing the exact speed. In general, we require additional restrictions to ensure the uniqueness of traveling wave solutions that satisfy boundary and initial conditions. This paper is based on the theoretical results that the bistable three-species competition system has a unique traveling wave solution on the premise of the monotonicity of the solution. Since the original monotonic neural networks are not smooth functions, they are not suitable for representing solutions of differential equations. We propose a method of approximating a monotone solution via a neural network representing a primitive function of another positive function. In the numerical integration, the operator learning-based neural network resolved the issue of differentiability by replacing the quadrature rule. We also provide theoretical results that a small training loss implies a convergence to a real solution. The set of functions neural networks can represent is dense in the solution space, so the results suggest the convergence of neural networks with appropriate training. We validate that the proposed method works successfully for the cases where the wave speed is identical to zero. Our monotonic neural network achieves a small error, suggesting that an accurate speed and solution can be estimated when the sign of wave speed is known.

Energy based control of a bi-stable and underactuated soft robotic system based on dielectric elastomer actuators*

In this paper, we propose an energy based control approach for a class of underactuated soft robotic systems. The considered case study consists of an elastic structure driven by soft dielectric elastomer actuators, and is able to achieve large bending displacement thanks to a bi-stable design concept. The bi-stability feature, however, causes the system to exhibit an unstable behavior in open-loop. After providing a port-Hamiltonian description of the soft robotic system, sufficient conditions for the existence of an energy based stabilizing controller are provided. A linear matrix inequality approach is then proposed to practically address the design of the controller gain. The effectiveness of the method is verified by means of simulation studies, conducted on an experimentally validated model of the real-life device.

Incipient Bearing Fault Diagnosis Based on the Two-State Theory for Stochastic Resonance Systems

The principle of stochastic resonance (SR) has been intensively used in designing algorithms for incipient bearing fault diagnosis based on stochastic simulation. In order to directly apply the semi-analytic result to the laboratory designs, an incipient bearing fault diagnosis algorithm based on the method of moments for stochastic resonant systems was proposed, with its real-time monitoring efficiency being assured by an available table of the involving stationary moments. Nevertheless, since the noise that buried the fault signal can only be estimated after sampling, calculation of such stationary moments might be an inconvenience. To fix this inconvenience, a brand-new fault diagnosis algorithm is designed in this paper based on the two-state theory for a biased underdamped bistable stochastic resonant system. The capability and the efficiency of the algorithm in identifying the characteristic frequency of the hidden fault signal are confirmed with the experimental data sets. This investigation actually opens a new way for improving the existing SR-based diagnosis algorithms.

Backward bifurcation, basic reinfection number and robustness of an SEIRE epidemic model with reinfection

Recent evidences show that individuals who recovered from COVID-19 can be reinfected. However, this phenomenon has rarely been studied using mathematical models. In this paper, we propose an SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds [Formula: see text] (the basic reproduction number) and [Formula: see text] (a threshold less than one). Our investigations show that when [Formula: see text], the system has an endemic equilibrium, which is globally asymptotically stable. When [Formula: see text], the epidemic system exhibits bistable dynamics. That is, the system has backward bifurcation and the disease cannot be eradicated. In order to eradicate the disease, we must ensure that the basic reproduction number [Formula: see text] is less than [Formula: see text]. The basic reinfection number is obtained to measure the reinfection force, which turns out to be a new tipping point for disease dynamics. We also give definition of robustness, a new concept to measure the difficulty of completely eliminating the disease for a bistable epidemic system. Numerical simulations are carried out to verify the conclusions.

Energy Harvesting Using a Nonlinear Resonator with Asymmetric Potential Wells

This paper presents the results of numerical simulations of a nonlinear bistable system for harvesting energy from ambient vibrating mechanical sources. Detailed model tests were carried out on an inertial energy harvesting system consisting of a piezoelectric beam with additional springs attached. The mathematical model was derived using the bond graph approach. Depending on the spring selection, the shape of the bistable potential wells was modified including the removal of wells’ degeneration. Consequently, the broken mirror symmetry between the potential wells led to additional solutions with corresponding voltage responses. The probability of occurrence for different high voltage/large orbit solutions with changes in potential symmetry was investigated. In particular, the periodicity of different solutions with respect to the harmonic excitation period were studied and compared in terms of the voltage output. The results showed that a large orbit period-6 subharmonic solution could be stabilized while some higher subharmonic solutions disappeared with the increasing asymmetry of potential wells. Changes in frequency ranges were also observed for chaotic solutions.

