动态Bertrand模型的分岔研究与混沌控制  

Bifurcation Research and Chaos Control of Dynamic Bertrand Model

作　　者：刘荣荣 周伟[1] 王文瑞 柏恩鹏 LIU Rong-rong;ZHOU Wei;WANG Wen-rui;BAI En-peng(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China;School of Economics and Management,Lanzhou Jiaotong University,Lanzhou 730070,China;School of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学数理学院,兰州730070 [2]兰州交通大学经济管理学院,兰州730070 [3]西北师范大学数学与统计学院,兰州730070

出　　处：《兰州交通大学学报》2020年第2期105-113,共9页Journal of Lanzhou Jiaotong University

基　　金：国家自然科学基金(61364001);兰州交通大学青年科学研究基金(2015029)。

摘　　要：通过技术差距将R&D溢出与产品替代程度联系起来,建立了具有R&D溢出的动态Bertrand双寡头模型.利用稳定性判据对系统进行理论分析,基于数值仿真,研究不同技术差距下,系统的稳定域以及R&D调整速度和产品替代程度的变化对R&D投入的影响.研究表明:对R&D调整速度的变化,系统在技术差距较小时要比差距较大时更先失去稳定性,但技术差距不论大小,系统都会随着调整速度的增大而失去稳定性;对产品替代程度的变化,不同技术差距下系统都呈现出稳定-不稳定-稳定的状态.最后用延迟反馈控制法对混沌运动进行控制,使系统重新回到稳定的Nash均衡状态.A Bertrand duopoly dynamic model with R&D spillover is established by connecting R&D spillover with product substitution degree through the technology gap.The system is analyzed theoretically by using the stability criterion.Through numerical simulation,the influence of stability region,the speed of adjustment and product substitution degree on R&D investment under different technological gaps was studied.The result shows that the shape of the stability region will be different under different technology gap.For the change of the speed of adjustment,the system will lose stability first when the technical gap is smaller than the larger one,but the system will lose stability with the increase of the speed of adjustment regardless of the technical gap.For the change of product substitution degree and different technical gaps,the system presents a steady-unsteady-stable state.At last,delayed feedback control is used to control the chaotic motion to help the system return to the stable Nash equilibrium state.

关 键 词：R&D投入 R&D溢出 产品替代程度 技术差距 分岔 混沌控制 

分 类 号：F224[经济管理—国民经济]