Equilibrium Solution Research Articles

An explicit well-balanced scheme on staggered grids for barotropic Euler equations

In this paper, we introduce a specific modification of the numerical fluxes in order to insure the well-balanced property of schemes on staggered grids for the Euler equations. This property is crucial for the numerical representation of equilibrium solutions of balance laws with source terms, like when describing flows subjected to gravity and a complex topography. We propose first and second order versions of the well-balanced scheme. The performances of the method are evaluated through a series of 1D and 2D benchmarks.

Stochastic adaptive linear quadratic nonzero-sum differential games

This paper focuses on solving stochastic linear quadratic nonzero-sum differential games with completely unknown system matrices and long-time average costs. Firstly, for the case that the system matrices are known, we design a model-based value iteration (VI) algorithm and a model-based robust VI algorithm to solve the coupled algebraic Riccati equations (AREs), which are directly related to the construction of the feedback Nash equilibrium solution. Then, we present a model-free VI algorithm, based on which feedback control strategies for players are designed using only the data of states and inputs, without requiring the prior knowledge of the system matrices. Moreover, we show that the designed feedback control strategies make the closed-loop game system stable and reach the Nash equilibrium. Finally, a numerical simulation is given to demonstrate the effectiveness of the proposed method.

Dynamic Bayesian game optimization method for multi-energy hub systems with incomplete load information

The interconnection of Energy Hub can enhance the energy efficiency and reliability associated with the independent operation of energy systems. However, the traditional optimal scheduling methods are hard to tackle the situation with the incomplete load information and competitive constraints of a multi-Energy Hub. This paper proposes a dynamic Bayesian game optimization scheduling strategy considering incomplete information on users’ load-side demand. First, a load forecast error coefficient is introduced to handle the conditional probability problem caused by incomplete information through the Bayesian rule. The constraint relationship among multi-Energy Hub under the dynamic price mechanism is analyzed to meet the economic and environmental coordination goals under load uncertainty. Second, extending from Nash equilibrium to Bayes Nash equilibrium proves the existence and uniqueness of Nash equilibrium solutions in Bayesian game models. The decision game algorithm is used to optimize the scheduling of the multi-Energy Hub and subsequently ensure the autonomous scheduling of the system and solve the information barriers. Finally, an Integrated Energy System composed of three Energy Hubs was used to validate the effectiveness and superiority of the proposed optimization method. Results show that under the Bayesian game decision, the multi-Energy Hub game model has reduced the total cost of the Integrated Energy System by 0.78 %, which can improve the economic benefit and address the environmental pollution problem.

[formula omitted]-player game formulation of the majority-vote model of opinion dynamics

From a self-centered perspective, it can be assumed that people only hold opinions that can benefit them. If opinions have no intrinsic value, and acquire their value when held by the majority of individuals in a discussion group, then we have a situation that can be modeled as an N-player game. Here we explore the dynamics of (binary) opinion formation using a game-theoretic framework to study an N-player game version of Galam’s local majority-vote model. The opinion dynamics is modeled by a stochastic imitation dynamics in which the individuals copy the opinion of more successful peers. In the infinite population limit, this dynamics is described by the classical replicator equation of evolutionary game theory. The equilibrium solution shows a threshold separating the initial frequencies that lead to the fixation of one opinion or the other. A comparison with Galam’s deterministic model reveals contrasting results, especially in the presence of inflexible individuals, who never change their opinions. In particular, the N-player game predicts a polarized equilibrium consisting only of extremists. Using finite-size scaling analysis, we evaluate the critical exponents that determine the population size dependence of the opinion’s fixation probability and mean fixation times near the threshold. The results underscore the usefulness of combining evolutionary game theory with opinion dynamics and the importance of statistical physics tools to summarize the results of Monte Carlo simulations.

Nonlinear Stability of Equilibria in Hamiltonian Systems with Multiple Resonances without Interactions

In this work, we advance in the study of the Lyapunov stability and instability of equilibrium solutions of Hamiltonian flows. More precisely, we study the nonlinear stability in the Lyapunov sense of equilibrium solutions in autonomous Hamiltonian systems with n-degrees of freedom, assuming the existence of two resonance vectors k1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{k}^1$$\\end{document} and k2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{k}^2$$\\end{document} without interaction (|k1|≤|k2|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\ extbf{k}^1|\\le |\ extbf{k}^2|$$\\end{document}). We provide conditions to obtain a type of formal stability, called Lie stability. In particular, we need to normalize the Hamiltonian function to any arbitrary order, and our results take into account the sign of the components of the resonance vectors. Subsequently, we guarantee some sufficient conditions to obtain exponential stability in the sense of Nekhoroshev for Lie stable systems. In addition, we give sufficient conditions for the instability in the Lyapunov sense of the full system. For this, it is necessary to normalize the Hamiltonian function to an adequate order, and assuming that the components of at least one resonance vector change of sign.

