Multiple Stable Equilibria Research Articles

Multistability and fixed-time multisynchronization of switched neural networks with state-dependent switching rules

This paper presents theoretical results on the multistability and fixed-time synchronization of switched neural networks with multiple almost-periodic solutions and state-dependent switching rules. It is shown herein that the number, location, and stability of the almost-periodic solutions of the switched neural networks can be characterized by making use of the state-space partition. Two sets of sufficient conditions are derived to ascertain the existence of 3n exponentially stable almost-periodic solutions. Subsequently, this paper introduces the novel concept of fixed-time multisynchronization in switched neural networks associated with a range of almost-periodic parameters within multiple stable equilibrium states for the first time. Based on the multistability results, it is demonstrated that there are 3n synchronization manifolds, wherein n is the number of neurons. Additionally, an estimation for the settling time required for drive–response switched neural networks to achieve synchronization is provided. It should be noted that this paper considers stable equilibrium points (static multisynchronization), stable almost-periodic orbits (dynamical multisynchronization), and hybrid stable equilibrium states (hybrid multisynchronization) as special cases of multistability (multisynchronization). Two numerical examples are elaborated to substantiate the theoretical results.

Dynamics of a Cournot game with bounded rational firms and various scale effects.

In this paper, we focus on a dynamic Cournot game in the market with a nonlinear (isoelastic) demand function. In our model, there are N competing firms featured by nonlinear cost functions, which enhances our model's resemblance to real-world scenarios. Firstly, we focus on the homogeneous case where firms' marginal costs change at equal rates. We establish analytical expressions of the market supply at equilibrium and perform comparative static analysis. In addition, we investigate the local stability under different economies of scale and show that there could be multiple stable equilibria if firms face economies of scale. The heterogeneous case where firms' marginal costs change at distinct rates is much more complex, thus we investigate the duopoly game with only two firms involved. Methods of symbolic computations such as triangular decomposition and partial cylindrical algebraic decomposition are employed in the analytical investigations of the equilibrium, which is nearly impossible by using the pencil-and-paper approach since the closed-form equilibrium is quite complicated. According to the computational results, we derive that two stable positive equilibria may coexist if both firms face economies of scale. Additionally, we conduct preliminary numerical simulations and find two different types of complex dynamics of the model considered in this paper: complex trajectories such as periodic and chaotic orbits may appear; the topological structure of the basins of attraction may be complex.

Stability and Sommerfeld effect in a multi-resonant types vibrating system with isolated rigid frame driven by four exciters

The previous studies on the vibrating system with multiple exciters less considered the Sommerfeld effect, which were mainly focused on synchronization and stability problems of the system in the whole resonant region. A dynamical model is proposed in this paper to study the synchronization, stability and the Sommerfeld effect of four exciters in the multi-resonant types vibrating system (MRTVS). In order to improve the power of the system, the vibrating system with two rigid frames is driven by four especially distributed exciters. The motion differential equations of the system are given by Lagrange’s equations, and the dimensionless coupling equations and synchronization criterion are constructed as well by the average method. The theory condition for stability of four exciters in synchronous states is derived from the Hamilton principle. The coupling dynamic characteristics, stability and Sommerfeld effect of the system are discussed numerically. The whole resonant region of the system is divided into three segments by natural frequencies, and the synchronous and stable states in the different resonant types are analyzed respectively. The diversity phenomenon of the nonlinear system is observed, in other words, the system exists multiple stable equilibrium solutions at a certain particular resonant region. Then the Sommerfeld effect around the natural frequencies (NFs) is further revealed. The correctness of theoretical analyses is examined by simulation and the experiment. It’s shown that the reasonable working point in engineering should be selected in the sub-resonant region with respect to the natural frequency of the main vibrating system. The present work can provide theoretical guidance for designing some new types of vibrating machines with high driving power.

"Doing what others do" does not stabilize continuous norms.

