scholarly journals Delay Cournot Duopoly Game with Gradient Adjustment: Berezowski Transition from a Discrete Model to a Continuous Model

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 32
Author(s):  
Akio Matsumoto ◽  
Ferenc Szidarovszky ◽  
Keiko Nakayama
Keyword(s):  
Stability Region ◽  
Period Doubling ◽  
Continuous Model ◽  
Delay Model ◽  
Chaotic Oscillations ◽  
Cournot Duopoly ◽  
Numerical Studies ◽  
Inertia Coefficient ◽  
The Stability ◽  
Theoretical Results

This paper investigates the asymptotical behavior of the equilibrium of linear classical duopolies by reconsidering the two-delay model with two different positive delays. In a two-dimensional analysis, the stability switching curves were first analytically determined. Numerical studies verified and illustrated the theoretical results. In the sensitivity analysis it was demonstrated that the inertia coefficient has a twofold effect: enlarges the stability region as well as simplifies the complicated dynamics with period-halving cascade. In contrary, the adjustment speed contracts the stability region and complicates simple dynamics with period-doubling bifurcation. In addition, for various values of τ1 and τ2, a wide variety of dynamics appears ranging from simple cycle via a Hopf bifurcation to chaotic oscillations.

scholarly journals Further Investigations on the Dynamics and Multistability Coexisted in a Memory-Based Cobweb Model

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
S. S. Askar
Keyword(s):  
Fixed Point ◽  
Equilibrium Point ◽  
Profit Maximization ◽  
Period Doubling ◽  
Speed Of Adjustment ◽  
Discrete Dynamic ◽  
Cobweb Model ◽  
Numerical Studies ◽  
Market Clearing Price ◽  
The Stability

Based on a nonlinear demand function and a market-clearing price, a cobweb model is introduced in this paper. A gradient mechanism that depends on the marginal profit is adopted to form the 1D discrete dynamic cobweb map. Analytical studies show that the map possesses four fixed points and only one attains the profit maximization. The stability/instability conditions for this fixed point are calculated and numerically studied. The numerical studies provide some insights about the cobweb map and confirm that this fixed point can be destabilized due to period-doubling bifurcation. The second part of the paper discusses the memory factor on the stabilization of the map’s equilibrium point. A gradient mechanism that depends on the marginal profit in the past two time steps is adopted to incorporate memory in the model. Hence, a 2D discrete dynamic map is constructed. Through theoretical and numerical investigations, we show that the equilibrium point of the 2D map becomes unstable due to two types of bifurcations that are Neimark–Sacker and flip bifurcations. Furthermore, the influence of the speed of adjustment parameter on the map’s equilibrium is analyzed via numerical experiments.

scholarly journals Numerical Exploration of Kaldorian Macrodynamics: Enhanced Stability and Predominance of Period Doubling and Chaos with Flexible Exchange Rates

2008 ◽  
Vol 2008 ◽  
pp. 1-23 ◽  
Author(s):  
Toichiro Asada ◽  
Christos Douskos ◽  
Panagiotis Markellos
Keyword(s):  
Exchange Rates ◽  
Stability Region ◽  
Extreme Values ◽  
Period Doubling ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Bifurcation Curve ◽  
The Stability ◽  
Flexible Exchange Rates ◽  
Enhanced Stability

We explore a discrete Kaldorian macrodynamic model of an open economy with flexible exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods marketα, and the degree of capital mobilityβ. We determine by a numerical grid search method the stability region in parameter space and find that flexible rates cause enhanced stability of equilibrium with respect to variations of the parameters. We identify the Hopf-Neimark bifurcation curve and the flip bifurcation curve, and find that the period doubling cascades which leads to chaos is the dominant behavior of the system outside the stability region, persisting to large values ofβ. Cyclical behavior of noticeable presence is detected for some extreme values of a state parameter. Bifurcation and Lyapunov exponent diagrams are computed illustrating the complex dynamics involved. Examples of attractors and trajectories are presented. The effect of the speed of adaptation of the expected rate is also briefly discussed. Finally, we explore a special model variation incorporating the “wealth effect” which is found to behave similarly to the basic model, contrary to the model of fixed exchange rates in which incorporation of this effect causes an entirely different behavior.

