Equilibrium Solution Research Articles

Bifurcations of a Laminated Circular Cylindrical Shell

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions.

Location and Price Competition on a Uniform Path with Different Pricing Policies

AbstractThis paper models duopolistic competition between an online retailer and a physical store retailer with the online retailer modelled as a firm with uniform delivered pricing policy and the physical store as a firm with a mill pricing policy. Both firms seek to maximize their respective profits through an appropriate choice of location and price. The market is assumed to be given by a “uniform path” - a tree whose node weights and arc lengths are equal. Modelling reality, we consider two alternate types of transportation costs faced by the online retailer: either dependent on the relative location of the firm and a customer or independent of it. Beginning with the framework of a Stackelberg, i.e., sequential game, optimal location and price strategies are analytically derived for both sequences of market entry by the two competitors. Cases in which the leader faces the first entry paradox or can become a monopolist by strategically deterring entry by the follower are delineated. Thereafter, Nash Equilibrium solutions to the simultaneous game are identified. Salient insights from the results include: (a) the competitive pressure faced by the physical store in the presence of online competition (b) the inability of an online retailer to compete in “large” markets under the first type of transportation cost and (c) the advantage to the physical retailer of being a market leader.

Robust distributed Nash equilibrium solution for multi‐agent differential graphical games

AbstractThis paper studies the differential graphical games for linear multi‐agent systems with modelling uncertainties. A robust optimal control policy that seeks the distributed Nash equilibrium solution and guarantees the leader‐following consensus is designed. The weighting matrices rely on modelling uncertainties, leading to the Nash equilibrium solution, and the solution can be obtained by solving a decoupled algebraic Riccati equation. Simulation studies are finally reported to illustrate the effectiveness of proposed policy.

Optimization of steady spin prediction by nonlinear iterative inversion algorithm

PurposeSince the inception of aircraft, the phenomenon of spin has persistently accompanied aircraft, and research into spin has been ongoing. This paper aims to introduce an optimization technique to enhance the traditional geometric method for predicting steady spin, aiming to achieve more precise predictive outcomes.Design/methodology/approachTo begin with, the force and moment equations used for motion analysis are initially presented, followed by the establishment of the motion model. Subsequently, the forward problem is set up, and the equilibrium solutions for the left and right spins of the aircraft are determined using the geometric method under the basic and wingtip configurations, thus solving the forward problem. In the final stage, nonlinear inversion is applied, and the inversion objective function is formulated based on the least squares approach. Through iterative processes, the measured data are interpolated, leading to the acquisition of the accurate equilibrium solution.FindingsThe findings indicate that the utilization of the nonlinear iterative inversion algorithm has effectively optimized the geometric method, yielding favorable outcomes. Postoptimization, the prediction accuracy has been enhanced, and the error has significantly diminished when compared to the preoptimization results.Originality/valueThe nonlinear inversion algorithm is used to refine the steady spin prediction for general aviation aircraft. This approach significantly mitigates the precision issues inherent in the forward problem. As demonstrated through the simulations provided, the application of the nonlinear iterative algorithm to resolve the inversion function yields promising optimization outcomes.

Analysis of Labor Relations in Private Enterprises A Three-party Game Based on Overtime Work

At present, China has successively introduced multiple laws and regulations to prevent employees from being forced to work overtime. However, this situation is still serious in private enterprises, which has led to gradually tense labor relations. Starting from the issue of overtime, this article constructs a game model between government regulatory departments, enterprises, and employees, explores the impact of government regulation on the labor management relationship of private enterprises, and uses evolutionary games to analyze the stability of the strategies of the three parties. The result shows that there is a stable equilibrium solution in the tripartite game: private enterprises arrange employees to work overtime normally, employees take the initiative to work overtime, and the government's project management department strictly supervises them. Among them, strict supervision by labor supervision agencies plays an important role in balancing labor management relations, therefore, certain incentive measures should be given to labor supervision agencies to ensure their strict supervision; Private enterprises should strictly abide by market order, proactively reduce the burden on employees, and avoid forcing employees to work overtime excessively; Employees should maintain a good work attitude and promote the development of common interests with private enterprises.

