scholarly journals THE LONG RUN OUTCOMES AND GLOBAL DYNAMICS OF A DUOPOLY GAME WITH MISSPECIFIED DEMAND FUNCTIONS

2004 ◽  
Vol 06 (03) ◽  
pp. 343-379 ◽  
Author(s):  
GIAN-ITALO BISCHI ◽  
CARL CHIARELLA ◽  
MICHAEL KOPEL
Keyword(s):  
Demand Function ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Steady States ◽  
Model Parameters ◽  
Global Dynamics ◽  
Market Demand ◽  
Economic Significance ◽  
Reaction Functions ◽  
Long Run

In this paper we study a model of a quantity-setting duopoly market where firms lack knowledge of the market demand. Using a misspecified demand function firms determine their profit-maximizing choices of their corresponding perceived market game. For illustrative purposes we assume that the (true) demand function is linear and that the reaction functions of the players are quadratic. We then investigate the global dynamics of this game and characterize the number of steady states and their welfare properties. We study the basins of attraction of these steady states and present situations in which global bifurcations of their basins occur when model parameters are varied. The economic significance of our result is to show that in situations where players choose their actions based on a misspecified model of the environment, additional self-confirming steady states may emerge, despite the fact that the Nash-equilibrium of the game under perfect knowledge is unique. As a consequence the long run outcome of the game and overall welfare is highly dependent upon initial conditions.

scholarly journals Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Toufik Khyat ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Initial Conditions ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
The Difference

In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation xn+1=fxn,xn−1, n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f, n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f>0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.

BIFURCATION STRUCTURE AT 1/3-SUBHARMONIC RESONANCE IN AN ASYMMETRIC NONLINEAR ELASTIC OSCILLATOR

1996 ◽  
Vol 06 (08) ◽  
pp. 1529-1546 ◽  
Author(s):  
G. REGA ◽  
A. SALVATORI
Keyword(s):  
Invariant Manifolds ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Subharmonic Resonance ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Bifurcation Structure ◽  
Response Curves ◽  
Nonlinear Elastic ◽  
Insight Into

The attractor-basin bifurcation structure in an asymmetric nonlinear oscillator representative of the planar finite forced dynamics of elastic structural systems with initial curvature is studied at the 1/3-subharmonic resonance regime. Local and global analyses are made by means of different computational tools to obtain frequency-response curves of coexisting regular solutions, bifurcation diagrams ensuing from different sets of initial conditions, manifolds structure of direct and inverse saddles corresponding to unstable periodic solutions, basins of attraction at different values of the control parameter. Deep insight into the global dynamics of the system and its evolution is achieved through the analysis of synthetic attractor-basin-manifold phase portraits. The topological mechanisms which entail onset and disappearance of various attractors, and the main and secondary evolutions to chaos, are identified. Special attention is devoted to the analysis of sudden bifurcational events characterizing the system global dynamics, associated with the topological behavior of the invariant manifolds of several direct and inverse saddles. Features of basin metamorphosis, attractor-basin accessibility, and window occurrence are examined. The approach followed, consisting in combined bifurcation analysis of the attractor-basin structure and of the manifold structure, is thought to be useful for a variety of dynamical systems.

scholarly journals Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
M. Garić-Demirović ◽  
M. R. S. Kulenović ◽  
M. Nurkanović
Keyword(s):  
Difference Equation ◽  
Unique Feature ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Minimal Period ◽  
Fractional Difference ◽  
The Difference

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the formxn+1=xn-12/(axn2+bxnxn-1+cxn-12),n=0,1,2,…,where the parameters  a,  b, and  c  are positive numbers and the initial conditionsx-1andx0are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.

scholarly journals Bifurcations and Multistability in a Model of Cytokine-Mediated Autoimmunity

2019 ◽  
Vol 29 (03) ◽  
pp. 1950034 ◽  
Author(s):  
Farzad Fatehi ◽  
Yuliya N. Kyrychko ◽  
Robert Molchanov ◽  
Konstantin B. Blyuss
Keyword(s):  
T Cells ◽  
Chronic Infection ◽  
Initial Conditions ◽  
Cell Activity ◽  
Basins Of Attraction ◽  
Activity Analysis ◽  
Steady States ◽  
System Parameters ◽  
Stability And Bifurcations

This paper investigates the dynamics of immune response and autoimmunity with particular emphasis on the role of regulatory T cells (Tregs), T cells with different activation thresholds, and cytokines in mediating T cell activity. Analysis of the steady states yields parameter regions corresponding to regimes of normal clearance of viral infection, chronic infection, or autoimmune behavior, and the boundaries of stability and bifurcations of relevant steady states are found in terms of system parameters. Numerical simulations are performed to illustrate different dynamical scenarios, and to identify basins of attraction of different steady states and periodic solutions, highlighting the important role played by the initial conditions in determining the outcome of immune interactions.

scholarly journals Global Dynamics of Three Anticompetitive Systems of Difference Equations in the Plane

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
M. DiPippo ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
General Linear ◽  
Global Dynamics ◽  
Rational Difference Equations ◽  
Special Cases ◽  
Fractional System

We investigate the global dynamics of several anticompetitive systems of rational difference equations which are special cases of general linear fractional system of the forms ., where all parameters and the initial conditions are arbitrary nonnegative numbers, such that both denominators are positive. We find the basins of attraction of all attractors of these systems.

scholarly journals Basins of Attraction for Two-Species Competitive Model with Quadratic Terms and the Singular Allee Effect

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
A. Brett ◽  
M. R. S. Kulenović
Keyword(s):  
Singular Point ◽  
Allee Effect ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
System Of Difference Equations ◽  
Competitive Model ◽  
Basins Of Attractions

We consider the following system of difference equations:xn+1=xn2/B1xn2+C1yn2, yn+1=yn2/A2+B2xn2+C2yn2,  n=0, 1, …,  whereB1,C1,A2,B2,C2are positive constants andx0, y0≥0are initial conditions. This system has interesting dynamics and it can have up to seven equilibrium points as well as a singular point at(0,0), which always possesses a basin of attraction. We characterize the basins of attractions of all equilibrium points as well as the singular point at(0,0)and thus describe the global dynamics of this system. Since the singular point at(0,0)always possesses a basin of attraction this system exhibits Allee’s effect.

