scholarly journals Stability, Multistability, and Complexity Analysis in a Dynamic Duopoly Game with Exponential Demand Function

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Hui Li ◽  
Wei Zhou ◽  
Tong Chu
Keyword(s):  
Complexity Analysis ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Time Dynamic ◽  
Critical Curves ◽  
Dynamic Duopoly ◽  
Stable Structures

In this paper, a discrete-time dynamic duopoly model, with nonlinear demand and cost functions, is established. The properties of existence and local stability of equilibrium points have been verified and analyzed. The stability conditions are also given with the help of the Jury criterion. With changing of the values of parameters, the system shows some new and interesting phenomena in terms to stability and multistability, such as V-shaped stable structures (also called Isoperiodic Stable Structures) and different shape basins of attraction of coexisting attractors. The eye-shaped structures appear where the period-doubling and period-halving bifurcations occur. Finally, by utilizing critical curves, the changes in the topological structure of basin of attraction and the reason of “holes” formation are analyzed. As a result, the generation of global bifurcation, such as contact bifurcation or final bifurcation, is usually accompanied by the contact of critical curves and boundary.

scholarly journals A Dynamic Duopoly Model: When a Firm Shares the Market with Certain Profit

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1826
Author(s):  
Sameh S. Askar
Keyword(s):  
Equilibrium Point ◽  
Market Share ◽  
Period Doubling ◽  
Analytical Investigation ◽  
Basins Of Attraction ◽  
Stability Conditions ◽  
Cournot Duopoly ◽  
Demand Functions ◽  
Duopoly Model ◽  
Dynamic Duopoly

The current paper analyzes a competition of the Cournot duopoly game whose players (firms) are heterogeneous in a market with isoelastic demand functions and linear costs. The first firm adopts a rationally-based gradient mechanism while the second one chooses to share the market with certain profit in order to update its production. It trades off between profit and market share maximization. The equilibrium point of the proposed game is calculated and its stability conditions are investigated. Our studies show that the equilibrium point becomes unstable through period doubling and Neimark–Sacker bifurcation. Furthermore, the map describing the proposed game is nonlinear and noninvertible which lead to several stable attractors. As in literature, we have provided an analytical investigation of the map’s basins of attraction that includes lobes regions.

scholarly journals Robustness of Supercavitating Vehicles Based on Multistability Analysis

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Yipin Lv ◽  
Tianhong Xiong ◽  
Wenjun Yi ◽  
Jun Guan
Keyword(s):  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Small Range ◽  
Practical Application ◽  
Stable States ◽  
Supercavitating Vehicles ◽  
Control Gain ◽  
Stable Periodic ◽  
Setting Parameters

Supercavity can increase speed of underwater vehicles greatly. However, external interferences always lead to instability of vehicles. This paper focuses on robustness of supercavitating vehicles. Based on a 4-dimensional dynamic model, the existence of multistability is verified in supercavitating system through simulation, and the robustness of vehicles varying with parameters is analyzed by basins of attraction. Results of the research disclose that the supercavitating system has three stable states in some regions of parameters space, namely, stable, periodic, and chaotic states, while in other regions it has various multistability, such as coexistence of two types of stable equilibrium points, coexistence of a limit cycle with a chaotic attractor, and coexistence of 1-periodic cycle with 2-periodic cycle. Provided that cavitation number varies within a small range, with increase of the feedback control gain of fin deflection angle, size of basin of attraction becomes smaller and robustness of the system becomes weaker. In practical application, robustness of supercavitating vehicles can be improved by setting parameters of system or adjusting initial launching conditions.

scholarly journals INVARIANT MANIFOLDS FOR COMPETITIVE DISCRETE SYSTEMS IN THE PLANE

2010 ◽  
Vol 20 (08) ◽  
pp. 2471-2486 ◽  
Author(s):  
M. R. S. KULENOVIĆ ◽  
ORLANDO MERINO
Keyword(s):  
Difference Equations ◽  
Invariant Manifolds ◽  
Equilibrium Points ◽  
Global Bifurcation ◽  
Discrete Systems ◽  
Basins Of Attraction ◽  
Competitive Systems ◽  
Invariant Regions ◽  
Competitive Map ◽  
Hyperbolic Equilibrium

Let T be a competitive map on a rectangular region [Formula: see text], and assume T is C1 in a neighborhood of a fixed point [Formula: see text]. The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from [Formula: see text] when both eigenvalues of the Jacobian of T at [Formula: see text] are nonzero and at least one of them has absolute value less than one, and establish that [Formula: see text] is an increasing curve that separates [Formula: see text] into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. These results, known in hyperbolic case, have been used to determine the basins of attraction of hyperbolic equilibrium points and to establish certain global bifurcation results when switching from competitive coexistence to competitive exclusion. The emphasis in applications in this paper is on planar systems of difference equations with nonhyperbolic equilibria, where we establish a precise description of the basins of attraction of finite or infinite number of equilibrium points.

