scholarly journals Dynamics of a Heterogeneous Constraint Profit Maximization Duopoly Model Based on an Isoelastic Demand

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
S. S. Askar ◽  
A. Ibrahim ◽  
A. A. Elsadany
Keyword(s):  
Basin Of Attraction ◽  
Profit Maximization ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Cournot Duopoly ◽  
Coexistence Of Attractors ◽  
Duopoly Model ◽  
Periodic Attractors ◽  
Initial Results ◽  
The Basin Of Attraction

A Cournot duopoly game is a two-firm market where the aim is to maximize profits. It is rational for every company to maximize its profits with minimal sales constraints. As a consequence, a model of constrained profit maximization (CPM) occurs when a business needs to be increased with profit minimal sales constraints. The CPM model, in which companies maximize profits under the minimum sales constraints, is an alternative to the profit maximization model. The current study constructs a duopoly game based on an isoelastic demand and homogeneous goods with heterogeneous strategies. In the event of sales constraint and no sales constraint, the local stability conditions of the Cournot equilibrium are derived. The initial results show that the duopoly model would be easier to stabilize if firms were to impose certain minimum sales constraints. Two routes to chaos are analyzed by numerical simulation using 2D bifurcation diagram, one of which is period doubling bifurcation and the other is Neimark–Sacker bifurcation. Four forms of coexistence of attractors are demonstrated by the basin of attraction, which is the coexistence of periodic attractors and chaotic attractors, the coexistence of periodic attractors and quasiperiodic attractors, and the coexistence of several chaotic attractors. Our findings show that the effect of game parameters on stability depends on the rules of expectations and restriction of sales by firms.

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 Chaotic Dynamics of a Mixed Rayleigh–Liénard Oscillator Driven by Parametric Periodic Damping and External Excitations

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yélomè Judicaël Fernando Kpomahou ◽  
Laurent Amoussou Hinvi ◽  
Joseph Adébiyi Adéchinan ◽  
Clément Hodévèwan Miwadinou
Keyword(s):  
Chaotic Dynamics ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Melnikov Method ◽  
Constant Term ◽  
Geometrical Shape ◽  
Analytical Prediction ◽  
Coexistence Of Attractors ◽  
The Basin Of Attraction

In this paper, chaotic dynamics of a mixed Rayleigh–Liénard oscillator driven by parametric periodic damping and external excitations is investigated analytically and numerically. The equilibrium points and their stability evolutions are analytically analyzed, and the transitions of dynamical behaviors are explored in detail. Furthermore, from the Melnikov method, the analytical criterion for the appearance of the homoclinic chaos is derived. Analytical prediction is tested against numerical simulations based on the basin of attraction of initial conditions. As a result, it is found that for ω = ν , the chaotic region decreases and disappears when the amplitude of the parametric periodic damping excitation increases. Moreover, increasing of F 1 and F 0 provokes an erosion of the basin of attraction and a modification of the geometrical shape of the chaotic attractors. For ω ≠ ν and η = 0.8 , the fractality of the basin of attraction increases as the amplitude of the external periodic excitation and constant term increase. Bifurcation structures of our system are performed through the fourth-order Runge–Kutta ode 45 algorithm. It is found that the system displays a remarkable route to chaos. It is also found that the system exhibits monostable and bistable oscillations as well as the phenomenon of coexistence of attractors.

scholarly journals Localization of Hidden Attractors in Chua’s System With Absolute Nonlinearity and Its FPGA Implementation

2021 ◽  
Vol 9 ◽  
Author(s):  
Xianming Wu ◽  
Huihai Wang ◽  
Shaobo He
Keyword(s):  
Basin Of Attraction ◽  
Digital Circuit ◽  
Fpga Implementation ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Hidden Attractor ◽  
Hidden Attractors ◽  
Chua's Circuit ◽  
The Mathematical Model ◽  
The Basin Of Attraction

Investigation of the classical self-excited and hidden attractors in the modified Chua’s circuit is a hot and interesting topic. In this article, a novel Chua’s circuit system with an absolute item is investigated. According to the mathematical model, dynamic characteristics are analyzed, including symmetry, equilibrium stability analysis, Hopf bifurcation analysis, Lyapunov exponents, bifurcation diagram, and the basin of attraction. The hidden attractors are located theoretically. Then, the coexistence of the hidden limit cycle and self-excited chaotic attractors are observed numerically and experimentally. The numerical simulation results are consistent with the FPGA implementation results. It shows that the hidden attractor can be localized in the digital circuit.

Finding Method and Analysis of Hidden Chaotic Attractors for Plasma Chaotic System From Physical and Mechanistic Perspectives

2020 ◽  
Vol 30 (05) ◽  
pp. 2050072 ◽  
Author(s):  
Yingjuan Yang ◽  
Guoyuan Qi ◽  
Jianbing Hu ◽  
Philippe Faradja
Keyword(s):  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
The Self ◽  
Numerical Continuation ◽  
Chaotic Attractors ◽  
Phase Portrait Analysis ◽  
The Stability ◽  
Two Parameters ◽  
The Basin Of Attraction

A method for finding hidden chaotic attractors in the plasma system is presented. Using the Routh–Hurwitz criterion, the stability distribution associated with two parameters is identified to find the region around the equilibrium points of the stable nodes, stable focus-nodes, saddles and saddle-foci for the purpose of investigating hidden chaos. A physical interpretation is provided of the stability distribution for each type of equilibrium point. The basin of attraction and parameter region of hidden chaos are identified by excluding the self-excited chaotic attractors of all equilibrium points. Homotopy and numerical continuation are also employed to check whether the basin of chaotic attraction intersects with the neighborhood of a saddle equilibrium. Bifurcation analysis, phase portrait analysis, and basins of different dynamical attraction are used as tools to distinguish visually the self-excited chaotic attractor and hidden chaotic attractor. The Casimir power reflects the error power between the dissipative energy and the energy supplied by the whistler field. It explains physically, analytically, and numerically the conditions that generate the different dynamics, such as sinks, periodic orbits, and chaos.