Differences in social norms are a key source of behavioral variation among human populations. It is widely assumed that a vast range of behaviors, even deleterious ones, can persist as long as they are locally common because deviants suffer coordination failures and social sanctions. Previous models have confirmed this intuition, showing that different populations may exhibit different norms even if they face similar environmental pressures or are linked by migration. Crucially, these studies have modeled norms as having a few discrete variants. Many norms, however, have a continuous range of variants. Here we present a mathematical model of the evolutionary dynamics of continuously varying norms and show that when the social payoffs of the behavioral options vary continuously the pressure to do what others do does not result in multiple stable equilibria. Instead, factors such as environmental pressure, individual preferences, moral beliefs, and cognitive attractors determine the outcome even if their effects are weak, and absent such factors populations linked by migration converge to the same norm. The results suggest that the content of norms across human societies is less arbitrary or historically constrained than previously assumed. Instead, there is greater scope for norms to evolve towards optimal individual or group-level solutions. Our findings also suggest that cooperative norms such as those that increase contributions to public goods might require evolved moral preferences, and not just social sanctions on deviants, to be stable.

Sonomaglev: Combining acoustic and diamagnetic levitation

Acoustic levitation and diamagnetic levitation are experimental methods that enable the contact-free study of both liquid droplets and solid particles. Here, we combine both the techniques into a single system that takes advantage of the strengths of each, allowing for the manipulation of levitated spherical water droplets (30 nl–14 μl) under conditions akin to weightlessness, in the laboratory, using a superconducting magnet fitted with two low-power ultrasonic transducers. We show that multiple droplets, arranged horizontally along a line, can be stably levitated with this system and demonstrate controlled contactless coalescence of two droplets. Numerical simulation of the magnetogravitational and acoustic potential reproduces the multiple stable equilibrium points observed in our experiments.

Faster network disruption from layered oscillatory dynamics.

Nonlinear complex network-coupled systems typically have multiple stable equilibrium states. Following perturbations or due to ambient noise, the system is pushed away from its initial equilibrium, and, depending on the direction and the amplitude of the excursion, it might undergo a transition to another equilibrium. It was recently demonstrated [M. Tyloo, J. Phys. Complex. 3 03LT01 (2022)] that layered complex networks may exhibit amplified fluctuations. Here, I investigate how noise with system-specific correlations impacts the first escape time of nonlinearly coupled oscillators. Interestingly, I show that, not only the strong amplification of the fluctuations is a threat to the good functioning of the network but also the spatial and temporal correlations of the noise along the lowest-lying eigenmodes of the Laplacian matrix. I analyze first escape times on synthetic networks and compare noise originating from layered dynamics to uncorrelated noise.

Learning and equilibrium transitions: Stochastic stability in discounted stochastic fictitious play

In this paper I study how adaptive learning leads to switches between multiple stable equilibria, and I develop tools to characterize the rate of transition. While the methods I develop are relatively general, I focus on a two-player model of stochastic fictitious play, where agents’ payoffs are subject to random shocks. Each player forecasts his opponent’s future play as a discounted average of past play. I analyze the behavior of agents’ beliefs as the discount rate on past information becomes small, but the payoff shock variance remains fixed. I show that agents tend to be drawn toward an equilibrium, but occasionally the stochastic shocks lead to endogenous shifts between equilibria. I then calculate the limiting transition rates and the invariant distribution of players’ beliefs, and use it to determine the most likely outcome observed in long run. I show this stochastically stable equilibrium satisfies an important continuity condition which allows for relatively direct characterization.

Hidden multiwing chaotic attractors with multiple stable equilibrium points

PurposeThe purpose of this paper is to construct a multiwing chaotic system that has hidden attractors with multiple stable equilibrium points. Because the multiwing hidden attractors chaotic systems are safer and have more dynamic behaviors, it is necessary to construct such a system to meet the needs of developing engineering.Design/methodology/approachBy introducing a multilevel pulse function into a three-dimensional chaotic system with two stable node–foci equilibrium points, a hidden multiwing attractor with multiple stable equilibrium points can be generated. The switching behavior of a hidden four-wing attractor is studied by phase portraits and time series. The dynamical properties of the multiwing attractor are analyzed via the Poincaré map, Lyapunov exponent spectrum and bifurcation diagram. Furthermore, the hardware experiment of the proposed four-wing hidden attractors was carried out.FindingsNot only unstable equilibrium points can produce multiwing attractors but stable node–foci equilibrium points can also produce multiwing attractors. And this system can obtain 2N + 2-wing attractors as the stage pulse of the multilevel pulse function is N. Moreover, the hardware experiment matches the simulation results well.Originality/valueThis paper constructs a new multiwing chaotic system by enlarging the number of stable node–foci equilibrium points. In addition, it is a nonautonomous system that is more suitable for practical projects. And the hardware experiment is also given in this article which has not been seen before. So, this paper promotes the development of hidden multiwing chaotic attractors in nonautonomous systems and makes sense for applications.