Effect of Delay on the Boundary of the Basin of Attraction in a Self-Excited Single Graded-Response Neuron

1997 ◽  
Vol 9 (2) ◽  
pp. 319-336 ◽  
Author(s):  
K. Pakdaman ◽  
C. P. Malta ◽  
C. Grotta-Ragazzo ◽  
J.-F. Vibert
Keyword(s):  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Transmission Delays ◽  
The Past ◽  
Graded Response ◽  
The Stability ◽  
Delay Dependent ◽  
The Basin Of Attraction

Little attention has been paid in the past to the effects of interunit transmission delays (representing a xonal and synaptic delays) ontheboundary of the basin of attraction of stable equilibrium points in neural networks. As a first step toward a better understanding of the influence of delay, we study the dynamics of a single graded-response neuron with a delayed excitatory self-connection. The behavior of this system is representative of that of a family of networks composed of graded-response neurons in which most trajectories converge to stable equilibrium points for any delay value. It is shown that changing the delay modifies the “location” of the boundary of the basin of attraction of the stable equilibrium points without affecting the stability of the equilibria. The dynamics of trajectories on the boundary are also delay dependent and influence the transient regime of trajectories within the adjacent basins. Our results suggest that when dealing with networks with delay, it is important to study not only the effect of the delay on the asymptotic convergence of the system but also on the boundary of the basins of attraction of the equilibria.

Synchronization and Global Dynamics of a Cournot Model with Nonlinear Demand and R&D Spillovers

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Wei Zhou ◽  
Mengfan Cui
Keyword(s):  
Invariant Manifolds ◽  
Global Bifurcation ◽  
Basins Of Attraction ◽  
Feasible Region ◽  
Global Dynamics ◽  
Cournot Model ◽  
One Dimensional ◽  
Critical Curves ◽  
Dynamical Behaviors ◽  
The One

In this paper, a dynamical Cournot model with nonlinear demand and R&D spillovers is established. The system is symmetric when the duopoly firms have same economic environments, and it is proved that both the diagonal and the coordinate axes are the one-dimensional invariant manifolds of system. The results show that Milnor attractor of system can be found through calculating the transverse Lyapunov exponents. The synchronization phenomenon is verified through basins of attraction. The effects of adjusting speed and R&D spillovers on the dynamical behaviors of the system are discussed. The topological structures of basins of attraction are analyzed through critical curves, and the evolution process of “holes” in the feasible region is numerically simulated. In addition, various global bifurcation behaviors, such as two kinds of contact bifurcation and the blowout bifurcation, are shown.

scholarly journals On the Stability and Multistability in a Duopoly Game with R&D Spillover and Price Competition

2019 ◽  
Vol 2019 ◽  
pp. 1-20 ◽  
Author(s):  
Wei Zhou ◽  
Xiao-Xue Wang
Keyword(s):  
Price Competition ◽  
Equilibrium Points ◽  
Research Result ◽  
Basin Of Attraction ◽  
Profit Maximization ◽  
Feasible Region ◽  
Multiple Attractors ◽  
Critical Curves ◽  
Coexistence Of Multiple Attractors ◽  
The Stability

In this paper, a dynamical two-stage game with R&D competition and joint profit maximization is built. The stability of all the equilibrium points is discussed through Jury condition, and the stability region of the Nash equilibrium point is then given. The influence of the parameters on the system is discussed, and we find that the firm can even benefit from chaos, when it has higher innovation efficiency and higher adjusting speed. And then the coexistence of multiple attractors is studied using basin of attraction. Our research result shows that the coexisting attractors can be observed in the two-parameter bifurcation diagram. At last, the boundary of feasible region, global bifurcations, and formation mechanism of fractal structure of attracting basin are analyzed through critical curves and noninvertible map theory.

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.

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.

LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1617-1636 ◽  
Author(s):  
SOMA DE ◽  
PARTHA SHARATHI DUTTA ◽  
SOUMITRO BANERJEE ◽  
AKHIL RANJAN ROY
Keyword(s):  
Three Dimensional ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Numerical Continuation ◽  
Homoclinic Tangency ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Border Collision Bifurcation ◽  
Smooth Map

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

Exploiting Global Dynamics of a Noncontact Atomic Force Microcantilever to Enhance Its Dynamical Robustness via Numerical Control

2016 ◽  
Vol 26 (07) ◽  
pp. 1630018 ◽  
Author(s):  
Valeria Settimi ◽  
Giuseppe Rega
Keyword(s):  
Global Bifurcation ◽  
Numerical Procedure ◽  
Basins Of Attraction ◽  
Operating Conditions ◽  
Numerical Control ◽  
Control Technique ◽  
Global Dynamics ◽  
Analytical Treatment ◽  
Topological Characterization ◽  
Safe Region

A control technique exploiting the global dynamical features is applied to a reduced order model of noncontact AFM, aiming to obtain an enlargement of the system’s safe region in parameters space. The method consists of optimally modifying the shape of the system excitation by adding controlling superharmonics, to delay the occurrence of the global events (i.e. homo/heteroclinic bifurcations of some saddle) which trigger the erosion of the basins of attraction leading to loss in safety. The system’s main saddles and the bifurcations involving the relevant manifolds are detected through accurate numerical investigations, and their topological characterization allows the determination of the global event responsible for the sharp reduction in the system dynamical integrity. Since an analytical treatment is impossible in applying the control, a fully numerical procedure is implemented. Besides being effective in detecting the value of the optimal superharmonic to be added for shifting the global bifurcation to a higher value of forcing amplitude, the method also proves to succeed in delaying the drop down of the erosion profile, thus increasing the overall robustness of the system during operating conditions.

Sign in / Sign up

Export Citation Format

Share Document