A Continuous Time Bertrand Duopoly Game With Fractional Delay and Conformable Derivative: Modeling, Discretization Process, Hopf Bifurcation, and Chaos

Abstract

The purpose of this paper is three-fold. First, we present a discretization process to obtain numerical solutions of a conformable fractional-order system with delays. Second, we extend the classical Bertrand duopoly game with integer delays to that with fractional delays. Third, we extend the game based on ordinary differential derivative to that based on conformable fractional-order derivative. Finally, we analyze the local stability, Hopf bifurcation, and chaos of the proposed game model.

A Bertrand Duopoly Game with Long-Memory Effects

“…As we know, the game theory is one of hottest academic topics for a long time. ere are many types of games, such as the Cournot game [1][2][3][4][5][6][7], the Bertrand game [8][9][10][11][12][13][14], the Stackelberg game [15][16][17], the evolutionary game [18], and the mixed game [19]. Many researchers have studied duopoly games with some interesting characteristics, such as bounded rationality [1,[15][16][17], technology licensing [3,20], and R&D [6,13].…”

Bifurcation Analysis of a Duopoly Game with R&D Spillover, Price Competition and Time Delays

“…Li and Ma [10] considered a small rational dual-channel game and simulate their model's complex dynamic behaviour in their research. Many researchers have explored the complex dynamical behaviours of this type of models from different aspects, such as differentiated goods [11][12][13][14][15], bounded rationality [16], heterogeneous firms [7,[17][18][19], delayed decisions [20][21][22][23][24][25] and other factors [26][27][28].…”

The effect of caputo fractional difference operator on a novel game theory model

“…In [18] a 4D fractional chaotic financial system characterized by investment incentives is presented, whereas in [21] the dynamics of a fractional economic system characterized by transient chaos are studied. Unlike continuous-time systems, very few papers regarding chaotic phenomena in economic systems described by discrete-time dynamics have been published to date [11,19,2]. These economic systems, which usually involve concepts from game theory applied to oligopolistic markets, generate complex dynamics (described by fractional-order difference equations) that lead to the existence of bifurcations and chaos [11].…”

“…These economic systems, which usually involve concepts from game theory applied to oligopolistic markets, generate complex dynamics (described by fractional-order difference equations) that lead to the existence of bifurcations and chaos [11]. For example, in [19] the Hopf bifurcations and the chaotic attractors of the discrete-time version of the Bertrand duopoly game with fractional delay is studied. In [2] chaotic phenomena discovered in a fractionalorder discrete-time Cournot duopoly game are analysed.…”