Chaotic and regular dynamics of a morphing shell with a vanishing-stiffness mode

Thin elastic shells are almost inextensible but easy to bend. In the presence of prestresses, geometric frustrations can produce complex elastic energetic landscapes, which have been tailored for the design of morphing structures with multiple stable equilibria or neutrally stable manifolds. We show that the co-existence of stiff and floppy modes leads to unexploited dynamical features. We build a neutrally stable cylindrical shell that under dynamical excitation alternates a chaotic behavior with a surprisingly regular regimes with a continuous precession of the curvature axis at a constant speed. We explain the experimental findings with a minimal model, showing how the intriguing dynamics is due to the subtle coupling between the prestress, geometrical nonlinearity, material anisotropy and inertial effects. Our results shed a new light on morphing structures dynamics and can be exploited in engineering applications such as energy harvesting.

Capturing of particles in suspension flow through a micro cross-shaped channel

Capturing of microscale particles through microfluidics has been a flourishing research area in chemical and biological engineering applications. This work follows our previous report (Zhang et al., AIChE J., vol. 66, 2019, e16822) by using micro plane laser-induced fluorescence (μPLIF) and high-speed photography technology to study the flow of a diluted suspension of buoyant particles in a micro cross-shaped channel at 30≤Re≤350. Particular attention is paid to particle accumulation behaviors caused by vortex breakdown within the steady engulfment flow and vortex shedding flow. Three particle capture modes are identified in accumulation regions including the particle chain mode, the steady and unsteady particle cluster mode at 100≤Re≤300. Multiple stable equilibrium orbits formed by captured particles in accumulation regions are attributed to the axial force balance. In addition, perturbations in particle trajectories induced by the vortex shedding flow or particle-particle interactions contribute to the particle release.

On a mathematical model of tumor-immune system interactions with an oncolytic virus therapy

<p style='text-indent:20px;'>We investigate a mathematical model of tumor–immune system interactions with oncolytic virus therapy (OVT). Susceptible tumor cells may become infected by viruses that are engineered specifically to kill cancer cells but not healthy cells. Once the infected cancer cells are destroyed by oncolysis, they release new infectious virus particles to help kill surrounding tumor cells. The immune system constructed includes innate and adaptive immunities while the adaptive immunity is further separated into anti-viral or anti-tumor immune cells. The model is first analyzed by studying boundary equilibria and their stability. Numerical bifurcation analysis is performed to investigate the outcomes of the oncolytic virus therapy. The model has a unique tumor remission equilibrium, which is unlikely to be stable based on the parameter values given in the literature. Multiple stable positive equilibria with tumor sizes close to the carrying capacity coexist in the system if the tumor is less antigenic. However, as the viral infection rate increases, the OVT becomes more effective in the sense that the tumor can be dormant for a longer period of time even when the tumor is weakly antigenic.</p>

Principle of Limiting Factors-Driven Piecewise Population Growth Model I: Qualitative Exploration and Study Cases on Continuous-Time Dynamics

We examine the comportment of the global trajectory of a piecewisely conceived single species population growth model. Formulation relies on what we develop as the principle of limiting factors for population growth, adapted from the law of the minimum of Liebig and the law of the tolerance of Shelford. The ensuing paradigm sets natality and mortality rates to express through extreme values of population growth determining factor. Dynamics through time occur over different growth phases. Transition points are interpreted as thresholds of viability, starvation, and intraspecific competition. In this delivery, we focus on the qualitative study of the global trajectory expressed on continuous time and on exploring the feasibility of analytical results against data on populations growing under experimental or natural conditions. All study cases sustained fittings of high reproducibility both at empirical and interpretative slants. Possible phase configurations include regimes with multiple stable equilibria, sigmoidal growth, extinction, or stationarity. Here, we also outline that the associating discrete-time piecewise model composes the logistic map applied over a particular region of the phase configuration. Preliminary exploratory analysis suggests that the logistic map’s chaos onset could surpass once the orbit enters a contiguous phase region.