BASIN CONFIGURATION OF A SIX-DIMENSIONAL MODEL OF AN ELECTRIC POWER SYSTEM

2002 ◽  
Vol 12 (06) ◽  
pp. 1333-1356 ◽  
Author(s):  
YOSHISUKE UEDA ◽  
HIROYUKI AMANO ◽  
RALPH H. ABRAHAM ◽  
H. BRUCE STEWART
Keyword(s):  
Power Systems ◽  
Power System ◽  
Initial Conditions ◽  
Electrical Power ◽  
Practical Importance ◽  
Intervention Strategy ◽  
Basins Of Attraction ◽  
Electric Power System ◽  
Point Of View ◽  
Electrical Power System

As part of an ongoing project on the stability of massively complex electrical power systems, we discuss the global geometric structure of contacts among the basins of attraction of a six-dimensional dynamical system. This system represents a simple model of an electrical power system involving three machines and an infinite bus. Apart from the possible occurrence of attractors representing pathological states, the contacts between the basins have a practical importance, from the point of view of the operation of a real electrical power system. With the aid of a global map of basins, one could hope to design an intervention strategy to boot the power system back into its normal state. Our method involves taking two-dimensional sections of the six-dimensional state space, and then determining the basins directly by numerical simulation from a dense grid of initial conditions. The relations among all the basins are given for a specific numerical example, that is, choosing particular values for the parameters in our model.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

Kinetic Model of Adiabatic Temperature Rise of Mass Concrete

2021 ◽  
Vol 13 (5) ◽  
pp. 771-780
Author(s):  
Shou-Kai Chen ◽  
Bo-Wen Xu
Keyword(s):  
Temperature Field ◽  
Temperature Rise ◽  
Initial Conditions ◽  
Dynamic Change ◽  
Adiabatic Temperature ◽  
Model Parameters ◽  
Hydration Reaction ◽  
Mass Concrete ◽  
Adiabatic Temperature Rise ◽  
Proposed Model

The adiabatic temperature rise model of mass concrete is very important for temperature field simulation, same to crack resistance capacity and temperature control of concrete structures. In this research, a thermal kinetics analysis was performed to study the exothermic hydration reaction process of concrete, and an adiabatic temperature rise model was proposed. The proposed model considers influencing factors, including initial temperature, temperature history, activation energy, and the completion degree of adiabatic temperature rise and is theoretically mature and definitive in physical meaning. It was performed on different initial temperatures for adiabatic temperature rise test; the data were employed in a regression analysis of the model parameters and initial conditions. The same function was applied to describe the dynamic change of the adiabatic temperature rise rates for different initial temperatures and different temperature changing processes and subsequently employed in a finite element analysis of the concrete temperature field. The test results indicated that the proposed model adequately fits the data of the adiabatic temperature rise test, which included different initial temperatures, and accurately predicts the changing pattern of adiabatic temperature rise of concrete at different initial temperatures. Compared with the results using the traditional age-based adiabatic temperature rise model, the results of a calculation example revealed that the simulated calculation results using the proposed model can accurately reflect the temperature change pattern of concrete in heat dissipation conditions.

Numerical Study of an Impact Oscillator

1995 ◽  
Author(s):  
Kannan Marudachalam ◽  
Faruk H. Bursal
Keyword(s):  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Initial Conditions ◽  
Numerical Study ◽  
Harmonic Excitation ◽  
General Purpose ◽  
Constrained Systems ◽  
Global Dynamics ◽  
Impact Oscillator ◽  
Stick Slip

Abstract Systems with discontinuous dynamics can be found in diverse disciplines. Meshing gears with backlash, impact dampers, relative motion of components that exhibit stick-slip phenomena axe but a few examples from mechanical systems. These form a class of dynamical systems where the nonlinearity is so severe that analysis becomes formidable, especially when global behavior needs to be known. Only recently have researchers attempted to investigate such systems in terms of modern dynamical systems theory. In this work, an impact oscillator with two-sided rigid constraints is used as a paradigm for studying the characteristics of discontinuous dynamical systems. The oscillator has zero stiffness and is subjected to harmonic excitation. The system is linear without impacts. However, the impacts introduce nonlinearity and dissipation (assuming inelastic impacts). A numerical algorithm is developed for studying the global dynamics of the system. A peculiar type of solution in which the trajectories in phase space from a certain set of initial conditions merge in finite time, making the dynamics non-invertible, is investigated. Also, the effect of “grazing,” a behavior common to constrained systems, on the dynamics of the system is studied. Based on the experience gained in studying this system, the need for an efficient general-purpose numerical algorithm for solving discontinuous dynamical systems is motivated. Investigation of stress, vibration, wear, noise, etc. that are associated with impact phenomena can benefit greatly from such an algorithm.

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