Price competition has become a universal commercial phenomenon nowadays. This paper considers a dynamic Bertrand price game model, in which enterprises have heterogeneous expectations. By the stability theory of the dynamic behavior of the Bertrand price game model, the instability of the boundary equilibrium point and the stability condition of the internal equilibrium point are obtained. Furthermore, bifurcation diagram, basin of attraction, and critical curve are introduced to investigate the dynamic behavior of this game. Numerical analysis shows that the change of model parameters in a dynamic system has a significant impact on the stability of the system and can even lead to complex dynamic behaviors in the evolution of the entire economic system. This kind of complex dynamic behavior will cause certain damage to the stability of the whole economic system, causing the market to fall into a chaotic state, which is manifested as a kind of market disorder competition, which is very unfavorable to the stability of the economic system. Therefore, the chaotic behavior of the dynamical system is controlled by time-delay feedback control and the numerical analysis shows that the effective control of the dynamical system can be unstable behavior and the rapid recovery of the market can be stable and orderly.

中文翻译：

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基于差异产品和异构期望的Bertrand三寡头动态分析和混沌控制

价格竞争已成为当今普遍的商业现象。本文考虑了一个动态的Bertrand价格博弈模型，其中企业具有不同的期望。通过Bertrand价格博弈模型动力学行为的稳定性理论，得出边界平衡点的不稳定性和内部平衡点的稳定性。此外，引入分叉图，吸引盆和临界曲线来研究该游戏的动态行为。数值分析表明，动态系统中模型参数的变化对系统的稳定性有重要影响，甚至可能导致整个经济系统演化过程中复杂的动态行为。这种复杂的动态行为会对整个经济体系的稳定性造成一定的破坏，使市场陷入混乱状态，表现为一种市场无序竞争，对经济的稳定性非常不利。系统。因此，通过时滞反馈控制来控制动力系统的混沌行为，数值分析表明，动力系统的有效控制可能是不稳定的行为，市场的快速恢复将是稳定有序的。