Critic learning control via zero-sum differential game for affine formation maneuver of multi-agent systems with cyber-attacks

The affine formation maneuver (AFM) control problem using the zero-sum differential game is studied for a second-order multi-agent system (MAS) against cyber-injection attacks on the actuators. First and foremost, the distributed control signals and the cyber-injected attack signals are viewed as two competing teams. Then, the leader-follower tracking error system is established where the followers aim to track the AFMs of the leaders. The leader-follower distributed optimal control policies and the cyber-attack attenuation policies are derived from the Nash equilibrium solution of the Hamilton-Jacobi-Isaac (HJI) equation. The solution of the HJI equation is estimated with the aid of the critic neural network. The weights of the critic system are learned online based on the adaptive dynamic programming (ADP) scheme. The uniformly ultimately bounded stability of the closed-loop system under the formulated zero-sum differential game control strategy has been proven by the Lyapunov stability theorem. A simulation example illustrates that the presented control method can attenuate cyber-injection attacks on the actuators and maintain the AFMs of the agents.

The Incentive to Join in Customer-to-Manufacturer (C2M) Mode: Competition and Quality Differentiation

Customer-to-Manufacturer (C2M) is an emerging business mode in which the platform analyzes customers’ information and provides suggestions to manufacturers so that the manufacturers can understand customers’ needs much better. This paper investigates C2M by considering two competing firms that sell products to customers through an e-commerce platform. One produces high-quality products (firm H), and the other produces low-quality products (firm L). The platform sells a product improvement report to the firms, each firm decides whether or not to buy the report and join in C2M mode. The study results show that buying the product improvement report gives each firm the privilege of setting a higher product price. However, such privileges are weakened when both firms buy the reports. When the report price is small, the Nash equilibrium solution is that both firms buy the product improvement report, although it is not individually optimal for each firm. It indicates that the report price influences the firms’ profits and incentive to engage in C2M mode and one firm’s purchasing decision affects the other. Interestingly, we find that a moderate report price may make firm H worse off but benefit firm L. It indicates that the C2M mode is not always appealing to firms.

Solid-Liquid Phase Equilibria of a Quaternary Sulfate-Type System Containing Lithium, Rubidium, and Magnesium at 298.2 K.

The aim of this study was to obtain the relationship between ion interactions and the crystallization patterns of salt species in the lithium-rubidium-magnesium sulfate system at 298.2 K. The phase equilibria of the aqueous quaternary system Li+, Rb+, Mg2+//SO42--H2O were studied by the isothermal dissolution method at T = 298.2 K and p = 94.77 kPa. The density, refractive index, and composition of equilibrium solution were determined, on the basis of which solid-liquid phase diagrams and density/refractive index vs composition diagrams were drawn. The phase diagram consists of four quaternary invariant points and six crystallization regions, corresponding to the crystallization areas of single salts Rb2SO4, Li2SO4·H2O, and MgSO4·7H2O, as well as double salts 3Li2SO4·Rb2SO4·2H2O, Li2SO4·Rb2SO4, and Rb2SO4·MgSO4·6H2O. Notably, rubidium-containing double salts occupy more than 50% of the entire phase diagram area. The results indicate that the interactions between Li+ and Rb+ with coexisting Mg2+ and SO42- are complex, leading to the formation and precipitation of various lithium- and rubidium-bearing double salts, which hinder the effective concentrations of lithium and rubidium during the solar evaporation process in salt pans. Additionally, a multitemperature comparison of the solid-liquid phase diagrams at 273.2, 298.2, and 308.2 K reveals that temperature is also a significant factor influencing the solid-phase types and crystallization areas. For instance, the crystallization form of the double salt 3Li2SO4·Rb2SO4·2H2O changes to 3Li2SO4·Rb2SO4 at 308.2 K and the crystallization area of Li2SO4·Rb2SO4 gradually decreases, while the crystallization area of Rb2SO4·MgSO4·6H2O generally exhibits an increasing trend.

