scholarly journals The Influences of Asymmetric Market Information on the Dynamics of Duopoly Game

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1132 ◽  
Author(s):  
Sameh S. Askar
Keyword(s):  
Asymmetric Information ◽  
Dynamic Characteristics ◽  
Equilibrium Points ◽  
Flip Bifurcation ◽  
Complex Dynamic ◽  
Gradient Based ◽  
Different Shapes ◽  
Attraction Basins ◽  
The Stability ◽  
Periodic Cycles

We investigate the complex dynamic characteristics of a duopoly game whose players adopt a gradient-based mechanism to update their outputs and one of them possesses in some way certain information about his/her opponent. We show that knowing such asymmetric information does not give any advantages but affects the stability of the game’s equilibrium points. Theoretically, we prove that the equilibrium points can be destabilized through Neimark-Sacker followed by flip bifurcation. Numerically, we prove that the map describing the game is noninvertible and gives rise to several stable attractors (multistability). Furthermore, the dynamics of the map give different shapes of quite complicated attraction basins of periodic cycles.

scholarly journals Dynamics and Stability Analysis of a Stackelberg Mixed Duopoly Game with Price Competition in Insurance Market

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Longfei Wei ◽  
Haiwei Wang ◽  
Jing Wang ◽  
Jialong Hou
Keyword(s):  
Nash Equilibrium ◽  
Price Competition ◽  
Equilibrium Points ◽  
Private Insurance ◽  
Insurance Market ◽  
Delayed Feedback Control ◽  
Mixed Duopoly ◽  
Flip Bifurcation ◽  
Insurance Company ◽  
The Stability

This paper investigates the dynamical behaviors of a Stackelberg mixed duopoly game with price competition in the insurance market, involving one state-owned public insurance company and one private insurance company. We study and compare the stability conditions for the Nash equilibrium points of two sequential-move games, public leadership, and private leadership games. Numerical simulations present complicated dynamic behaviors. It is shown that the Nash equilibrium becomes unstable as the price adjustment speed increases, and the system eventually becomes chaotic via flip bifurcation. Moreover, the time-delayed feedback control is used to force the system back to stability.

scholarly journals Asymmetric Information on Price Can Affect Bertrand Duopoly Players with the Gradient-Based Mechanism

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
S. S. Askar
Keyword(s):  
Asymmetric Information ◽  
Equilibrium Points ◽  
Classical Model ◽  
Market Price ◽  
Basins Of Attraction ◽  
Time Step ◽  
Interior Equilibrium ◽  
Proposed Model ◽  
Gradient Based ◽  
Parameter Values

We study a Bertrand duopoly game in which firms adopt a gradient-based mechanism to update their prices. In this competition, one of the firms knows somehow the price adopted by the other firm next time step. Such asymmetric information of the market price possessed by one firm gives interesting results about its stability in the market. Under such information, we use the bounded rationality mechanism to build the model describing the game at hand. We calculate the equilibrium points of the game and study their stabilities. Using different sets of parameter values, we show that the interior equilibrium point can be destabilized through flip and Neimark–Sacker bifurcations. We compare the region of stability of the proposed model with a classical Bertrand model without asymmetric information. The results show that the proposed game’s map is noninvertible with type Z 0 − Z 2 or Z 1 − Z 3 , while the classical model is of type Z 0 − Z 2 only. This explains the quite complicated basins of attraction given for the proposed map.

scholarly journals A Dynamic Duopoly Cournot Model with R&D Efforts and Its Dynamic Behavior Analysis

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Wei Zhou ◽  
Jie Zhou ◽  
Tong Chu ◽  
Hui Li
Keyword(s):  
Dynamic Behavior ◽  
Equilibrium Points ◽  
Flip Bifurcation ◽  
Nash Equilibrium Point ◽  
Dynamical Behaviors ◽  
Coexistence Of Attractors ◽  
Route To Chaos ◽  
The Stability ◽  
The Given ◽  
Jury Criterion

