scholarly journals Complex Dynamics of Mixed Triopoly Game with Quantity and Price Competition

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Jing Wang ◽  
Zhenhua Bao ◽  
Junqing Huang ◽  
Yujing Song
Keyword(s):  
Nash Equilibrium ◽  
Price Competition ◽  
Control Method ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Private Firms ◽  
Public Firm ◽  
Feedback Control Method

This article investigates the dynamics of a mixed triopoly game in which a state-owned public firm competes against two private firms. In this game, the public firm and private firms are considered to be boundedly rational and naive, respectively. Based on both quantity and price competition, the game’s equilibrium points are calculated, and then the local stability of boundary points and the Nash equilibrium points is analyzed. Numerical simulations are presented to display the dynamic behaviors including bifurcation diagrams, maximal Lyapunov exponent, and sensitive dependence on initial conditions. The chaotic behavior of the two models has been stabilized on the Nash equilibrium point by using the delay feedback control method. The thresholds under price and quantity competition are also compared.

scholarly journals Analysis of a duopoly game with heterogeneous players participating in carbon emission trading

2014 ◽  
Vol 19 (1) ◽  
pp. 118-131 ◽  
Author(s):  
Lingrui Zhao ◽  
Jixiang Zhang
Keyword(s):  
Nash Equilibrium ◽  
Carbon Emission ◽  
Control Method ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Emission Trading ◽  
Heterogeneous Players ◽  
Stable Conditions ◽  
Carbon Emission Trading

In this paper, a price competition model with two heterogeneous players participating in carbon emission trading is formulated. The stable conditions of the equilibrium points of this system are discussed. Numerical simulations are used to show bifurcation diagrams, strange attractors, and sensitive dependence on initial conditions. We observe that the speed of adjustment of bounded rational player may change the stability of the Nash equilibrium and cause the system to behave chaotically. In addition, we find that the price of emission permits plays an important role in the duopoly game. The chaotic behavior of the system has been stabilized on the Nash equilibrium point by applying delay feedback control method.

scholarly journals On a New Cournot Duopoly Game

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
H. N. Agiza ◽  
A. A. Elsadany ◽  
M. M. El-Dessoky
Keyword(s):  
Nash Equilibrium ◽  
Control Method ◽  
Chaotic Behavior ◽  
Cournot Duopoly ◽  
Nash Equilibrium Point ◽  
Dynamical Behaviors ◽  
Feedback Control Method ◽  
Delay Feedback Control ◽  
Stability And Bifurcations ◽  
Theoretical Results

This paper presents a new Cournot duopoly game. The main advantage of this game is that the outputs are nonnegative for all times. We investigate the complexity of the corresponding dynamical behaviors of the game such as stability and bifurcations. Computer simulations will be used to confirm our theoretical results. It is found that the chaotic behavior of the game has been stabilized on the Nash equilibrium point by using delay feedback control method.

scholarly journals Dynamics Analysis of Game and Chaotic Control in the Chinese Fixed Broadband Telecom Market

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jia Liu ◽  
Guoliang Liu ◽  
Na Li ◽  
Hongliang Xu
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Control Method ◽  
Chaotic Behavior ◽  
Spillover Effect ◽  
Dynamic Game ◽  
Nash Equilibrium Point ◽  
Heterogeneous Players ◽  
Existence And Stability ◽  
Feedback Control Method

This paper considers a dynamic duopoly Cournot model based on nonlinear cost functions. The model with heterogeneous players and the spillover effect is applied to the Chinese fixed broadband telecom market. We have studied its dynamic game process. The existence and stability of the Nash equilibrium of the system have been discussed. Simulations are used to show the complex dynamical behaviors of the system. The results illustrate that altering the relevant parameters of system can affect the stability of the Nash equilibrium point and cause chaos to occur. With the use of the delay feedback control method, the chaotic behavior of the model has been stabilized at the Nash equilibrium point. The analysis and results will be of great importance for the Chinese fixed broadband telecom market.

scholarly journals Complexity Analysis of a 3-Player Game with Bounded Rationality Participating in Nitrogen Emission Reduction

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16 ◽  
Author(s):  
Jixiang Zhang ◽  
Xuan Xi
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Bounded Rationality ◽  
Control Method ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Stable Region ◽  
Nitrogen Emissions ◽  
Nash Equilibrium Point ◽  
The Stability

In this paper, a decision-making competition game model concerning governments, agricultural enterprises, and the public, all of which participate in the reduction of nitrogen emissions in the watersheds, is established based on bounded rationality. First, the stability conditions of the equilibrium points in the system are discussed, and the stable region of the Nash equilibrium is determined. Then, the bifurcation diagram, maximal Lyapunov exponent, strange attractor, and sensitive dependence on the initial conditions are shown through numerical simulations. The research shows that the adjustment speed of three players’ decisions may alter the stability of the Nash equilibrium point and lead to chaos in the system. Among these decisions, a government’s decision has the largest effect on the system. In addition, we find that some parameters will affect the stability of the system; when the parameters become beneficial for enterprises to reduce nitrogen emissions, the increase in the parameters can help control the chaotic market. Finally, the delay feedback control method is used to successfully control the chaos in the system and stabilize it at the Nash equilibrium point. The research of this paper is of great significance to the environmental governance decisions and nitrogen reduction management.