Monotonicity of wave profiles for a bistable system on 2-D lattice

This paper is concerned with the monotonicity problem of traveling wave solutions for a class of nonlocal bistable system on two-dimensional lattice. We prove that any traveling wave solution, including the standing wave solution (wave speed c = 0 c=0 ), is strictly monotone increasing for this nonlocal bistable lattice system.

Stochastic dynamics of a metal magnon parametron

A magnon parametron is a calculating element that carries binary information by a discretized oscillating phase: 0 and π. Owing to the strong non-linearity of the magnetization dynamics, the oscillation phase flips to another stochastically, which can give a rise to unconventional computing functionalities, including probabilistic computing. Here, we investigated the stochastic dynamics of ferromagnetic-metal magnon parametron, of which the precession phase is discretized into two (0 and π) by parametric pumping of magnons. We found that an AC magnetic field perpendicular to an external field can control the precession phase in the magnon parametron, of which amplitude dependence follows the sigmoid function, a requirement for probabilistic bit operation. We also found that the time scale for flipping between different precession phases grows exponentially as pumping microwave power. Our finding ensures that the magnon parametron can be used as a calculating element for unconventional computing schemes based on bistable systems.

A bistable and reconfigurable molecular system with encodable bonds.

Molecular systems with ability to controllably transform between different conformations play pivotal roles in regulating biochemical functions. Here, we report the design of a bistable DNA origami four-way junction (DOJ) molecular system that adopts two distinct stable conformations with controllable reconfigurability by using conformation-controlled base stacking. Exquisite control over DOJ's conformation and transformation is realized by programming the stacking bonds (quasi-blunt-ends) within the junction to induce prescribed coaxial stacking of neighboring junction arms. A specific DOJ conformation may be achieved by encoding the stacking bonds with binary stacking sequences based on thermodynamic calculations. Dynamic transformations of DOJ between various conformations are achieved by using specific environmental and molecular stimulations to reprogram the stacking codes. This work provides a useful platform for constructing self-assembled DNA nanostructures and nanomachines and insights for future design of artificial molecular systems with increasing complexity and reconfigurability.

Research on a Bearing Fault Enhancement Diagnosis Method with Convolutional Neural Network Based on Adaptive Stochastic Resonance.

As a powerful feature extraction tool, a convolutional neural network (CNN) has strong adaptability for big data applications such as bearing fault diagnosis, whereas the classification performance is limited when the quality of raw signals is poor. In this paper, stochastic resonance (SR), which provides an advanced feature enhancement approach for weak signals with strong background noise, is introduced as a data pre-processing method for the CNN to improve its classification performance. First, a multiparameter adjusting bistable Duffing system that can achieve SR under large-parameter weak signals is introduced. A hybrid optimization algorithm (HOA) combining the genetic algorithm (GA) and the simulated annealing (SA) is proposed to adaptively obtain the optimized parameters and output signal-to-noise ratio (SNR) of the Duffing system. Therefore, the data optimization based on the multiparameter-adjusting SR of Duffing system can be realized. An SR-based mapping method is further proposed to convert the outputs of the Duffing system into grey images, which can be further processed by a normal CNN with batch normalization (BN) layers and dropout layers. After verifying the feasibility of the HOA in multiparameter optimization of the Duffing system, the bearing fault data set from the CWRU bearing data center was processed by the proposed fault enhancement classification and identification method. The research showed that the weak features of the bearing signals could be enhanced significantly through the adaptive multiparameter optimization of SR, and classification accuracies for 10 categories of bearing signals could achieve 100% and those for 20 categories could achieve more than 96.9%, which is better than other methods. The influences of the population number on the classification accuracies and calculation time were further studied, and the feature map and network visualization are presented. It was demonstrated that the proposed method can realize high-performance fault enhancement classification and identification.