An uncertainty cognition-based game model for lane-changing process in mixed driving environment

The non-instantaneous nature of lane-changing demands real-time adaptability for autonomous vehicles (AVs) to respond continuously changing traffic conditions. In the mixed environment where AVs coexist with human-driven vehicles (HVs), the lack of inter-vehicle information exchange necessitates the Nash Equilibrium as best response. In addition, the unpredictable intentions of HV introduce uncertainty, posing a challenge for the solution of equilibrium. This paper introduces an aggressiveness parameter reflecting human drivers' yielding tendencies to autonomous vehicles and enables human-like uncertainty cognition during lane changes. To meet the practical solution requirements of the uncertainty cognition-based game model, we propose Proactive Equilibrium Strategy Algorithm (PESA) based on two-stage Nash equilibrium and anticipation of the opponent's next-stage strategy. Utilising Next Generation Simulation (NGSIM) as environmental data, PESA shows safer and more efficient lane-changing behaviour and leads to more favourable post-lane-changing traffic conditions compared to actual data outcomes.

A study on omnichannel retailers' return strategies considering showroom and consumer preference behavior

AbstractTo address omnichannel retailers' return challenges, considering consumers' brand and channel preferences. Four subgame models for retailers' Experience‐in‐Store‐and‐Buy‐Online (ESBO) return strategies in a double oligopoly market are developed. Using an optimization model and inverse induction method, equilibrium strategies under different conditions are determined. Finally, simulation analyzes parameter sensitivity and equilibrium strategies. Findings show that when consumers' brand preference is low, the (Y,Y) subgame is the unique equilibrium solution, but profits do not significantly increase and are always lower than the (N,N) subgame, leading to a Prisoner's Dilemma. With a clear brand preference, both retailers always obtain the maximum profit under the (Y,Y) subgame. In markets with a high proportion of omnichannel consumers and a large number of high‐cost‐to‐store consumers, the party implementing the ESBO strategy benefits more in a fiercely competitive market, while the profit situation of the party not implementing the ESBO strategy is less optimistic.

Competing Manufacturers Adopt Blockchain for Tracing Power Batteries: Is There a Win-Win Zone?

Blockchain-based battery tracking offers solutions to issues like information asymmetry, counterfeit battery risk, and technical barriers in assessing battery condition. This paper aims to identify the drivers behind manufacturers adopting blockchain for battery tracking and assess whether a mutually beneficial outcome exists. We develop a game model featuring two competing manufacturers, and extend it to include asymmetric competition and battery quality considerations. Equilibrium solutions reveal two main incentives for manufacturers to adopt blockchain: reverse profit compensation and enhancement of battery quality. Blockchain traceability facilitates retired battery recovery in a large-scale market, even when adoption costs outweigh reuse savings and collection prices are low. If one manufacturer implements blockchain, reducing blockchain costs or expanding the market can lead to a “win-win” outcome for competitors. Our findings offer novel managerial insights into manufacturers’ blockchain adoption decisions.

A generic approach for network defense strategies generation based on evolutionary game theory

The generation of optimal defense strategies in dynamic adversarial environments is crucial for cybersecurity. Recently, defense approaches based on evolutionary game theory have gained significant achievements. However, they would fail when facing complex networks and sophisticated attack strategies, due to the fatal drawbacks of defense strategy generation considering atomic attacks only. To relieve this issue, a generic approach for generating defense strategies using evolutionary game theory is proposed in this paper. Initially, a novel payoff quantification method for network attack-defense games based on attack graphs is designed. Innovatively, two factors concerning the decision-maker's degree of irrationality (DI) and the level of environmental security (LES) are introduced into the replicator dynamics equation to model the impacts on equilibrium solutions. Noting that Active Directory (AD) domain service is one of the most used and representative information security management system in Windows domains, from which attack graphs and paths can be plainly extracted and analyzed. Therefore, it is necessary and imperative to anchor AD to unfold the theoretical analyses and experiments validation based on a real environment. Case studies on a real-world AD network demonstrate that the proposed approach is effective and can generate stable and efficient defense strategies.

Stability of twisted states in power-law-coupled Kuramoto oscillators on a circle with and without time delay.