In this paper, a dynamic two-stage Cournot duopoly game with R&D efforts is built. Then, the local stability of the equilibrium points are discussed, and the stability condition of the Nash equilibrium point is also deduced through Jury criterion. The complex dynamical behaviors of the built model are investigated by numerical simulations. We found that the unique route to chaos is flip bifurcation, and the increase of adjusting speed will cause the system to lose stability and produce more complex dynamic behavior. In addition, we also found the phenomenon of multistability in the given model. Several kinds of coexistence of attractors are shown. In particular, we found that boundary attractors can coexist with internal attractors, which also aggravates the complexity of the system. At last, the chaotic state in the built system has been successfully controlled.

scholarly journals The asymptotic behavior of a dynamical process coming from the development in continuous fractions

2020 ◽  
Vol 34 ◽  
pp. 03003
Author(s):  
Maria-Liliana Bucur ◽  
Cristina-Gabriela Cerbulescu
Keyword(s):  
Asymptotic Behavior ◽  
Iterative Process ◽  
Continued Fraction ◽  
Equilibrium Points ◽  
Dynamical Process ◽  
Integer Part ◽  
Attraction Basins ◽  
Infinite Continued Fraction ◽  
The Stability

The aim of this paper is the study of a dynamical process generated by a sequence of maps: $${x_{n + 1}} = {f_n}\left( {{x_n}} \right)$$ where $${f_n}{\rm{ : }}\left( {0,\infty } \right){\rm{ }} \to \left( {0,\infty } \right){\rm{, }}{f_n}{\rm{ }}\left( x \right){\rm{ = }}{{{c_n}} \over {1 + x}}{\rm{ for all }}n{\rm{ }} \in {\rm{ }}N{\rm{ and }}{\left( {{c_n}} \right)_n}$$ is a a sequence of positive numbers. This process is generated similar to continuous fractions development. A continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the inverse of another number, then writing this other number as the sum of its integer part and another inverse, and so on. In a finite continued fraction (or terminated continued fraction), the iteration is terminated after finitely many steps by using an integer in stead of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers are called the coefficients or terms of the continued fraction. We will study the pre-equilibrium points for this process, the attraction basins and the stability.

scholarly journals Order and Chaos in a Prey-Predator Model Incorporating Refuge, Disease, and Harvesting

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Dahlia Khaled Bahlool ◽  
Huda Abdul Satar ◽  
Hiba Abdullah Ibrahim
Keyword(s):  
Equilibrium Points ◽  
Global Dynamics ◽  
Persistence Conditions ◽  
Local Bifurcation Analysis ◽  
Uniqueness Of The Solution ◽  
Harvesting Process ◽  
Predator Model ◽  
The Stability ◽  
Predator System ◽  
Type Ii Functional Response

In this paper, a mathematical model consisting of a prey-predator system incorporating infectious disease in the prey has been proposed and analyzed. It is assumed that the predator preys upon the nonrefugees prey only according to the modified Holling type-II functional response. There is a harvesting process from the predator. The existence and uniqueness of the solution in addition to their bounded are discussed. The stability analysis of the model around all possible equilibrium points is investigated. The persistence conditions of the system are established. Local bifurcation analysis in view of the Sotomayor theorem is carried out. Numerical simulation has been applied to investigate the global dynamics and specify the effect of varying the parameters. It is observed that the system has a chaotic dynamics.

scholarly journals Nonlinear Dynamics of a PI Hydroturbine Governing System with Double Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Hongwei Luo ◽  
Jiangang Zhang ◽  
Wenju Du ◽  
Jiarong Lu ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Bifurcation Theorem ◽  
Center Manifold Theorem ◽  
Governing System ◽  
Normal Form Method ◽  
The Stability ◽  
System Frequency ◽  
Realistic Significance ◽  
Theoretical Results

A PI hydroturbine governing system with saturation and double delays is generated in small perturbation. The nonlinear dynamic behavior of the system is investigated. More precisely, at first, we analyze the stability and Hopf bifurcation of the PI hydroturbine governing system with double delays under the four different cases. Corresponding stability theorem and Hopf bifurcation theorem of the system are obtained at equilibrium points. And then the stability of periodic solution and the direction of the Hopf bifurcation are illustrated by using the normal form method and center manifold theorem. We find out that the stability and direction of the Hopf bifurcation are determined by three parameters. The results have great realistic significance to guarantee the power system frequency stability and improve the stability of the hydropower system. At last, some numerical examples are given to verify the correctness of the theoretical results.