Reducing carbon emissions in prefabricated buildings supply chains: a focus on component manufacturing processes.

The construction sector accounts for 23% of emissions from global economic activity, with China responsible for nearly 41%. Although China has vigorously promoted the development of prefabricated buildings (PBS) in pursuit of cleaner production, the carbon emissions from prefabricated component factories (PCF) should not be underestimated. So, the focus of this research was on how to promote the decarbonization of PCF. Based on the carbon trading market mechanism, the carbon emissions trading tax and revenue tax collection, the authors established a differential game model consisting of the local government and the PCF, studied the equilibrium solutions under different decision models, and analyzed the roles of the two tax systems, carbon trading revenue, and market preferences. The results are as follows: (1) The PCF's low-carbon technology (LCT) innovation efforts can be directly affected by the carbon price, component price, and tax rate and indirectly affected by influencing the local government's efforts. Besides, the local government and the PCF strategies can be changed through the central government's regulation of carbon prices and tax rates. (2) PCF should set reasonable prices for components, improving economic efficiency and the LCT stocks. (3) Cost-sharing contracts can encourage PCF to increase their LCT innovation, which is conducive to increasing the optimal benefits of the PCF. (4) The local government cannot be motivated by cost-sharing contracts. They can increase their optimal benefits only if the cost-sharing coefficient is below a threshold or if the environmental benefits from low-carbon production are above a specific value.

Nonlinear vibrations, bifurcations and chaos of piezoelectric composite lattice sandwich plate with four simply supported edges

This paper is devoted to investigate the nonlinear dynamics, bifurcation and chaos of the piezoelectric composite lattice sandwich plate with 1:3 internal resonance. Galerkin method is used to discretize the nonlinear partial differential governing equations of motion into the ordinary differential equations. The multi-scale method is used to obtain the modulation equations of the piezoelectric composite lattice sandwich plate in the polar coordinates and Cartesian coordinates in the primary and parametric resonances, respectively. Using Newton-Raphson method, the bifurcation diagrams of the steady-state equilibrium solution are obtained for the modulation equation with the change of the system parameters. The stability of the steady-state equilibrium solution is analyzed for the modulation equation. The existence of the static bifurcations, such as the saddle-node bifurcation, pitchfork bifurcation and Hopf dynamic bifurcations are found. The complex nonlinear jump phenomena caused by the existence of the multiple nonlinear steady-state solution regions are analyzed in detail. Fourth-order Runge-Kutta method is used to continue tracing the nonlinear periodic solution of the modulation equation in Cartesian coordinates. The time-history diagram, phase portrait and Poincaré map are obtained for the piezoelectric composite lattice sandwich plate in the nonlinear resonance. The dynamic process of the nonlinear periodic solution is described for the modulation equation from the limit cycle oscillation to the chaotic attractor in detail.

Game theory-based optimization for efficient IoT task offloading in 6G network base stations

This study aims to optimise computing for intricate jobs within the overlapping coverage of 6G network base stations. A multi-access edge computing network model is created by solving the issues of task offloading. This model involves many base stations and IoT devices. It takes into account factors such as task delay, energy consumption, societal impacts, and economic incentives. The objective of joint optimization is to maximize the revenues of base stations and the utilities of IoT devices by considering base station price, IoT device base station selection, and job offloading mechanisms. The IoT device base station selection is addressed using a many-to-one matching game model, while pricing and task offloading interactions are managed by a two-stage approach based on Stackelberg game theory. The suggested Optimal Pricing and Best Response Algorithm (OBGT), which is based on Game Theory, successfully reaches equilibrium solutions. It demonstrates rapid convergence in simulations and improves both base station profits and IoT device utility. This study integrates data-driven mobile computing systems assurance for futuristic smart vertical networks, enhancing the efficiency of task offloading and optimization.