Other than the fully synchronized state, a twisted state can also be an equilibrium solution in the Kuramoto model and its variations. In the present work, we explore the stability of the twisted state in Kuramoto oscillators put on a rim of a planar circle in the two-dimensional space in the presence of power-law decaying interaction strength (∼r^{-α} with the distance r) and time delays due to a finite speed of information transfer. For example, our model can phenomenologically mimic a large sports stadium where many people try to sing or clap their hands in unison; the sound intensity decays with the distance and there can exist a time delay proportional to the distance due to the finiteness of sound speed. We first consider the case without the time delay effect and numerically find that stable twisted states emerge when the exponent α exceeds a critical value of α_{c}≈2. In other words, for α<α_{c}, the fully synchronized state, not the twisted state, is the only stable fixed point of the dynamics. In our analytic approach, we also derive an equation for α_{c} and discuss its solutions. In the presence of time delay, we find that it is possible that the synchronized state becomes unstable while twisted states are stable.

Triadic interaction in the background of a pairwise spin-glass.

Developing an equilibrium solution for a pairwise spin-glass with a quenched random infinite range shows a continuous phase transition. Models with p-spin interactions have been studied and the exact solution was provided that shows a continuous phase transition for p=2 and a first-order one for p>2. Although the p-spin interactions were studied individually without considering lower-order interactions, is it always feasible to ignore the lower ones? Here, we are interested in finding an analytical solution for considering a triadic interaction as a perturbation in the background of a pairwise interaction in the Sherrington-Kirkpatrick spin-glass model. Our results indicate a sudden phase transition as a consequence of considering triadic interactions that signal a switch from a continuous to an explosive phase transition.

Distributed time-varying Nash equilibrium in resilient multi-objective formation control for cyber–physical systems

This paper investigates the problem of multi-objective time-varying formation control in cyber–physical systems (CPSs) with saturated constraints and malicious agents. By incorporating the concept of multi-agent systems and employing objective weight allocation method, the problem is reformulated as the seeking for the Nash equilibrium solution in a time-varying non-cooperative game. To evaluate the trustworthiness of agent behavior, we propose an adaptive weight learning algorithm based on reinforcement learning. Furthermore, we develop a novel smooth saturation control strategy and a predefined time controller to design a distributed resilient algorithm for CPSs with both known and unknown parameters. Additionally, we apply the designed algorithm to the resilient distributed multi-objective time-varying formation control problem of unmanned aerial vehicles (UAVs) to verify its effectiveness.

Bifurcation analysis in the regularized four-sided cavity flow: Equilibrium states in a D2-symmetric fluid system

In this study, we investigate a handful of equilibrium states and their hydrodynamic stability for the incompressible flow in a square cavity driven by the tangential simultaneous motion of all of its lids at the same speed. This problem has been investigated in the recent past, albeit only partially and always disregarding the singular boundary conditions at the corners. The implications of the system symmetries have not been rigorously addressed, either. In our study, we introduce a regularized version of the boundary conditions to avoid the corner singularities, in a way that preserves the main features of the original setup. In addition, the Navier–Stokes equations are discretized by means of highly accurate Chebyshev spectral methods that provide exponential convergence of all computed flows and consistently eliminate any potential source of structural instability of the bifurcation scenario. We employ Newton–Krylov solvers, implemented within continuation algorithms, to accurately compute equilibrium solutions. Linear stability analysis of both the primary symmetric base flow and the secondary asymmetric states uncovers new branches of fully asymmetric steady states. The analysis has allowed identification of six previously undetected bifurcations, all of which are associated with the disruption of either the rotational invariance or the reflection symmetry. Some of these bifurcations have been found to be quite clustered in some regions of the parameter space, which points at the underlying action of higher codimension mechanisms. Notably, all the bifurcations reported occur within the range of low to moderate Reynolds numbers, making the regularized four-sided lid-driven cavity flow a reliable benchmark for assessing different numerical schemes in the context of Navier–Stokes equivariant bifurcation theory.

