scholarly journals Nonlinear Dynamics of Cournot Duopoly Game: When One Firm Considers Social Welfare

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
S. S. Askar ◽  
A. A. Elsadany
Keyword(s):  
Nonlinear Dynamics ◽  
Social Welfare ◽  
Fixed Points ◽  
Phase Plane ◽  
Complex Dynamics ◽  
Stability Conditions ◽  
Cournot Duopoly ◽  
Discrete Dynamic ◽  
Gradient Based ◽  
Dynamic Map

In this paper, we study the competition between two firms whose outputs are quantities. The first firm considers maximization of its profit while the second firm considers maximization of its social welfare. Adopting a gradient-based mechanism, we introduce a nonlinear discrete dynamic map which is used to describe the dynamics of this game. For this map, the fixed points are calculated and their stability conditions are analyzed. This includes investigating some attracting set and chaotic behaviors for the complex dynamics of the map. We have also investigated the types of the preimages that characterize the phase plane of the map and conclude that the game’s map is noninvertible of type Z 4 − Z 2 .

scholarly journals On Comparing between Two Nonlinear Cournot Duopoly Models

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
S. S. Askar
Keyword(s):  
Utility Function ◽  
Fixed Points ◽  
Bounded Rationality ◽  
Profit Maximization ◽  
Flip Bifurcation ◽  
Cournot Duopoly ◽  
Trade Off ◽  
Trade Offs ◽  
Gradient Based ◽  
Conditions Of Stability

The comparison between two nonlinear duopoly models constructed based on symmetric utility function that is derived from Cobb–Douglas is investigated in this paper. The first model consists of two firms which update their outputs using gradient-based mechanism called bounded rationality. The second model contains a bounded rational firm that is competing with a firm whose outputs depend on a trade-off between market share maximization and profit maximization. For the two models, the fixed points are calculated and their conditions of stability are analyzed. The obtained results show that the second model is more stabilizing provided that the second firm adopts low weights of trade-offs. We show that the two models can be destabilized via flip bifurcation only. Furthermore, the noninvertibility of the two models that can give rise to several stable attractors is discussed.

Nonlinear Dynamics of an Imperfect Microbeam Under an Axial Load and Electric Excitation

2011 ◽  
Author(s):  
Laura Ruzziconi ◽  
Mohammad I. Younis ◽  
Stefano Lenci
Keyword(s):  
Nonlinear Dynamics ◽  
Microelectromechanical Systems ◽  
Complex Dynamics ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Theoretical Point ◽  
Initial Shape ◽  
Electric Excitation

This study is motivated by the growing attention, both from a practical and a theoretical point of view, toward the nonlinear behavior of microelectromechanical systems (MEMS). We analyze the nonlinear dynamics of an imperfect microbeam under an axial force and electric excitation. The imperfection of the microbeam, typically due to microfabrication processes, is simulated assuming the microbeam to be of a shallow arched initial shape. The device has a bistable static behavior. The aim is that of illustrating the nonlinear phenomena, which arise due to the coupling of mechanical and electrical nonlinearities, and discussing their usefulness for the engineering design of the microstructure. We derive a single-mode-reduced-order model by combining the classical Galerkin technique and the Pade´ approximation. Despite its apparent simplicity, this model is able to capture the main features of the complex dynamics of the device. Extensive numerical simulations are performed using frequency response diagrams, attractor-basins phase portraits, and frequency-dynamic voltage behavior charts. We investigate the overall scenario, up to the inevitable escape, obtaining the theoretical boundaries of appearance and disappearance of the main attractors. The main features of the nonlinear dynamics are discussed, stressing their existence and their practical relevance. We focus on the coexistence of robust attractors, which leads to a considerable versatility of behavior. This is a very attractive feature in MEMS applications. The ranges of coexistence are analyzed in detail, remarkably at high values of the dynamic excitation, where the penetration of the escape (dynamic pull-in) inside the double well may prevent the safe jump between the attractors.

