Chaos Control and Anti-Control of the Heterogeneous Cournot Oligopoly Model

Marek Lampart,Alžběta Lampartová

doi:10.3390/math8101670

Abstract

The main aim of this paper focuses on chaos suppression (control) and stimulation (anti-control) of a heterogeneous Cournot oligopoly model. This goal is reached by applying the theory of dynamical systems, namely impulsive control. The main aim was to demonstrate, through massive numerical simulations and estimation of the maximal Lyapunov exponent, the 0-1test for chaos, and bifurcation analysis, that it is possible to control the dynamical behavior of the investigated model by finding injection values under which the desired phenomena are attained. Moreover, it was shown that there are injection values for which the injected system admits a self-excited cycle or chaotic trajectory.

Highlights

Chaosbut other types of movement patterns that come from experimental data and those generated by simulations of a given model reflect phenomena in mathematics and the sciences.Since it seems that chaos is something undesirable, one can try to avoid or prevent it [3].there are numerous situations where having random-looking and irregular patterns is desirable, e.g., the record of EEG
Economic dynamics has not been investigated for a long time because of the high mathematical computational requirements
The main focus was on dynamics investigation of a newly introduced injected two-dimensional discrete dynamical system

Summary

Introduction

Chaosbut other types of movement patterns (like period and quasiperiod) that come from experimental data and those generated by simulations of a given model reflect phenomena in mathematics and the sciences (including economics) (for motivation, see, e.g., Reference [1,2]) Since it seems that chaos is something undesirable, one can try to avoid or prevent it [3]. Mathematics 2020, 8, 1670 imperfect competition à la Cournot has been deeply researched for more than the past century (see, e.g., Reference [8,9,10,11]), and influence of demand type functions have been considered: linear in Reference [12], piecewise linear in Reference [11], for duopoly, iso-elastic in Reference [13], for triopoly and other, more sophisticated, types in Reference [14,15]) For this purpose, the impulsed dynamical system is introduced. The paper closes with concluding notes in the last part, Section 7

The Impulsive Control System

Chaos Control

Chaos Anti-Control

Chaos Control Versus Anti-Control

Alternative Impulse Control Approach

Conclusions