重要混沌经济学课件

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,二级,三级,四级,五级,*,Oligopoly theory,is one of the oldest branches of mathematical economics dated back to 1838 when its basic model was proposed by French economist,Cournot,The Cournot duopoly Kopel Model,In the recent literatures,it is also demonstrated that the,oligopolistic markets,may become chaotic.,9 T.Puu,Chaos in duopoly pricing,Chaos Solitons Fractals 1(1991)573-581.,10 T.Puu,The chaotic duopolists revisited,J.Econom,Behav.Organ.33(1998)385-394,Among the first who have shown the Cournot model may lead to complex,behavior,such as periodic and chaotic behavior,was Puu,910,.,In this work,we consider a general case of a duopoly model:,The Cournot duopoly Kopel Model,11,11 Kopel,M.,”Simplex and Complex Adjustment Dynamics in Cournot Duopoly Models,”Chaos,Solitions and Fractals,7,2031-2048,1996.,1,Oligopoly theory is one of the,The Cournot duopoly Kopel Model,Previous work:,Our work:,Analyzing chaotic behavior,numerically,A rigorous proof for existence of chaos from mathematical,point of view is given.,Two different types of intermittent chaos in this model are found,and analyzed.,2,The Cournot duopoly Kopel Mode,混沌经济学,Day,于,1982,年将非线性动态引入到经济学中，,引发了人们对传统经济学的反思，为人们提供,了崭新的视角,宏观经济,中存在混沌现象,在,微观经济学,领域，厂商或者其他经济个体,所经营产品的价格、生产或销售的产品数量都,可能产生波动，呈现出混沌动态,3,混沌经济学 Day于1982年将非线性动态引入到经济学中，,X,和,Y,代表两个寡头厂商,4.1,一个古诺双寡头经济模型描述,厂商,X,和,厂商,Y,在,t,+1,时间段生产的产品数量分别用,x,(,t,+1),和,y,(,t,+1),表示,Nash,平衡点：,古诺双寡头,Kopel,经济模型,7,7 Kopel M.Chaos,Solitons&Fractals,1996,7:20312048,4,X和Y代表两个寡头厂商 4.1 一个古诺双寡头经济模型,4.2,分形分析,分形图：,平衡点,周期,混沌,平衡点,混沌,周期,，,5,4.2 分形分析 分形图：平衡点周期混沌平衡点混沌周,光滑经济周期,非光滑,经济周期,混沌,由光滑经济周期演变为混沌：,4.2,分形分析,6,光滑经济周期非光滑经济周期混沌 由光滑经济周期演变为混沌,混沌,吸,引子共存现象,两个共存的混沌吸引子,吸引域,由于对称性，混沌吸引子共存现象普遍存在,4.2,分形分析,7,混沌吸引子共存现象两个共存的混沌吸引子吸引域由于对称性,4.3,混沌吸引子的计算机辅助,证明,将双寡头,Kopel,模型改写为向量形式：,其中：,研究映射 的动态（）：,定义为,，以此类推得到：,8,4.3 混沌吸引子的计算机辅助证明 将双寡头Kopel,定理,4.3,混沌吸引子的计算机辅助,证明,Kopel,经济模型具有如下性质：关于四边形 的映射 存在一个闭的不变集 ，使得,与,2,个符号的移位映射半共轭，且,因此,，,当 时，古诺双寡头,Kopel,经济,模型有,正拓扑熵,。,9,定理4.3 混沌吸引子的计算机辅助证明 Kopel经济模型,4.4,间歇混沌特性分析,PM-I,型间歇混沌,：,分形图,10,4.4 间歇混沌特性分析 PM-I型间歇混沌：分形图10,4.4,间歇混沌特性分析,分岔前后,x,的时间序列,分岔前，,分岔后，,过渡混沌,6,倍周期点,11,4.4 间歇混沌特性分析 分岔前后x的时间序列分岔前，分岔,4.4,间歇混沌特性分析,结论：服从幂指数为,-0.496,的幂律分布,幂指数特征值,:,-,0.5,PM-I,型间歇混沌：,层流态平均持续时间分布,12,4.4 间歇混沌特性分析结论：服从幂指数为-0.496的幂律,诱发激变导致的,间歇混沌,：,4.4,间歇混沌特性分析,分形图,发生激变前，,发生激变前，,发生激变后，,发生激变后，,13,诱发激变导致的间歇混沌：4.4 间歇混沌特性分析分形图发生,4.4,间歇混沌特性分析,结论：服从幂指数为,-0.65,的幂律分布,幂指数特征值,:,-3/2,-1/2,诱发激变导致的,间歇混沌,：,层流态平均持续时间分布,14,4.4 间歇混沌特性分析结论：服从幂指数为-0.65的幂律分,在经济学系统中出现的间歇混沌现象可以,解释为系统本身具有调节机制，不借助于任,何外力，系统总是能够将混乱的市场调整回,(,相对,),平稳状态,或者解释为系统有记忆机制，总是能够记,住混乱前的状态并恢复,4.4,间歇混沌特性分析,15,在经济学系统中出现的间歇混沌现象可以4.4 间歇混沌特性分析,4.5,长期平均利润分析,混沌能否带来更多的利润,？