Abstract
In this article, a novel discrete system based on an economic model is introduced. Conditions for local stability of the model’s fixed points are obtained. Existence of supercritical Neimark–Sacker bifurcation is shown around the game’s Nash equilibrium. Existence of stable period-2 orbits resulting from flip bifurcation around the game’s Nash equilibrium is also proved. Existence of chaotic dynamics in the proposed game is also shown via two routes: Neimark–Sacker bifurcation and flip bifurcation. Based on the bifurcation theory of discrete-time systems, sufficient conditions of transcritical bifurcation are derived and applied to the proposed model. The interesting phenomenon of coexisting multi-chaotic attractors, such as coexistence of two, three, four, and five-piece chaotic attractors, is found in the proposed model. For this reason, numerical simulations of basins of attraction are performed to verify the appearance of this important phenomenon that reflects the unpredictability and higher complexity in the proposed game.
Highlights
Studying the dynamics of economic games has recently become the focus.[1,2,3,4,5,6,7,8,9,10] As a small number of firms produces homogeneous products and dominate the trade, the universal market is said to be oligopolistic
The complex dynamics of oligopolistic games arise from the fact that reaction of competitors must be considered by firms
A novel triopoly game with bounded rationality has been modeled according to local profit maximization
Summary
Studying the dynamics of economic games has recently become the focus.[1,2,3,4,5,6,7,8,9,10] As a small number of firms produces homogeneous products and dominate the trade, the universal market is said to be oligopolistic. The fixed point E3 = (0, 0, 0:5(a À c3)) is LAS if all the following conditions hold (i) À 2\a11⁄2a À c1 + 0:5b13(c3 À a)\0 (ii) À 2\a21⁄2a À c2 + 0:5b23(c3 À a)\0 (iii) À 2\a3(c3 À a)\0. The sufficient conditions of local stability of the fixed points E5, E6 can be obtained They are summarized by the following theorems. B223 + 4) is LAS if all the following conditions hold (i) À 2\a11⁄2a À c1 À b12x3 À b13x4\0 (ii) 0\a2a3x3x4b223 À m3a2x4b23 + m31⁄2(1 + aa2) À (a2c2 + 4a2x3) + 1\2 (iii) a3x3x4b223 + (1 À m3)x4b23 + (a À c2 À 4x3)(m3 À 1).[0] (iv) a2a3x3x4b223 À (1 + m3)a2x4b23 + (m3 + 1) 1⁄2a2(a À c2 À 4x3) + 2.0 where m3=1+a3(aÀc3À4x4+b23x3),x3=(2(aÀc2)+ b23(c3 À a))=b223 + 4, andx4 = (2(a À c3) + b23(a À c2))= b223 + 4.
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