Bifurcations and Chaos in a PD-Controlled Pendulum

1998 ◽  
Vol 120 (1) ◽  
pp. 146-149 ◽  
Author(s):  
Joaqui´n Alvarez ◽  
Fernando Verduzco
Keyword(s):  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Equilibrium Points

The complex dynamics of a pendulum controlled by a Proportional-Derivative (PD) compensator are analyzed. A classification of equilibrium points and the characterization of their bifurcations is also presented. It is shown that the controlled pendulum may exhibit a chaotic behavior when the desired position is periodic and the proportional gain and total dissipation are small enough.

scholarly journals Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley–Martin Functional Response Incorporating Proportional Prey Refuge

2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
P. K. Santra ◽  
G. S. Mahapatra ◽  
G. R. Phaijoo
Keyword(s):  
Functional Response ◽  
Control Method ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Phase Portraits ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
And Control

The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.

scholarly journals Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1803
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Initial Point ◽  
Electrical Circuit ◽  
Nonlinear Resistor ◽  
Voltage Current Characteristic ◽  
Time Waveform

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

Control of Discrete Time Chaotic Systems via Combination of Linear and Nonlinear Dynamic Programming

2014 ◽  
Vol 10 (1) ◽  
Author(s):  
Kaveh Merat ◽  
Jafar Abbaszadeh Chekan ◽  
Hassan Salarieh ◽  
Aria Alasty
Keyword(s):  
Discrete Time ◽  
Control Method ◽  
Controller Design ◽  
Chaotic Behavior ◽  
Noise Signal ◽  
Delayed Feedback Control ◽  
Optimal Controller ◽  
Clustering Method ◽  
Control Approach ◽  
Feedback Control Method

In this article by introducing and subsequently applying the Min–Max method, chaos has been suppressed in discrete time systems. By using this nonlinear technique, the chaotic behavior of Behrens–Feichtinger model is stabilized on its first and second-order unstable fixed points (UFP) in presence and absence of noise signal. In this step, a comparison has also been carried out among the proposed Min–Max controller and the Pyragas delayed feedback control method. Next, to reduce the computation required for controller design, the clustering method has been introduced as a quantization method in the Min–Max control approach. To improve the performance of the acquired controller through clustering method obtained with the Min–Max method, a linear optimal controller is also introduced and combined with the previously discussed nonlinear control law. The resultant combined controller has been applied on the Henon map and through comparison with both Pyragas controller, and the linear optimal controller alone, its advantages are discussed.

scholarly journals Study on Triopoly Dynamic Game Model Based on Different Demand Forecast Methods in the Market

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Junhai Ma ◽  
Lijian Sun ◽  
Xueli Zhan
Keyword(s):  
Control Method ◽  
Complex Dynamics ◽  
Dynamic Game ◽  
Basins Of Attraction ◽  
Attraction Domain ◽  
Stable Region ◽  
Demand Forecast ◽  
Feedback Control Method ◽  
Set Up ◽  
The Impact

The impact of inaccurate demand beliefs on dynamics of a Triopoly game is studied. We suppose that all the players make their own estimations on possible demand with errors. A dynamic Triopoly game with such demand belief is set up. Based on this model, existence and local stable region of the equilibriums are investigated by 3D stable regions of Nash equilibrium point. The complex dynamics, such as bifurcation scenarios and route to chaos, are displayed in 2D bifurcation diagrams, in which e1 and α are negatively related to each other. Basins of attraction are investigated and we found that the attraction domain becomes smaller with the increase in price modification speed, which indicates that all the players’ output must be kept within a certain range so as to keep the system stable. Feedback control method is used to keep the system at an equilibrium state.

scholarly journals Adaptive Control of Chaos in Chua's Circuit

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Weiping Guo ◽  
Diantong Liu
Keyword(s):  
Feedback Control ◽  
Control Method ◽  
Equilibrium Points ◽  
Lyapunov Theory ◽  
Chua’S Circuit ◽  
Control Of Chaos ◽  
Chua's Circuit ◽  
Adaptive Technology ◽  
Feedback Control Method ◽  
And Control

A feedback control method and an adaptive feedback control method are proposed for Chua's circuit chaos system, which is a simple 3D autonomous system. The asymptotical stability is proven with Lyapunov theory for both of the proposed methods, and the system can be dragged to one of its three unstable equilibrium points respectively. Simulation results show that the proposed methods are valid, and control performance is improved through introducing adaptive technology.

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