A new method for solving the linearized 1D Vlasov–Poisson system yielding a new class of solutions

We describe a new method for solving the linearized 1D Vlasov–Poisson system by using properties of Cauchy-type integrals. Our method remedies critical flaws of the two standard methods, reveals a previously unrecognized Gaussian-in-time-like decay, and can also account for an externally applied electric field. The Landau approximation involves deforming the Bromwich contour around the poles closest to the real axis due to the analytically continued dielectric function, finding the long-time behavior for a stable system: Landau damping. Jackson's generalization encircles all poles while sending the contour to infinity, assuming its contribution vanishes, which is not true in general. This gives incorrect solutions for physically reasonable configurations and can exhibit pathological behavior, of which we show examples. The van Kampen method expresses the solution for a stable equilibrium as a continuous superposition of waves, resulting in an opaque integral. Case's generalization includes unstable systems and predicts a decaying discrete mode for each growing discrete mode, an apparent contradiction to both the Jackson solution and ours. We show, without imposing additional constraints, that the decaying modes are never present in the time evolution due to an exact cancellation with part of the continuum. Our solution is free of integral expressions, is obtained using algebra and Laurent series expansions, does not rely on analytic continuations, and results in a correct asymptotically convergent form in the case of infinite sums. The analysis used can be readily applied in higher-dimensional, electromagnetic systems and also provides a new technique for evaluating certain inverse Laplace transforms.

Low-carbon economic operation strategy for multi-agent integrated energy system considering uncertainty of renewable energy power generation

The uncertainty of renewable energy output threatens the operation safety of multi-agent integrated energy system (MAIES), which makes it difficult to balance the low-carbon economic operation demands of various stakeholders. However, the existing research solely focuses on the operational strategy of multi-agent game involving integrated energy suppliers and users in deterministic scenarios, overlooking the complementary supporting role and game interaction of shared energy storage and wind farm as independent entities of interest under the instability of renewable energy power generation. Hence, this paper first establishes the optimal operation models for integrated energy system operator (IESO), user aggregator (UA), shared energy storage operator (SESO), and wind farm operator (WFO) considering the stepped carbon trading. Second, in the face of the actual situation of uncertainty of photovoltaic and wind power output, fuzzy chance-constrained programming is adopted for processing. Then, a bi-layer game equilibrium model with IESO as a leader and UA, SESO, and WFO as followers is proposed, and the existence and uniqueness of Stackelberg equilibrium solution are proved. Finally, simulation calculation is carried out based on the YALMIP toolbox in the Matlab R2023a software, and the improved particle swarm optimization algorithm and CPLEX solver are used to solve the model. The results demonstrate that the participation of SESO and WFO as independent stakeholders in the game interaction can improve the economic and environmental benefits of MAIES. The iterative optimization of demand response subsidy prices can effectively motivate users to participate in demand response, improve the ability of MAIES to cope with the uncertain risks of renewable energy generation and load, and reduce the power grid dispatch pressure.

Safe tracking in games: Achieving optimal control with unknown dynamics and constraints

AbstractThis paper introduces mix‐zero‐sum differential (MZSD) game theory to address multi‐player tracking systems, offering a better understanding of the coexistence of cooperation and competition among players. Within this framework, we present an optimal safety tracking control (OSTC) method, which incorporates a control barrier function (CBF) into the value function to ensure that the tracking error remains within a specified range, thus guaranteeing safety while achieving optimization. Simultaneously, to eliminate the need for system dynamics, we propose a novel approach leveraging off‐policy integral reinforcement learning (IRL) technology to obtain the Nash equilibrium solution of the MZSD games. We establish a unique critics–actors neural network (NN) structure that updates concurrently. Furthermore, we analyze stability and convergence using the Lyapunov method. We conduct two simulations to demonstrate the effectiveness of the proposed algorithm.

Stability of viscoelastic film on a slippery inclined plane

The linear and weakly nonlinear stability of viscoelastic film flowing down a slippery inclined plane is investigated analytically. Under the assumption of the long wave approximation, the first-order Benny equation of Oldroyd-B fluid thin film with slip condition is obtained. Through the normal mode analysis, the neutral stability curve and the temporal growth rates are calculated to explore the linear stability of the film. Linear results show that the critical Reynolds number decreases with the increase in slip length and viscoelastic parameter and that the liquid film may exhibit pure elastic instability. For the nonlinear stability analysis, both hydrodynamic instability and elastic instability are discussed. The primary bifurcations in the phase plane are identified by calculating the Landau coefficient, i.e., the unconditional stable region, the supercritical region, the subcritical region, and the explosive region. The dependence of primary bifurcation regions upon the slip length and Deborah number are studied, and the results indicate that the slip boundary and viscoelasticity destabilizes the flow. According to the Ginzburg–Landau equation, the threshold amplitude of the nonlinear equilibrium solution is analyzed as well.