A PHASE-PLANE DESCRIPTION OF NONLINEAR TRAVELING WAVES IN BUBBLY LIQUIDS

1999 ◽  
Vol 07 (02) ◽  
pp. 71-82
Author(s):  
A. NADIM ◽  
D. GOLDMAN ◽  
J. J. CARTMELL ◽  
P. E. BARBONE
Keyword(s):  
Equation Of State ◽  
Fixed Points ◽  
Traveling Wave ◽  
Phase Plane ◽  
Volume Fraction ◽  
Low Frequency ◽  
Traveling Wave Solutions ◽  
Bubbly Liquid ◽  
Phase Plane Analysis ◽  
Bubbly Liquids

One-dimensional traveling wave solutions to the fully nonlinear continuity and Euler equations in a bubbly liquid are considered. The elimination of velocity from the two equations leaves a single nonlinear algebraic relation between the pressure and density profiles in the mixture. On assuming the bubbles to have identical size and taking the volume fraction of bubbles in the medium to be small, an equation of state which relates the mixture pressure to the density and its first two material time-derivatives is derived. When this equation of state is linearized and combined with the laws of conservation of mass and momentum, a nonlinear, second-order, ordinary differential equation is obtained for the density as a function of the single traveling wave coordinate. A phase-plane analysis of this equation reveals the existence of two fixed points, one of which is a saddle and the other a node. A single trajectory connects the two fixed points and corresponds to a traveling shock wave solution when the Mach number of the wave, defined as the ratio of traveling wave speed to the low-frequency speed of sound in the bubbly liquid, exceeds unity. The analysis provides a qualitative explanation of the oscillations behind shocks seen in experiments on bubbly liquids.

scholarly journals Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 101
Author(s):  
Alicia Cordero ◽  
Marlon Moscoso-Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
Complex Dynamics ◽  
Convergence Properties ◽  
Parameter Spaces ◽  
Numerical Tests ◽  
The Family ◽  
Dynamics Stability

In this paper, we present a new parametric family of three-step iterative for solving nonlinear equations. First, we design a fourth-order triparametric family that, by holding only one of its parameters, we get to accelerate its convergence and finally obtain a sixth-order uniparametric family. With this last family, we study its convergence, its complex dynamics (stability), and its numerical behavior. The parameter spaces and dynamical planes are presented showing the complexity of the family. From the parameter spaces, we have been able to determine different members of the family that have bad convergence properties, as attracting periodic orbits and attracting strange fixed points appear in their dynamical planes. Moreover, this same study has allowed us to detect family members with especially stable behavior and suitable for solving practical problems. Several numerical tests are performed to illustrate the efficiency and stability of the presented family.

On the Motion of the Pendulum in an Alternating, Sawtooth Force Field

2020 ◽  
Vol 30 (09) ◽  
pp. 2050135
Author(s):  
Alexander A. Burov ◽  
Vasily I. Nikonov
Keyword(s):  
Nonlinear Dynamics ◽  
Iterative Method ◽  
Force Field ◽  
Periodic Solutions ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Melnikov Method ◽  
Stability Conditions ◽  
Poincaré Maps ◽  
Separatrix Splitting

The motion of the pendulum in a variable sawtooth force field is considered. For the “lower” equilibrium, the necessary stability conditions are investigated numerically, the results are presented in the form of an Ince–Strutt diagram. Using the Poincaré–Melnikov method separatrix splitting is studied analytically. Numerically, for some values of parameters, the nonlinear dynamics is studied using Poincaré maps, the regions of regular and chaotic behavior are revealed. The iterative method earlier proposed is used for the localization of periodic solutions, located inside the numerically identified “invariant tori”.

Managerial Delegation Contracts, “Green” R&D and Emissions Taxation

2018 ◽  
Vol 19 (2) ◽  
Author(s):  
Joanna Poyago-Theotoky ◽  
Soo Keong Yong
Keyword(s):  
Social Welfare ◽  
Managerial Compensation ◽  
Cournot Duopoly ◽  
Emissions Tax ◽  
Managerial Delegation ◽  
Sales Compensation ◽  
Firm Owners

Abstract We introduce an explicit environmental incentive into a managerial compensation contract in the context of a Cournot duopoly with pollution externalities under an emissions tax regime. We show that, depending on the effectiveness of “green” R&D, compared to a standard sales compensation contract, the explicit environmental focused contract results in more abatement. As a consequence, the regulator sets a lower emissions tax, and social welfare is higher. Moreover, in general, firm owners earn higher profits when adopting the environmental delegation contract.

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.

Sign in / Sign up

Export Citation Format

Share Document