Stochastic delay differential equations: Analysis and simulation studies

Stochastic delay differential equations play an important role in modelling scientific and engineering systems. In this paper the partial differential equation satisfying the time evolution of the probability density function of the state variable governed by the stochastic delay differential equation (SDDE) with additive white noise and coloured noise perturbation is obtained under small time delay and small correlation time approximation using path integral formalism. This partial differential equation reduces to a Fokker–Planck equation for particular linear and non-linear SDDE. Fokker–Planck equation is then solved with reflecting boundary conditions to get the analytical solutions for Stationary Probability Density Function (SPDF). The analytical solutions for SPDF is compared with the SPDF obtained using simulation. Further the Mean First Passage Time (MFPT) of the bistable system is calculated for various values of time delay and noise strength and compared with that of the simulation. The MFPT for the time delayed system for the case of cubic potential driven by Gaussian and Levy noise as well as asymmetric bistable potential driven by a noise and periodic driving force has been investigated.

Uniqueness and global stability of V-shaped fronts for the buffered bistable system in [formula omitted

This paper is devoted to V-shaped calcium concentration waves of the buffered bistable system in R2. We prove that multiple immobile buffers (where all buffers do not diffuse) do not affect the global asymptotic stability and uniqueness of V-shaped calcium concentration waves. The existence of such curved fronts has been proved in our previous work (Jia et al., 2022). However, the study of its stability is difficult since the buffers do not diffuse. We first apply the sliding method to construct a crucial sub-solution. Then, based on the super- and sub-solution method, and comparison principle, we achieve the global asymptotic stability of the V-shaped fronts.

Stochastic resonance of a high-order-degradation bistable system and its application in fault diagnosis with variable speed condition

Rolling element bearings play a critical role in rotating machines with variable speed conditions, and their fault detection has attracted considerable interest from researchers in recent decades. In this paper, a high-order-degradation bistable system is proposed, and the corresponding stochastic resonance (SR) based on a new measurement index is investigated for fault detection with variable speed conditions. It successfully achieves fault detection without prior knowledge. Firstly, the output signal-to-noise ratio (SNR) of the high-order-degradation bistable stochastic resonance (HBSR) is theoretically analyzed using the approximate adiabatic theory. Secondly, a novel measurement index for determining the occurrence of stochastic resonance is proposed. Thirdly, the experimental data are used to validate the efficiency of the proposed nonlinear system and measurement index when the fault types and the actual fault order are unknown. Finally, the ground test data from a civil aircraft engine is also used to further validate the performance of the HBSR based on the novel measurement index.

Coherence and stochastic resonance in the fractional-birhythmic self-sustained system subjected to fractional time-delay feedback and Lévy noise

This paper treats the nonlinear dynamics of an enzymatic-substrate modeled by the fractional multi-limit cycles Van der Pol oscillator with fractional time-delay feedback device subjected to Lévy noise perturbation and periodic excitation. The fractional electronic circuit has been used to model the system and the oscillations are described by a nonlinear fractional differential equation and show a new bifurcation parameter. The robustness of the stochastic resonance is examined by the use of standard measures within a continuous and a two-state description of the system. Firstly, the electronic circuit with the fractional-order operator and fractional delay feedback is used as a prototype of a fractional self-sustained system. Secondly, based on the minimum mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of the damping force and restoring force, and the original system is further simplified to an equivalent integer order system. The stochastic bifurcation of a bistable Van der Pol system with fractional-order and time delay without and under Lévy noise excitation is studied where we show the considerable effect of these parameters on birhythmic region, escape time and energy barriers. Additionally, we study the multi-effects of fractional order and fractional time-delay feedback in the self-sustained system driven by Lévy noise. The effects of fractional-order parameter, time delay feedback parameter on the autocorrelation function, power spectral density and signal-to-noise-ratio used in this investigation are shown to be maximized for an appropriate choice of the Lévy noise intensity and for a convenient choice of fractional-order and time delay feedback parameters. For a choice of a control parameter in the birhythmic region, by varying the other parameters of the system like fractional and time delay parameters, it appears that, for a fixed value of skewness Lévy noise parameter, the initial selection of the attractor seems to have a large effect on the resonance and coherence.