Dynamic analysis and control of an active engine mount system

2002 ◽  
Vol 216 (11) ◽  
pp. 921-931 ◽  
Author(s):  
Y-W Lee ◽  
C-W Lee
Keyword(s):  
Transfer Function ◽  
Dynamic Characteristics ◽  
Performance Test ◽  
Adaptive Controller ◽  
Lms Algorithm ◽  
Equivalent Mass ◽  
Engine Mount ◽  
Active Engine ◽  
The Stability ◽  
Mass Spring

Dynamic characteristics of a prototype active engine mount (AEM), designed on the basis of a hydraulic engine mount, have been investigated and an adaptive controller for the AEM has been designed. An equivalent mass-spring-damper AEM model is proposed, and the transfer function that describes the dynamic characteristics of the AEM is deduced from mathematical analysis of the model. The damping coefficient of the model is derived by considering the non-linear flow effect in the inertia track. Experiments confirmed that the model precisely describes the dynamic characteristics of the AEM. An adaptive controller using the filtered-X LMS algorithm is designed to cancel the force transmitted through the AEM. The stability of the LMS algorithm is guaranteed by using the secondary path transfer function derived on the basis of the dynamic model of the AEM. The performance test in the laboratory shows that the AEM system is capable of significantly reducing the force transmitted through the AEM.

On the stability of a mathematical model for HIV(AIDS) - cancer dynamics

2021 ◽  
Vol 8 (4) ◽  
pp. 783-796
Author(s):  
H. W. Salih ◽  
◽  
A. Nachaoui ◽  
Keyword(s):  
Mathematical Model ◽  
Highly Active Antiretroviral Therapy ◽  
Lyapunov Method ◽  
Equilibrium Points ◽  
Hiv Treatment ◽  
Biochemical Processes ◽  
Highly Active ◽  
Biological Phenomena ◽  
The Stability ◽  
The Impact

In this work, we study an impulsive mathematical model proposed by Chavez et al. [1] to describe the dynamics of cancer growth and HIV infection, when chemotherapy and HIV treatment are combined. To better understand these complex biological phenomena, we study the stability of equilibrium points. To do this, we construct an appropriate Lyapunov function for the first equilibrium point while the indirect Lyapunov method is used for the second one. None of the equilibrium points obtained allow us to study the stability of the chemotherapeutic dynamics, we then propose a bifurcation of the model and make a study of the bifurcated system which contributes to a better understanding of the underlying biochemical processes which govern this highly active antiretroviral therapy. This shows that this mathematical model is sufficiently realistic to formulate the impact of this treatment.

Bifurcation and stability of resonance of electric vehicle powertrain: Focus on the influence of electromagnetic parameters

2021 ◽  
pp. 095440702110450
Author(s):  
Qingzhen Han ◽  
Shiqin Niu ◽  
Lei He
Keyword(s):  
Electric Vehicle ◽  
Electromechanical Coupling ◽  
Equilibrium Points ◽  
Resonance Curve ◽  
Drive Shaft ◽  
Electromagnetic Parameters ◽  
Stability And Bifurcation ◽  
Fold Bifurcation ◽  
The Stability ◽  
Vehicle Powertrain

The influence of the electromagnetic parameters on the torsional dynamics of the electric vehicle powertrain is studied by considering the electromechanical coupling effect. By adding the electromagnetic torque on the drive side, the powertrain is simplified as nonlinear drive-shaft model. The number, stability, and bifurcation conditions of the equilibrium points of the nonlinear drive-shaft model are deduced. Based on the averaged equations and the amplitude-frequency response equation, the stability and bifurcation conditions, such as fold bifurcation and Hopf bifurcation, of the resonance curve are discussed. The influence of electromagnetic parameters on the torsional dynamics is studied by simulation. It is shown that with the change of the parameters, the number as well as the stability of the equilibrium points may be changed which is affected by fold bifurcation. It is also shown that the resonance curve may lose its stability when fold bifurcation happens. By limiting the parameters in the region without fold bifurcation, the unstable dynamics of the resonance curve can be controlled.

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