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This study reconsiders a duopoly model with advertisement introduced earlier by Ahmed, Agiza and Hassan [1999]. It demonstrates three major findings. The first is that the model can be destabilized via either flip bifurcation or Hopf bifurcation. The second is that a half-pitchfork bifurcation of the output occurs when the advertisement dynamics is...
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Context 1
... contrast, the same bifurcation scenario takes place in case (2) under more plausible cost structure if the advertisement outlay is higher, as observed in Figure 4B. Figure 5 depicts the two-dimensional bifurcation diagram of (9) in the (c, a) plane in which c ranges from 1.8 to 2.3 and a from 2+r to 3+r. As in Figure 4, the regions of periodic and aperiodic behavior are indicated with different colors and the solutions are infeasible in the white region. ...
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... simulations are performed with different values of c there. In the first simulation, a is increased along the vertical segment AB of Figure 5 while c = 1.85 and r = 0.6 are fixed. The resultant bifurcation diagram against a is depicted in Figure 6A. ...
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... can also be observed: the range of q 1 as well as the diameter of one of the period-2 cycles become much smaller as c decreases with all other parameter values kept unchanged. Alternatively, if we choose c as the bifurcation parameter and fix a = 3 and r = 0.6, then we have a different type of bifurcation scenario as illustrated in Figure 7 in which we increase c along the horizontal segment CD from 1.8 to 2.0935 as shown in Figure 5. Since a is unchanged along this segment, the advertisement map keeps generating a two period cycle. ...
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Citations
... It is necessary to establish a dynamic game model based on bounded rationality. In the proposed approach, we will improve and supplement the model according to the processing method of reference [34,39,40,[42][43][44][45][46][47] and then analyse the dynamic system. In the second stage of the game, the respective pro t function is derived, and the marginal pro t of the enterprise in period is: e product takes production as the decision variable. ...
At present, the problems of homogenization and low quality in China’s iron and steel industry are particularly prominent and the ability of the enterprises to cope with change is insufficient. Adopting product differentiation strategy and dynamic adjustment strategy can allow steel enterprises and the industry to better adapt to future changes. By introducing the product differentiation degree (substitution coefficient) and the bounded rationality strategy to simulate these two strategic means, this paper constructs an extended two-stage dynamic game model to analyse the dynamic game scenarios and steel market stability in China. As new findings, we report the following: (1) The system is more likely to fall into an unbalanced state when multiple enterprises adopt the policy of dynamic output adjustment simultaneously. (2) Enterprises with large output and small output have different output adjustment policies. When enterprises with big-scale output adopt a bit larger adjustment policies, enterprises with small output will be strongly impacted, and the available adjustment space will be sharply compressed. (3) The gradual increase in the difference between products reduces the stability of the market. (4) When product differentiation and bounded rationality strategies coexist, the steel market may fall into an unbalanced state when the degree of product difference increases excessively and the enterprise adopts more drastic output adjustment policies. Therefore, there are pros and cons to product differentiation strategy and bounded rationality adjustment strategy. When each steel oligopoly enterprise formulates a production plan, it needs to comprehensively consider the output changes of the other enterprises and carefully weigh the strategic issues.