The dynamics of cooperation, power, and inequality in a group-structured society

Most human societies are characterized by the presence of different identity groups which cooperate but also compete for resources and power. To deepen our understanding of the underlying social dynamics, we model a society subdivided into groups with constant sizes and dynamically changing powers. Both individuals within groups and groups themselves participate in collective actions. The groups are also engaged in political contests over power which determines how jointly produced resources are divided. Using analytical approximations and agent-based simulations, we show that the model exhibits rich behavior characterized by multiple stable equilibria and, under some conditions, non-equilibrium dynamics. We demonstrate that societies in which individuals act independently are more stable than those in which actions of individuals are completely synchronized. We show that mechanisms preventing politically powerful groups from bending the rules of competition in their favor play a key role in promoting between-group cooperation and reducing inequality between groups. We also show that small groups can be more successful in competition than large groups if the jointly-produced goods are rivalrous and the potential benefit of cooperation is relatively small. Otherwise large groups dominate. Overall our model contributes towards a better understanding of the causes of variation between societies in terms of the economic and political inequality within them.

Design of a multi-stable piezoelectric energy harvester with programmable equilibrium point configurations

In this paper, a novel multiple-stable piezoelectric energy harvester with programmable equilibrium point configurations is proposed. Currently, multi-stable energy harvesters have been widely investigated due to possible improvement in performance. However, there are two main problems in the existing literature. The first is that obtaining multiple stable equilibrium points for energy harvesters, such as penta-stable, hepta-stable points, requires a highly complex structural design. The second is that the current design is difficult to arbitrarily specify the number and the coordinates of the equilibrium points, which is helpful for optimizing the performance of the energy harvester. Therefore, we solve these two problems by using programmable nonlinear force technology. As examples, a tri-stable and a hepta-stable energy harvester with specified coordinates of equilibrium points are designed. Both simulations and experiments show that the designs are feasible. Under the acceleration of 2 m/s2 at 7 Hz, the tri-stable energy harvester can charge a 4.7 μF capacitor to 28.6 V, and the hepta-stable energy harvester can charge the capacitor to 21.9 V in 40 s. These voltages are sufficient to power some sensor nodes. In a word, the proposed method is promising in the performance optimization of energy harvesters.

Design and Optimal Control of a Multistable, Cooperative Microactuator

In order to satisfy the demand for the high functionality of future microdevices, research on new concepts for multistable microactuators with enlarged working ranges becomes increasingly important. A challenge for the design of such actuators lies in overcoming the mechanical connections of the moved object, which limit its deflection angle or traveling distance. Although numerous approaches have already been proposed to solve this issue, only a few have considered multiple asymptotically stable resting positions. In order to fill this gap, we present a microactuator that allows large vertical displacements of a freely moving permanent magnet on a millimeter-scale. Multiple stable equilibria are generated at predefined positions by superimposing permanent magnetic fields, thus removing the need for constant energy input. In order to achieve fast object movements with low solenoid currents, we apply a combination of piezoelectric and electromagnetic actuation, which work as cooperative manipulators. Optimal trajectory planning and flatness-based control ensure time- and energy-efficient motion while being able to compensate for disturbances. We demonstrate the advantage of the proposed actuator in terms of its expandability and show the effectiveness of the controller with regard to the initial state uncertainty.

Sustainability over sets and the business cycle

Many rigorous works have examined the behavior of Kaldor business cycle models. In this paper, utilizing the sustainability over sets (SOS) concept, a general theorem is proven establishing necessary and sufficient conditions for the existence and identification of rectangular sustainable sets for a general Kaldor model. These conditions can be readily verified, either in the two-dimensional space of lower and upper bounds on income, or in the one-dimensional space of upper bound on income, thus enabling identification of the region of economic sustainability for the considered Kaldor model. The Theorem’s power is illustrated with a case study of a particular Kaldor business cycle model from the literature that can exhibit rich dynamic behavior, and can give rise to economic sustainability regions that encompass multiple stable equilibrium points. Furthermore, a methodology is also presented that identifies non-rectangular sustainable sets, when the imposition of tight lower and upper bounds on one of the economic system variables (e.g., income) prevents the existence of a sustainable rectangular set.

Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability.

We deduce a one-dimensional model of elastic planar rods starting from the Föppl-von Kármán model of thin shells. Such model is enhanced by additional kinematical descriptors that keep explicit track of the compatibility condition requested in the two-dimensional parent continuum, that in the standard rods models are identically satisfied after the dimensional reduction. An inextensible model is also proposed, starting from the nonlinear Koiter model of inextensible shells. These enhanced models describe the nonlinear planar bending of rods and allow to account for some phenomena of preeminent importance even in one-dimensional bodies, such as formation of singularities and localization (d-cones), otherwise inaccessible by the classical one-dimensional models. Moreover, the effects of the compatibility translate into the possibility to obtain multiple stable equilibrium configurations.

Categorical Screening with Rational Inattention

We consider a rationally inattentive screener who evaluates pools of candidates from distinct and observable social categories. Candidates choose how much effort to invest before being screened, with a payoff in a post-screening market that depends on the screening outcome. This model has multiple stable equilibria, and even ex ante identical categories can receive asymmetric equilibrium treatment. The imposition of a quota, a lower bound for acceptances from a category can result in increases in both screening intensity and skill investment, by destabilizing the least active equilibrium. The endogeneity of screening enables powerful comparative statics analyses with respect to differences across categories in the cost of being screened, the bias faced in the screening process, and the distribution of the costs of investment, allowing us to unify and extend several strands in the literature on categorical inequality.

Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractional-order Cohen–Grossberg neural networks**Project supported by the Natural Science Foundation of Zhejiang Province of China (Grant Nos. LY18F030023, LY17F030016, LQ18F030015, and LY18F020028) and the National Natural Science Foundation of China (Grant Nos. 61503338, 61773348, and 61972354).

This paper addresses the coexistence and local stability of multiple equilibrium points for fractional-order Cohen–Grossberg neural networks (FOCGNNs) with time delays. Based on Brouwer’s fixed point theorem, sufficient conditions are established to ensure the existence of equilibrium points for FOCGNNs. Through the use of Hardy inequality, fractional Halanay inequality, and Lyapunov theory, some criteria are established to ensure the local Lagrange stability and the local Lyapunov asymptotical stability of equilibrium points for FOCGNNs. The obtained results encompass those of integer-order Hopfield neural networks with or without delay as special cases. The activation functions are nonlinear and nonmonotonic. There could be many corner points in this general class of activation functions. The structure of activation functions makes FOCGNNs could have a lot of stable equilibrium points. Coexistence of multiple stable equilibrium points is necessary when neural networks come to pattern recognition and associative memories. Finally, two numerical examples are provided to illustrate the effectiveness of the obtained results.

Hybrid multisynchronization of coupled multistable memristive neural networks with time delays

In this paper, we focus on synchronization issue of coupled multistable memristive neural networks (CMMNNs) with time delay under multiple stable equilibrium states. First, we build delayed CMMNNs consisting of one master subnetwork without controller and N−1 identical slave subnetworks with controllers, and every subnetwork has n nodes. Moreover, this paper investigates multistability of delayed CMMNNs with continuous nonmonotonic piecewise linear activation function (PLAF) owning 2r+2 corner points. By using the theorems of differential inclusion and fixed point, sufficient conditions are derived such that master subnetwork of CMMNNs can acquire (r+2)n exponentially stable equilibrium points, stable periodic orbits or hybrid stable equilibrium states. Then, this paper proposes hybrid multisynchronization of delayed CMMNNs related with various external inputs under multiple stable equilibrium states for the first time. There exist (r+2)n hybrid multisynchronization manifolds in CMMNNs with different initial conditions and external inputs. Finally, two numerical simulations are given to illustrate the effectiveness of the obtained results.