Abstract
In this paper, a discrete-time dynamic duopoly model, with nonlinear demand and cost functions, is established. The properties of existence and local stability of equilibrium points have been verified and analyzed. The stability conditions are also given with the help of the Jury criterion. With changing of the values of parameters, the system shows some new and interesting phenomena in terms to stability and multistability, such as V-shaped stable structures (also called Isoperiodic Stable Structures) and different shape basins of attraction of coexisting attractors. The eye-shaped structures appear where the period-doubling and period-halving bifurcations occur. Finally, by utilizing critical curves, the changes in the topological structure of basin of attraction and the reason of “holes” formation are analyzed. As a result, the generation of global bifurcation, such as contact bifurcation or final bifurcation, is usually accompanied by the contact of critical curves and boundary.
Highlights
Nonlinear dynamical system can describe many complicated and curious phenomena, and the theory of nonlinear dynamics is widely applied in many fields of scientific research studies
It is worth noticing that a group of scholars, Peng et al [9] considered a quadratic form of the cost function to establish a remanufacturing duopoly model and to study its complex dynamic characteristics
On the basis of considering the exponential demand function, the cost function is improved to quadratic cost. e firms considered in the market both have incomplete market information. erefore, they are boundedly rational. en, a discrete-time evolution model of the duopoly game with bounded rationality is built. e gradient adjustment mechanism is introduced to adjust the production strategy
Summary
Nonlinear dynamical system can describe many complicated and curious phenomena, and the theory of nonlinear dynamics is widely applied in many fields of scientific research studies. E primary purpose of this paper is to establish a discrete-time dynamic duopoly game model with bounded rationality, where both demand and cost functions are considered in nonlinear forms. By changing the values of the parameters, there are much complex phenomena that may arise, and we can further study their influences on the stability of the system or the structure of the basin of attraction through utilizing the theory of critical curves in subsequent sections.
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