重要混沌经济学.pptx
I.I.The Cournot duopoly Kopel ModelThe 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.第1页/共33页混沌经济学 Day于于1982年将非线性动态引入到经济学中，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了崭新的视角 宏观经济宏观经济中存在混沌现象中存在混沌现象 在在微观经济学微观经济学领域，厂商或者其他经济个体领域，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，呈现出混沌动态可能产生波动，呈现出混沌动态第2页/共33页 X和和Y代表两个寡头厂商代表两个寡头厂商 4.1 一个古诺双寡头经济模型描述 厂商厂商X和和厂商厂商Y在在t+1时间段生产的产品数量分别时间段生产的产品数量分别用用x(t+1)和和y(t+1)表示表示 Nash平衡点：平衡点：古诺双寡头古诺双寡头 Kopel经济模型经济模型777 Kopel M.Chaos,Solitons&Fractals,1996,7:20312048 第3页/共33页4.2 分形分析 分形图：分形图：平衡点平衡点周期周期混沌混沌平衡点平衡点混沌混沌周期周期，第4页/共33页光滑经济周期光滑经济周期非光滑非光滑经济周期经济周期混沌混沌 由光滑经济周期演变为混沌：由光滑经济周期演变为混沌：4.2 分形分析 第5页/共33页 混沌混沌吸吸引子共存现象引子共存现象两个共存的混沌吸引子两个共存的混沌吸引子吸引域吸引域由于对称性，混沌吸引子共存现象普遍存在由于对称性，混沌吸引子共存现象普遍存在4.2 分形分析 第6页/共33页4.3 混沌吸引子的计算机辅助证明 将双寡头将双寡头Kopel模型改写为向量形式：模型改写为向量形式：其中：其中：研究映射研究映射 的动态（的动态（）：）：定义为定义为 ，以此类推得到：，以此类推得到：第7页/共33页 定理定理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：关于四边形经济模型具有如下性质：关于四边形 的映射的映射 存在一个闭的不变集存在一个闭的不变集 ，使得，使得 与与2 2个符号的移位映射半共轭，且个符号的移位映射半共轭，且因此因此，当当 时，古诺双寡头时，古诺双寡头Kopel经济经济模型有模型有正拓扑熵正拓扑熵。第8页/共33页4.4 间歇混沌特性分析l PM-I型间歇混沌型间歇混沌：分形图分形图第9页/共33页4.4 间歇混沌特性分析l 分岔前后分岔前后x的时间序列的时间序列分岔前，分岔前，分岔后，分岔后，过渡混沌过渡混沌6倍周期点倍周期点第10页/共33页4.4 间歇混沌特性分析结论：服从幂指数为结论：服从幂指数为-0.496-0.496的幂律分布的幂律分布幂指数特征值幂指数特征值:-0.5l PM-I型间歇混沌：型间歇混沌：层流态平均持续时间分布层流态平均持续时间分布第11页/共33页l 诱发激变导致的诱发激变导致的间歇混沌间歇混沌：4.4 间歇混沌特性分析分形图分形图发生激变前，发生激变前，发生激变前，发生激变前，发生激变后，发生激变后，发生激变后，发生激变后，第12页/共33页4.4 间歇混沌特性分析结论：服从幂指数为结论：服从幂指数为-0.65-0.65的幂律分布的幂律分布幂指数特征值幂指数特征值:-3/2,-1/2l 诱发激变导致的诱发激变导致的间歇混沌间歇混沌：层流态平均持续时间分布层流态平均持续时间分布第13页/共33页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 间歇混沌特性分析第14页/共33页4.5 长期平均利润分析l 混沌能否带来更多的利润混沌能否带来更多的利润？混沌动态的平均利润混沌动态的平均利润:非零平衡点非零平衡点:第15页/共33页4.5 长期平均利润分析结论：混沌市场并不是完全有害的结论：混沌市场并不是完全有害的第16页/共33页4.6 控制混沌到Nash平衡点 考虑受控的古诺双寡头考虑受控的古诺双寡头Kopel经济模型，经济模型，平衡点平衡点 是是局部渐近稳定的，当且仅当局部渐近稳定的，当且仅当 定理定理l 稳健的经济市场仍然是人们最需要的稳健的经济市场仍然是人们最需要的第17页/共33页4.6 控制混沌到Nash平衡点令（）l 仿真结果仿真结果第18页/共33页重点研究了一个古诺双寡头经济模型中的各种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析了混沌对利润的影响，得到了混沌并非完全有害的结论有害的结论小结小结：四、经济系统中的混沌动态研究第19页/共33页n 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:I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model Consider two firms X and Y:(1)Where,denote the goods quantities that firm X and firm Y producein period t,respectively.第20页/共33页I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model n 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).第21页/共33页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 I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model n Bifurcation analysis bifurcation diagram:rich and complex dynamics Fig.1 Bifurcation diagram.(a)Fix ,and .(b)Fix and (b)(a)第22页/共33页I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model Observation of chaotic attractors and basins of attraction Smooth CycleLost of SmoothnessChaoticFig.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)第23页/共33页I.I.The Cournot duopoly Kopel ModelThe 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)第24页/共33页I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model n Horseshoe Chaos in the model A convenient expression for the Kopel model is described as follows:where(3)After a great many trial attempts,we will discuss the dynamics of system(3)with under the map and obtain that there exists a horseshoe in this attractor.第25页/共33页We take a proper quadrangle|ABCD|to be a subset in the plane with its four vertices being I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model Fig.4 The attractor when and the proper quadrangle .第26页/共33页Fig.5.The quadrangle and its image .I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model Theorem 1.For the map corresponding to the quadrangle ,there exists a closed invariant set for which is semi-conjugate to the 2-shift map.Hence,.第27页/共33页I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model Proof.We select two appropriate subsets in .The first one is denoted by a as shown the yellow quadrangle in Fig.6,with and be its left and right edge,respectively.Then under the map ,is mapped to which is on the right side of the edge ,and is mapped to which is on the left side of the edge .To prove the above theorem,we should find two mutually disjointed subsets of such that there exists a -connected family with respect to them.Fig.6 The subset and its image under the map with .第28页/共33页I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model The second subset is the purple quadrangle,denoted by b,as shown in Fig.7.Take and to be the left and right edge of b,respectively.Then the image is on the left side of the edge ,and the image is on the right side of the edge .Fig.7 The subset and its image under the map with .第29页/共33页I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model Upon the above simulation results,it is easy to see that the subset a and b are disjointedand it follows that for every connection v with respect to a and b,the images and lie wholly across the quadrangles ,that is to say,the images and are still connections with respect to a and b.According to topological horseshoe Theorem,there exists a -connected family,which means that is semi-conjugate to the 2-shift map.Hence,based on the Lemma,we know that the entropy of is not less than ,which implies system(1)has positive topological entropy with .The proof is thus completed.第30页/共33页I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model n Intermittency chaos dynamics A typical intermittency phenomenonIntermittent events are characterized by time series that display time intervals with low variabilities interrupted by bursts of very high variabilitiesLaminar stateBurst state第31页/共33页I.I.The Cournot duopoly Kopel ModelThe Cournot duopoly Kopel Model n Type of intermittency:PM type-I IntermittencyPM type-II IntermittencyPM type-III Intermittency(1)Pomeau-Manneville intermittency(2)Crisis-induced intermittency:(3)In-out IntermittencyOn-off IntermittencyAttractor wideningAttractor merging第32页/共33页感谢您的观看！第33页/共33页