Constructing a Chaotic System with Any Number of Attractors

2017 ◽  
Vol 27 (08) ◽  
pp. 1750118 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Vector Field ◽  
Geometric Structure ◽  
Autonomous System ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Original System ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
The Basin Of Attraction ◽  
New System

Applying some transformation to an autonomous system, we obtain a new system, which might keep the dynamical behavior of the original system or generate different dynamics. But this is often accompanied by the appearance of discontinuous points, where the vector field for the new system is not continuous at these points. We discuss the effects of the discontinuous points, and provide two methods to construct systems with any preassigned number of chaotic attractors via some transformation. The first one does not change the geometric structure of the attractors, since the discontinuous points are out of the basin of attraction. The second one might make the new systems have different dynamics, like multiscroll chaotic attractors, or infinitely many chaotic attractors. These results illustrate that both the equilibria and the discontinuous points affect the global dynamics.

THE EFFECT OF DAMPING ON THE STEADY STATE AND BASIN BIFURCATION PATTERNS OF A NONLINEAR MECHANICAL OSCILLATOR

1992 ◽  
Vol 02 (01) ◽  
pp. 81-91 ◽  
Author(s):  
MOHAMED S. SOLIMAN ◽  
J.M.T. THOMPSON
Keyword(s):  
Steady State ◽  
Basin Of Attraction ◽  
Period Doubling ◽  
Qualitative And Quantitative ◽  
Periodically Driven ◽  
Resonance Response ◽  
Nonlinear Mechanical ◽  
Damped Oscillator ◽  
The Basin Of Attraction

This paper examines the role of damping on both the steady state and basin behavior of a periodically driven damped oscillator with the ability to escape from a potential well. We examine the effect of damping on both the qualitative and quantitative resonance response of the system. Particular attention is paid to how the damping scales the main steady state bifurcations; saddle-nodes, period-doubling flips, cascades to chaos, boundary crises, etc. We also investigate how the damping level effects the main homoclinic and heteroclinic basin bifurcations that may result in a rapid erosion and stratification of the basin of attraction and hence a loss of engineering integrity of the system.

Chaotic phenomena triggering the escape from a potential well

1989 ◽  
Vol 421 (1861) ◽  
pp. 195-225 ◽  
Keyword(s):  
Harmonic Balance ◽  
Scaling Laws ◽  
Basin Of Attraction ◽  
Period Doubling ◽  
Potential Well ◽  
Homoclinic Tangle ◽  
Balance Analysis ◽  
Melnikov Analysis ◽  
Chaotic Transients ◽  
The Basin Of Attraction

This paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold. The aim is to show how the well-known atoms of dissipative dynamics (saddle-node folds, period-doubling flips, cascades to chaos, boundary crises, etc.) assemble to form molecules of overall response (hierarchies of cusps, incomplete Feigenbaum trees, etc.). Particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle, the latter being accurately predicted by a Melnikov analysis. After escape, chaotic transients are shown to conform to recent scaling laws. Analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible.

scholarly journals Strategic trade policy with interlocking cross-ownership

2021 ◽  
Author(s):  
Luciano Fanti ◽  
Domenico Buccella
Keyword(s):  
Trade Policy ◽  
Free Trade ◽  
Strategic Trade Policy ◽  
Cournot Duopoly ◽  
Public Intervention ◽  
Tax Subsidy ◽  
Strategic Trade ◽  
Duopoly Model ◽  
Product Competition ◽  
Cross Ownership

AbstractBy analysing interlocking cross-ownership, this work reconsiders the inefficiency of activist governments that set subsidies for their exporters (Brander and Spencer, J Int Econ 18:83–100). Making use of a third-market Cournot duopoly model, we show that the implementation of strategic trade policy in the form of a tax (subsidy) when goods are differentiated (complements) is Pareto-superior to free trade within precise ranges of firms’ cross-ownership, richly depending on the degree of product competition. These results challenge the conventional ones in which public intervention (1) is always the provision of a subsidy and (2) always leads to a Pareto-inferior (resp. Pareto-superior) equilibrium when products are substitutes (resp. complements).

scholarly journals Optimization and Stability of Heat Engines: The Role of Entropy Evolution

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 865 ◽  
Author(s):  
Julian Gonzalez-Ayala ◽  
Moises Santillán ◽  
Maria Santos ◽  
Antonio Calvo Hernández ◽  
José Mateos Roco
Keyword(s):  
Local Stability ◽  
Production Efficiency ◽  
Basin Of Attraction ◽  
Heat Engine ◽  
Time Constraints ◽  
Thermal Gradients ◽  
Heat Engines ◽  
Low Dissipation ◽  
The Basin Of Attraction

Local stability of maximum power and maximum compromise (Omega) operation regimes dynamic evolution for a low-dissipation heat engine is analyzed. The thermodynamic behavior of trajectories to the stationary state, after perturbing the operation regime, display a trade-off between stability, entropy production, efficiency and power output. This allows considering stability and optimization as connected pieces of a single phenomenon. Trajectories inside the basin of attraction display the smallest entropy drops. Additionally, it was found that time constraints, related with irreversible and endoreversible behaviors, influence the thermodynamic evolution of relaxation trajectories. The behavior of the evolution in terms of the symmetries of the model and the applied thermal gradients was analyzed.

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