Nash-equilibrium strategies of orbital Target-Attacker-Defender game with a non-maneuvering target

This study investigates the orbital Target-Attacker-Defender (TAD) game problem in the context of space missions. In this game, the Attacker and the Defender compete for a Target that is unable to maneuver due to its original mission constraints. This paper establishes three TAD game models based on the thrust output capabilities: unconstrained thrust output, thrust constrained by an upper bound, and fixed thrust magnitude. These models are then solved using differential game theory to obtain Nash equilibrium solutions for the game problems, and the correctness and effectiveness of the solution methods are verified through simulations. Furthermore, an analysis of the winning mechanisms of the game is conducted, identifying key factors that influence the game’s outcomes, including weight coefficients in payoffs, the maximum thrust acceleration limit, and the initial game state. Considering the unique characteristics of space missions, a specific focus is given to the analysis of the Defender’s initial states in the hovering formation and in-plane circling formation, revealing overall success patterns for defense strategies from these two formations. In summary, this study provides valuable insights into the control strategies and winning mechanisms of orbital TAD games, deepening our understanding of these games and offering practical guidance to improve success rates in real-world scenarios.

Logistics service decision for e-commerce supply chain with fairness concern under uncertain environment

In this paper, we construct a two-echelon supply chain composed of an e-commerce platform and a manufacturer. We investigate the impact of manufacturer’s fairness concern (MFC) on e-commerce supply chain logistics service decisions under an uncertain environment. Given the uncertainty of the market environment, we depict unit cost of product and logistics service elasticity as uncertain variables. We construct four game-theoretical models, i.e. the manufacturer or the platform undertakes logistics service without and with considering the MFC, and compare the equilibrium solutions through numerical analysis. The results demonstrate that providing logistics services through an e-commerce platform increases the profits of all supply chain participants, regardless of whether the MFC is considered or not. Furthermore, we discover that the manufacturer will not always benefit from the MFC, the MFC can decrease the expected profits of supply chain participants. Interestingly, an increase in the level of environmental uncertainty does not necessarily imply a threat to the interests of supply chain participants. Under an uncertain environment, every supply chain participant needs to consider its fairness concern behavior and coordinate with logistics service strategies.

SIRS Epidemic Models with Delays, Partial and Temporary Immunity and Vaccination

The basic reproduction, or reproductive number, is a useful index that indicates whether or not there will be an epidemic. However, it is also very important to determine whether an epidemic will eventually decrease and disappear or persist as an endemic. Different infectious diseases have different behaviors and mathematical models used to simulated them should capture the most important processes; however, the models also involve simplifications. Influenza epidemics are usually short-lived and can be modeled with ordinary differential equations without considering demographics. Delays such as the infection time can change the behavior of the solutions. The same is true if there is permanent or temporary immunity, or complete or partial immunity. Vaccination, isolation and the use of antivirals can also change the outcome. In this paper, we introduce several new models and use them to find the effects of all the above factors paying special attention to whether the model can represent an infectious process that eventually disappears. We determine the equilibrium solutions and establish the stability of the disease-free equilibrium using various methods. We also show that many models of influenza or other epidemics with a short duration do not have solutions with a disappearing epidemic. The main objective of the paper is to introduce different ways of modeling immunity in epidemic models. Several scenarios with different immunities are studied since a person may not be re-infected because he/she has total or partial immunity or because there were no close contacts. We show that some relatively small changes, such as in the vaccination rate, can significantly change the dynamics; for example, the existence and number of the disease-free equilibria. We also illustrate that while introducing delays makes the models more realistic, the dynamics have the same qualitative behavior.

A self-consistent model for dust settling and the vertical shear instability in protoplanetary disks

Abstract The spatial distribution of dust particles in protoplanetary disks affects dust evolution and planetesimal formation processes. The vertical shear instability (VSI) is one of the candidate hydrodynamic mechanisms that can generate turbulence in the outer disk region and affect dust diffusion. Turbulence driven by the VSI has a predominant vertical motion that can prevent dust settling. On the other hand, the dust distribution controls the spatial distribution of the gas cooling rate, thereby affecting the strength of VSI-driven turbulence. Here, we present a semi-analytic model that determines the vertical dust distribution and the strength of VSI-driven turbulence in a self-consistent manner. The model uses an empirical formula for the vertical diffusion coefficient in VSI-driven turbulence obtained from our recent hydrodynamical simulations. The formula returns the vertical diffusion coefficient as a function of the vertical profile of the cooling rate, which is determined by the vertical dust distribution. We use this model to search for an equilibrium vertical dust profile where settling balances with turbulent diffusion for a given maximum grain size. We find that if the grains are sufficiently small, there exists a stable equilibrium dust distribution where VSI-driven turbulence is sustained at a level of αz ∼ 10−3, where αz is the dimensionless vertical diffusion coefficient. However, as the maximum grain size increases, the equilibrium solution vanishes because the VSI can no longer stop the settling of the grains. This runaway settling may explain highly settled dust rings found in the outer part of some protoplanetary disks.

