Stable cycles in a Cournot duopoly model of Kopel

Abstract

We consider a discrete map proposed by M. Kopel that models a nonlinear Cournot duopoly consisting of a market structure between the two opposite cases of monopoly and competition. The stability of the fixed points of the discrete dynamical system is analyzed. Synchronization of two dynamics parameters of the Cournot duopoly is considered in the computation of stability boundaries formed by parts of codim-1 bifurcation curves. We discover more on the dynamics of the map by computing numerically the critical normal form coefficients of all codim-1 and codim-2 bifurcation points and computing the associated two-parameter codim-1 curves rooted in some codim-2 points. It enables us to compute the stability domains of the low-order iterates of the map. We concentrate in particular on the second, third and fourth iterates and their relation to the period doubling, 1:3 and 1:4 resonant Neimark-Sacker points.

Neimark–Sacker bifurcation and the generate cases of Kopel oligopoly model with different adjustment speed

“…For the case of fold bifurcation, some facts can be found in Lemma 2.1, that is, there exist two newborn fixed points when μ varies in the neighbourhood of 1 or 3, respectively. For the case of flip bifurcation, theoretical analysis and numerical simulation can be carried out as those done in [12]. 21 , it is nonhyperbolic and a flip bifurcation occurs at E 2 ; (vi) if μ = μ 22 , it is nonhyperbolic and a Neimark-Sacker bifurcation occurs at E 2 .…”

“…More attention was paid to bifurcation and chaos of map (1) by Agiza [1], and OGY method was introduced to control the chaos for improving the market's performance. With the help of MATCONTM, a set of package on the basis of MATLAB developing platform, Govaerts and Khoshsiar Ghaziani [12] used the method of bifurcation continuation to study the one (two)-parameter bifurcations. A revised Kopel oligopoly model with extrapolative foresight was constructed by Gao, Zhong and Mei [11], and Neimark-Sacker bifurcation analysis showed the complex dynamics and the transitions between different dynamical systems.…”

“…This paper is outlined as follows. The existence and local dynamics of fixed points for map (2) are investigated as the general case of results in [12], especially the conditions of Neimark-Sacker bifurcation, 1:3 and 1:4 resonances in Sect. 2.…”

Complex Dynamic Analysis for Game Model under Different Regulatory Levels in China’s Housing Rental Market

“…In traditional game theory, it is often assumed that participants are bounded rational and that participants are fully informed; however, the problem of incomplete information and bounded rationality is evident due to the complexity of the economic environment [19]. Evolutionary game is studied with a bounded rational group of participants [20][21][22][23][24][25][26]. In the continuous dynamic gaming, the lowyield strategy choice is constantly eliminated by the highyield strategy choice; after repeated gaming, elimination, and choice, the gaming parties achieve stability at the relative maximum benefit [27].…”

Bifurcation and Chaos in a Price Game of Irrigation Water in a Coastal Irrigation District

“…Analyzing bifurcation and chaos is not an easy task for most of researchers. Fortunately, there are many powerful methods for us to study bifurcation and chaos, such as 0-1 test algorithm for chaos [33][34][35][36][37][38], MATLAB package MatCont series [39][40][41][42][43] for the bifurcation of discrete, and continuous dynamical systems. This paper is organized as follows.…”