scholarly journals Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

2020 ◽  
Vol 24 (3) ◽  
pp. 137-151
Author(s):  
Z. T. Zhusubaliyev ◽  
D. S. Kuzmina ◽  
O. O. Yanochkina
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Stability Region ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Continuous Map ◽  
The Stability ◽  
Smooth Map

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

Configuration and Robustness in Visual Servo

2004 ◽  
Vol 16 (2) ◽  
pp. 178-185 ◽  
Author(s):  
Graziano Chesi ◽  
◽  
Koichi Hashimoto
Keyword(s):  
Local Stability ◽  
Stability Region ◽  
Visual Servoing ◽  
Eye Position ◽  
Visual Servo ◽  
Control Laws ◽  
Extrinsic Parameters ◽  
Point To Point ◽  
The Stability ◽  
Theoretical Results

Effects of camera calibration errors on the point-to-point task are investigated in static-eye and hand-eye visual servoing realized with position-based and image-based control laws. For these four configurations, the effect of uncertainty on the extrinsic parameters is analyzed. The results show local stability for all configurations under small calibration errors. However, a steady state error is found in the hand-eye position-based configuration. Simulations have been done to confirm the theoretical results and evaluate the effects of the uncertainty in terms of stability region. Another contribution of the paper consists of providing a method for estimating the stability region robust against uncertainty directions for the static-eye position-based case with uncertainty on the camera centers.

The chaotic dynamics of a quantum Cournot duopoly game with bounded rationality

2020 ◽  
Vol 18 (06) ◽  
pp. 2050029
Author(s):  
Xinli Zhang ◽  
Deshan Sun ◽  
Wei Jiang
Keyword(s):  
Quantum Entanglement ◽  
Stability Region ◽  
Chaotic Dynamics ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Largest Lyapunov Exponent ◽  
Cournot Duopoly ◽  
The Stability ◽  
The Impact ◽  
Rational Players

This paper analyzes the chaotic dynamics of a quantum Cournot duopoly game with bounded rational players by applying quantum game theory. We investigate the impact of quantum entanglement on the stability of the quantum Nash equilibrium points and chaotic dynamics behaviors of the system. The result shows that the stability region decreases with the quantum entanglement increasing. The adjustment speeds of bounded rational players can lead to chaotic behaviors, and quantum entanglement accelerates the bifurcation and chaos of the system. Numerical simulations demonstrate the chaotic features via stability region, bifurcation, largest Lyapunov exponent, strange attractors, sensitivity to initial conditions and fractal dimensions.

The influence of the Stokes and Archimedes forces on the stability of a two-phase flow

2003 ◽  
Vol 3 ◽  
pp. 266-270
Author(s):  
B.H. Khudjuyerov ◽  
I.A. Chuliev
Keyword(s):  
Spectral Method ◽  
Stability Region ◽  
Gas Flow ◽  
Two Phase Flow ◽  
Solid Phase ◽  
Stability Characteristic ◽  
Phase Flow ◽  
Two Phase ◽  
The Stability

The problem of the stability of a two-phase flow is considered. The solution of the stability equations is performed by the spectral method using polynomials of Chebyshev. A decrease in the stability region gas flow with the addition of particles of the solid phase. The analysis influence on the stability characteristic of Stokes and Archimedes forces.