Real-time coordinated control strategy for the hybrid electric propulsion system of a flying car with model adaptation and game theory

Coordination among the multiple power components is essential for the hybrid electric propulsion systems since the decision made by one impacts the state and decisions of others due to the coupling mechanical and electrical dynamics. Traditional model-based approaches with weight-sum objectives may result in misleading optimization or even unstable situations because of the inherent poor scalability and synergy. To overcome this issue, this paper proposes a novel game theory-based control strategy with model adaptation mechanism for the hybrid electric flying car to enhance the control performance, system stability, and robustness. Firstly, a control-oriented model of the system is derived with the utilization of the recursive least square parameter estimation method. The non-cooperative game framework is then established with the turboshaft engine subsystem and electric supply subsystem treated as two independent players. The performance of Nash equilibrium solutions with closed-loop and open-loop information structures are investigated where the problem is iteratively solved by exploiting Pontryagin’s Minimum Principle and dynamic programming, respectively. The simulation results demonstrate that the game theory-based controllers can outperform MPC with the weight-sum objectives in terms of control efficiency and robustness improvement and the game theory-based controller with open-loop information structure shows excellent computation efficiency. Moreover, the result of the hardware-in-the-loop experiment demonstrates the real-time performance of the proposed controller.

Influence maximization based on simplicial contagion models

In recent years, the exploration of node centrality has received significant attention and extensive investigation, primarily fuelled by its applications in diverse domains such as product recommendations, opinion propagation, disease spread, and other scenarios requiring the maximization of node influence. Despite various perspectives emphasizing the indispensability of higher-order networks, research specifically delving into node centrality within the realm of hypergraphs has been relatively constrained. This study delves into the Simplicial Contagion Model (SCM) within the context of influence maximization (IM), specifically utilizing the susceptible-infected-recovered (SIR) model for illustration. To address the optimization challenge associated with IM on SCM, we present a comprehensive theoretical framework grounded in the message passing process. Additionally, we conduct a thorough stability analysis of equilibrium solutions within the self-consistent equations. Furthermore, we introduce a metric called collective influence and propose an adaptive algorithm, known as the Collective Influence Adaptive (CIA), to identify influential propagators in the spreading process. Notably, our algorithm distinguishes itself by prioritizing collective influence over individual influence, resulting in demonstrably superior performance, a characteristic substantiated by a comprehensive array of experiments.

Sourcing from supplier in the presence of financial service providers’ information asymmetry and quit probabilities

PurposeConsidering the financial service providers’ (FSPs) information asymmetry in evaluating the supplier and their distinct quit probabilities, we want to examine the supplier’s preference of the financing schemes if both the bank and the online platform exist and how the buyer sets the contract terms in the two financing schemes.Design/methodology/approachWe establish a Stackelberg game model to capture the interactions among three parties, i.e. a supplier, a capital-sufficient buyer and an FSP (either a bank or an online platform), within a first-time contract.FindingsIn the non-FSPs’ quit case, the buyer’s profit is higher under the bank loan scenario, while the supplier’s profit performs adversely. The supply chain’s profit is heavily dependent on the buyer’s profit difference between the two financing schemes. Moreover, we find that the supplier borrows the money to exactly cover the production cost. The equilibrium solutions of the FSPs’ quit case and of the capital-sufficient supplier’s case are also derived.Originality/valueFirst, we assign different risk profiles to different FSPs in our setting so that modeling a previously ignored but practically significant problem. Second, we innovatively take the FSP’s quit probability into account in our model. Third, we elucidate how these factors can influence the relative efficiency of the two types of financing schemes and the settings of the contract, which further complements and extends the current SCF research.