In this paper, the local and global dynamics of a two-stage Bertrand dynamic duopoly model with R&D spillover are studied. This paper aims to study the complex dynamics of the built model by changing the speed of adjustment and other parameters. Combining analysis tools and numerical techniques, we discuss the stability of the system and calculate the stability conditions at first. Then we infer the dynamic phenomena of the system with different sets of parameters and get some interesting results. If two firms operate under different market environment, the system will exhibit two kinds of chaotic intermittent phenomena. While if they operate in same market environment, the system will exhibit the synchronization and on–off intermittency. Finally, through the study of complex basins and multistability, we see how the number of attractors and the size or shape of the basin will change with the change of parameters.

中文翻译：

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具有研发溢出和半合谋的 Bertrand 博弈中的分岔、全局动态和同步

在本文中，研究了具有 R&D 溢出的两阶段 Bertrand 动态双头垄断模型的局部和全局动态。本文旨在通过改变调整速度和其他参数来研究构建模型的复杂动力学。结合分析工具和数值技术，我们首先讨论了系统的稳定性并计算了稳定性条件。然后我们用不同的参数集推断系统的动态现象，并得到一些有趣的结果。如果两个企业在不同的市场环境下运作，系统会表现出两种混沌间歇现象。而如果它们在相同的市场环境中运行，则系统会表现出同步和开关的间歇性。最后，通过对复杂盆地和多稳性的研究，