Dynamic bargaining game DEA carbon emissions abatement allocation and the Nash equilibrium

Allocating carbon emissions abatement (CEA) is crucial for reducing carbon footprint of organizations (or decision-making units, DMUs) and mitigating global warming. However, discrepancies often arise among DMUs regarding CEA allocation. This study introduces a dynamic bargaining game approach to address the issue under the nonparametric production frontier analysis framework. Our approach employs an iterative process, allowing each DMU to propose its individual preferred CEA allocation proposal in each iteration. Subsequently, dynamic negotiations among the DMUs occur, leading to the eventual consensus on the CEA allocation result. Our theoretical analysis demonstrates that, at the end of the dynamic bargaining game, all DMUs will converge on the same CEA allocation result, termed as the consensus CEA allocation result. This agreement is reached despite each DMU customizing its individual CEA allocation proposal to suit its own interests. Moreover, we prove that this consensus CEA allocation result represents a Nash equilibrium solution, thereby ensuring its stability and acceptance among all DMUs. Finally, we provide a simple numerical example and a case study of CEA allocation across 27 European Union countries to illustrate the usefulness of our approach and compare it with prior CEA allocation approaches.

A unified framework for simplicial Kuramoto models

Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: “simple” models, “Hodge-coupled” models, and “order-coupled” (Dirac) models. Our framework is based on topology and discrete differential geometry, as well as gradient systems and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models.

A system dynamics-based model for the evolution of environmental group events

Based on the system dynamics theory, this paper establishes an environmental mass event evolution model and explores the evolution law of mass events caused by environmental problems. From a methodological point of view, the mixed-strategy evolutionary game principle and dynamic punishment measures are combined, and simulation analysis is carried out by Anylogic software, and the results show that there is no stable evolutionary equilibrium solution for the two sides of the game in the traditional asymmetric mixed-strategy game model, and after adjusting the game payoff matrix and incorporating the dynamic punishment strategy, stable evolutionary equilibrium solutions appear in the evolutionary game model, and the system begins to tend to be stabilized. The process and conclusions of the simulation experiment provide methodological reference and theoretical support for the analysis of the evolution of environmental mass events.

Systematic Literature Review: Analysis of Misconception Problems and Diagnostic Instruments for Learning Chemistry

This research aims to analyze research trends in Chemical misconceptions in chemistry learning and diagnostic instruments used. The research method used in this research is a systematic literature review (SLR). The database obtained was 16 articles selected from the Scopus and Google Scholar databases with Publish or Perish (PoP). The results of the research found were the majority of students experienced misconceptions in learning chemistry caused by errors in preconceptions and abstract thinking concepts in the material, the method used to analyze the misconceptions found was a two-level, three-level, four-level diagnostic instrument, multiple choice, and semi-open tests and The misconceptions found in chemistry materials are acid-base, reaction rate, chemical equilibrium, chemical bonding, salt hydrolysis, and buffer solution.

Responding to consumer environmental awareness: A variational inequality approach to green manufacturing optimization

In response to increasing consumer environmental awareness, our study introduces a sustainable supply chain optimization model that stimulates competition among multiple green manufacturers within an uncertain market context. Each manufacturer crafts a selection of products showcasing various degrees of greenness, purposefully designed to meet the expanding demand of environmentally conscious consumers. This strategic model emphasizes profit maximization for green manufacturers, necessitating the definition of ideal greenness levels and production quantities for an array of product types. By employing variational inequality theory, we transmute the preliminary optimization model into an interconnected variational inequality problem. We propose an improved Euler algorithm, a proficient tool for attaining equilibrium solutions. Empirical scenarios inspired by the electric vehicle industry lend credibility to our model, indicating that the surge in consumer environmental awareness compels manufacturers to diversify their offerings and accentuate their products’ greenness levels. Intriguingly, the research suggests that escalating competition can incentivize manufacturers to elevate their products’ greenness levels, strategically countering potential profit erosion. In addition, the study reveals that intensifying consumers’ green utility sensitivity doesn’t invariably augment demand for manufacturers, even with significant improvements in product greenness. This paradox highlights the necessity for manufacturers to strategically adjust their production approaches in light of varying consumer preferences. Moreover, when the level of consumer environmental awareness achieves a certain level, the fierce competition might not always be harmful to some manufacturers and the industry if they can adopt effective strategies to improve green utility sensitivity.