Horizontal Bi-Stable Vibration Energy Harvesting Using Electromagnetic Induction and Power Generation Efficiency Improvement via Stochastic Resonance

In this study, a vibration energy-harvesting system is developed by first proposing a horizontal bi-stable vibration model comprising an elastic spring and a mass block and then applying an electromagnetic induction power generation device composed of a magnet and a coil. Subsequently, based on a weight function that considers the mutual positional relationship between the magnet and conducting coil, a set of simultaneous governing equations that consider the elastic force of the elastic spring and the Lorentz force of electromagnetic induction is derived. Additionally, a numerical analysis method employing the Runge–Kutta method is utilized to obtain a numerical solution for the vibration response displacement and vibration power generation voltage simultaneously. Experiments are performed to verify the results yielded by the proposed bi-stable vibration energy-harvesting system. The results shows that the measured vibration response displacement and the vibration power generation voltage are consistent with the analytical results. Moreover, issues including the identification of damping coefficients that consider the mutual effects of normal kinetic friction and electromagnetic induction damping forces, as well as the effects of electromagnetic induction damping on the vibration response displacement, are discussed comprehensively. Simultaneously adding random and periodic signals to the bi-stable vibration model results in stochastic resonance and improves both the vibration amplification effect and vibration power generation.

A novel two-dimensional exponential potential bi-stable stochastic resonance system and its application in bearing fault diagnosis

It is difficult for classical stochastic resonance systems to extract weak signals under strong noise environment, therefore a novel two-dimensional exponential potential bi-stable stochastic resonance system (NTBSR) is proposed. First, the equivalent potential function, the mean first-pass time (MFPT) and the output signal-to-noise ratio (SNR) of NTBSR are derived under the adiabatic approximation theory. At the same time, the influence of different system parameters on them is explored. Then, NTBSR, the one-dimensional bi-stable stochastic resonance system (OBSR) and the two-dimensional classical bi-stable stochastic resonance system (TCBSR) are respectively simulated numerically, based on the fourth-order Runge–Kutta algorithm. It is found that the output SNR of NTBSR is the best. Finally, the NTBSR is applied to the fault signal diagnosis of different types of bearings, and the parameters are optimized through the adaptive genetic algorithm (AGA). The test results are compared with wavelet transform method, and TCBSR. The detection results on two sets of bearing fault data show that the NTBSR system has better effects on the enhancement and detection of bearing fault signals, and it is verified that the stochastic resonance method is superior to the traditional wavelet transform method in terms of signal detection and noise utilization. This provides good theoretical support and application value for practical engineering application.

A general hypothesis of multistable systems in pathophysiology

Despite intensive investigations numerous diseases remain etiologically puzzling and recalcitrant to treatments. A hypothesis is proposed here assuming that these difficulties are due to an unsuitable approach to the mechanisms of life, which is subjugated by an apparent complexity and fails to grasp the uniformity that lays behind. The stability of metabolism, despite the enormous complex of chemical reactions, suggests that reciprocal control is a prerequisite of life. Negative feedback loops have been known for a long time to maintain homeostasis, while more recently, different life processes involved in transitions or changes have been modeled by positive loops giving rise to bistable switches, also including various diseases. The present hypothesis makes a generalization, by assuming that any functional element of a biological system is involved in a positive or a negative feedback loop. Consequently, the hypothesis holds that the starting mechanism of any disease that affects a healthy human can be conceptually reduced to a bistable or multistationary loop system, thus providing a unifying model leading to the discovery of critical therapeutic targets.

Unknown fault diagnosis of planetary gearbox based on optimal rank nonnegative matrix factorization and improved stochastic resonance of bistable system

Unknown fault diagnosis of planetary gearbox based on optimal rank nonnegative matrix factorization and improved stochastic resonance of bistable system