,混沌动态的平均利润,:,非零平衡点,:,16,4.5 长期平均利润分析 混沌能否带来更多的利润？混沌动态的,4.5,长期平均利润分析,结论：混沌市场并不是完全有害的,17,4.5 长期平均利润分析结论：混沌市场并不是完全有害的17,4.6,控制混沌到,Nash,平衡点,考虑受控的古诺双寡头,Kopel,经济模型，,平衡点 是,局部渐近稳定的，当且仅当,定理,稳健的经济市场仍然是人们最需要的,18,4.6 控制混沌到Nash平衡点 考虑受控的古诺双寡头,4.6,控制混沌到,Nash,平衡点,令,（）,仿真结果,19,4.6 控制混沌到Nash平衡点令（,重点研究了一个古诺双寡头经济模型中的各种,混沌动态，从理论上证明了混沌存在性，并分,析了混沌对利润的影响，得到了混沌并非完全,有害的结论,小结,：,四、,经济,系统中的混沌动态研究,20,重点研究了一个古诺双寡头经济模型中的各种小结：四、经济系,Model description,(1).Both firms must consider the actions and reactions of the competitor,(2).The competitors have choose their actions simultaneously,(3).Each firm forms the expectation on the quantity of the other firm,which,depend on their own quantity and the quantity of the other firm both,produced in the previous period,in order to determine a profit maximizing,quantity to produce in the next period.,Remarks:,The Cournot duopoly Kopel Model,Consider two firms,X,and,Y,:,(1),Where,denote the goods quantities that firm,X,and firm,Y,produce,in period,t,respectively.,21,Model descriptionRemarks:The,The Cournot duopoly Kopel Model,Nash-equilibria of the Kopel model,The fixed points(Nash-equilibrium)of system(1)satisfy the equations:,The solutions of Eq.(2)give four equilibria,:,(2),for,for,Remark:,The fixed points depend on .In case,we should have (positive solution).,Also,in case and,we should have (real solution).,22,The Cournot duopoly Kopel Mode,bifurcation diagram provides a general view of the evolution process of the dynamical behaviors by plotting a state variable with the abscissa being one parameter,The Cournot duopoly Kopel Model,Bifurcation analysis,bifurcation diagram:rich and complex dynamics,Fig.1 Bifurcation diagram.(a)Fix ,and .(b)Fix and,(b),(a),23,bifurcation diagram provides a,The Cournot duopoly Kopel Model,Observation of chaotic attractors and basins of attraction,Smooth Cycle,Lost of Smoothness,Chaotic,Fig.2 Different attractors in Kopel model.(a)One smooth invariant cycle with ,.,(b)Invariant cycle loses its smoothness when ,.(c)Chaotic attractor with,.,(a),(b),(c),24,The Cournot duopoly Kopel Mode,The Cournot duopoly Kopel Model,Coexistence of two chaotic attractors:,Fig.3 Two chaotic attractors coexist with different initial conditions when .,(a)Phase portraits of the two chaotic attractors and the four Nash equilibria;(b)The basins of attractions.,(a),(b),25,The Courn