Enlarging the region of stability in robust model predictive controller based on dual-mode control

2021 ◽  
pp. 014233122110123
Author(s):  
Fatemeh Khani ◽  
Mohammad Haeri
Keyword(s):  
Stability Region ◽  
Linear Matrix ◽  
Input State ◽  
Dual Mode ◽  
Model Predictive Controller ◽  
Mode Control ◽  
Robust Model ◽  
Output Constraints ◽  
Predictive Controller ◽  
The Stability

Industrial processes are inherently nonlinear with input, state, and output constraints. A proper control system should handle these challenging control problems over a large operating region. The robust model predictive controller (RMPC) could be an linear matrix inequality (LMI)-based method that estimates stability region of the closed-loop system as an ellipsoid. This presentation, however, restricts confident application of the controller on systems with large operating regions. In this paper, a dual-mode control strategy is employed to enlarge the stability region in first place and then, trajectory reversing method (TRM) is employed to approximate the stability region more accurately. Finally, the effectiveness of the proposed scheme is illustrated on a continuous stirred tank reactor (CSTR) process.

Turning Dynamics: Part 1 — Experimental Analysis

2012 ◽  
Author(s):  
Eric B. Halfmann ◽  
C. Steve Suh ◽  
N. P. Hung
Keyword(s):  
Instantaneous Frequency ◽  
Period Doubling ◽  
Displacement Sensor ◽  
Laser Displacement Sensor ◽  
Turning Process ◽  
Time Frequency ◽  
Spectral Components ◽  
Tool Vibrations ◽  
Exact Moment ◽  
The Stability

The workpiece and tool vibrations in a lathe are experimentally studied to establish improved understanding of cutting dynamics that would support efforts in exceeding the current limits of the turning process. A Keyence laser displacement sensor is employed to monitor the workpiece and tool vibrations during chatter-free and chatter cutting. A procedure is developed that utilizes instantaneous frequency (IF) to identify the modes related to measurement noise and those innate of the cutting process. Instantaneous frequency is shown to thoroughly characterize the underlying turning dynamics and identify the exact moment in time when chatter fully developed. That IF provides the needed resolution for identifying the onset of chatter suggests that the stability of the process should be monitored in the time-frequency domain to effectively detect and characterize machining instability. It is determined that for the cutting tests performed chatters of the workpiece and tool are associated with the changing of the spectral components and more specifically period-doubling bifurcation. The analysis presented provides a view of the underlying dynamics of the lathe process which has not been experimentally observed before.

scholarly journals Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuai Yang ◽  
Haijun Jiang ◽  
Cheng Hu ◽  
Juan Yu ◽  
Jiarong Li
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Model Parameters ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Reductio Ad Absurdum ◽  
The Stability ◽  
The Impact ◽  
Theoretical Results ◽  
The Basic Reproduction Number

Abstract In this paper, a novel rumor-spreading model is proposed under bilingual environment and heterogenous networks, which considers that exposures may be converted to spreaders or stiflers at a set rate. Firstly, the nonnegativity and boundedness of the solution for rumor-spreading model are proved by reductio ad absurdum. Secondly, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. Whereafter, the global stability of rumor-prevailing equilibrium is explored by utilizing Lyapunov method and LaSalle’s invariance principle. Finally, the sensitivity analysis and the numerical simulation are respectively presented to analyze the impact of model parameters and illustrate the validity of theoretical results.

scholarly journals On the optimal control of coronavirus (2019-nCov) mathematical model; a numerical approach

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
N. H. Sweilam ◽  
S. M. Al-Mekhlafi ◽  
A. O. Albalawi ◽  
D. Baleanu
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Weighted Average ◽  
Necessary Optimality Conditions ◽  
Numerical Approach ◽  
Population Number ◽  
Infected People ◽  
The Stability ◽  
Novel Coronavirus ◽  
Theoretical Results

Abstract In this paper, a novel coronavirus (2019-nCov) mathematical model with modified parameters is presented. This model consists of six nonlinear fractional order differential equations. Optimal control of the suggested model is the main objective of this work. Two control variables are presented in this model to minimize the population number of infected and asymptotically infected people. Necessary optimality conditions are derived. The Grünwald–Letnikov nonstandard weighted average finite difference method is constructed for simulating the proposed optimal control system. The stability of the proposed method is proved. In order to validate the theoretical results, numerical simulations and comparative studies are given.

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