2010 年系统科学与复杂网络研讨会学术报告 Chaos Modelling and Applications in Financial Engineering 混沌动力学系统建模及在金融工程领域中的应用陈增强教授南开大学

2010年系统科学与复杂网络研讨会学术报告 Chaos Modelling and Applications in Financial Engineering 混沌动力学系统建模及在金融工程领域中的应用陈增强教授南开大学

Chaos Modelling and Applications in Financial Engineering ZENGQIANG CHEN Department of Automation Nankai University Email: chenzq@nankai.edu.cn

感谢上海理工大学许晓鸣校长香港城市大学陈关荣教授的热情邀请和上海系统工程研究院的支持

Outline • Introduction to Chaos • Topological Horseshoe Theory • Chaos in Economics • The Analysis of two Economic Systems

Introduction to Chaos What is Chaos? Chaos exists in nonlinear dynamical systems

Introduction to Chaos Basic properties of Chaos • sensitive dependence on initial conditions .506127 .506 1961, Lorenz’s experiment of weather prediction

Introduction to Chaos • The trajectory is bounded • and never repeats • Self-similar

Introduction to Chaos • Unpredictability Chaosisaperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions[1]. [1] “Nonlinear Dynamics and chaos”, Strogatz, S. H., Addison-Wesley Publishing Company, Boston, 1994.

Introduction to Chaos Classical Chaotic attractors: Lorenz attractor

Introduction to Chaos Classical Chaotic attractors: Rössler attractor

Introduction to Chaos Classical Chaotic attractors: Chen attractor

Introduction to Chaos How to determine chaos: • Lyapunov exponents • Topological entropy • Bifurcation, such as period-doubling route to chaos • Melnikov method • Ši’lnikov method • Topological horseshoe theory and symbolic dynamics

Topological Horseshoe Theory Smale horseshoe map: pioneering work Smale horseshoe map is the prototypical map possessing a chaotic invariant set Theorem. There is a closed invariant set for which is conjugate to a two-sided 2-shift.

Topological Horseshoe Theory Topological horseshoe: J. Kennedy and J.A. Yorke’s work[2] Assumptions: (1) is a separable metric space (2) is locally connected and compact (3) The map is continuous (4) The set and are disjoint and compact, and each component of intersects both and (5) has crossing number Theorem. There is a closed invariant set for which is semi-conjugate to a one-sided M-shift. (If f is homeomorphism, then is two-sided). Remark. (1) This theorem applies to invariant set with only one expanding direction; (2) Core concept , “The crossing number”, is useless in practical point of view [2] J. Kenney, and J. A. Yorke, “Topological horseshoes,” Trans. Amer. Math Soc., vol. 353, pp. 2513-2530, Feb. 2001.

Topological Horseshoe Theory Topological horseshoe: Yang Xiao-Song’s work Proposing a recent famous topological horseshoe theorem • Applicable for continuous system, piecewise continuous system, discrete system • Applicable for invariant set withmultiple expanding direction • Combines with computer numericalsimulations

Topological Horseshoe Theory Topological horseshoe: Yang Xiao-Song’s work[3] Definition: f-connected family Let be a metric space, is a compact subset of , and is map satisfying the assumption that there exist m mutually disjoint subsets and of , the restriction of to each , i.e., is continuous. Definition. Let be a compact subset of , such that for each is nonempty and compact, then is called a connection with respect to . Let be a family of connections s with respect to satisfying the following property: Then is said to be a -connected family with respect to . [3] X. S. Yang, and Y. Tang, “Horseshoe in piecewise continuous maps,” Chaos Solitons & Fractals, vol. 19, pp. 841–845, Apr. 2004.

Topological Horseshoe Theory • Theorem: Theorem.Suppose that there exists a f-connected family with respect to and . Then there exists a compact invariant set , such that is semi-conjugate to m-shift. Topological entropy=logm Remark. The semi-conjugacy is defined as follows. If there exists a continuous and onto map Such that , then is said to be semi-conjugate to . An important fact is the following statement. Lemma.Consider two dynamical systems and . If is semi-conjugate to ,then the topological entropy of is not less than that of , i.e. .

Topological Horseshoe Theory • Important Comment (1) Topological horseshoe theorem Continuous time system Poincaré Map Topological horseshoe theorem ~ the computer-assisted computation • more applicable, can be applied to many systems • provides a geometrical method to find the topological horseshoe (2)Every statement about existence of horseshoe can tolerate some fixed bounded errors , because of inevitable of errors in computer computation

Topological Horseshoe Theory • Steps for applying the Theorem : -- Continuous case • Construct Poincaré cross-section and the proper Poincaré map • Find an invariant set, such that the Poincaré map is semi-conjugate • to a m-shift map. -- Discrete case Find a proper map which is semi-conjugate to a m-shift map.

拓扑马蹄理论简介 研究背景 ● 从数学意义上严格地证明混沌吸引子的存在性是一项重要工作。 ●目前，对于连续系统，常用的证明混沌的方法是Šil’nikov方法。特点：应用过程繁琐、有一定的局限性 ● 近年来发展迅速的拓扑马蹄理论提供了一种较为简便的方法。特点：应用广泛、操作简单、充分利用了计算机数值计算

拓扑马蹄理论简介 (2) 一个重要的拓扑马蹄定理—Kennedy和Yorke[5] 以交叉数作为前提，不实用 • 关于拓扑马蹄理论的重要工作 (1)开拓性工作—Smale马蹄映射[4] Smale马蹄比较规范，条件也较为苛刻，它假设映射是一个微分同胚，从数值角度看，计算量太大，不便于应用 (3)Zgliczynski和Gidead的拓扑马蹄定理[6] 可以用来研究具体系统的拓扑马蹄存在性，有一定实用性 [4] Wiggins S. New York: Springer-Verlag, 1990 [5] Kennedy J, Yorke J. Tran. Amer. Manth. Soc., 2001, 353: 2513~2530 [6] Zgliczynski P, Gidea M. J. Differential Equations, 2004, 202: 32~58

3.1拓扑马蹄理论简介 拓扑马蹄：设X是一个度量空间。考虑一个（分段）连续映射。若存在一个紧致的不变集，使得限制在上的动态与移位映射（半）拓扑共轭，那么称具有拓扑马蹄。拓扑马蹄引理：假定存在一簇对应于和的连接簇，则存在一个紧致不变集，使得与一个移位映射半拓扑共轭。因此，的拓扑熵满足。 • Yang提出的拓扑马蹄引理[7] 符号动力学与计算机数值计算相结合适用于离散系统，（分段）连续系统；整数阶系统，分数阶系统；混沌系统，超混沌系统 [7] Yang X S, Tang Y. Chaos Solitons & Fractals, 2004, 19(4): 841~845

拓扑马蹄理论简介 • （1）找到合适的Poincaré截面； • （2）在截面上定义合适的子集( 和 )； • （3）定义合适的回归次数的Poincaré映射。 • 连续系统中寻找拓扑马蹄的步骤三大技术难点已有工作： Rössler系统、改进的Chen系统、Lorenz系统、 Hopfield神经网络等我们的工作：将该理论推广应用，证明典型经济系统的混沌吸引子存在性

Chaos in Economics Chaotic economics (Nonlinear economics): Day is among the pioneers of chaotic research in economics as this field was becoming increasingly popular in the early 1980s. • Ref. [4]: Wandering growth cycles: Chaos emerge • Nowadays, chaotic economics includes almost every fields of economics: • Economic cycle, Monetary, Finance, Stock market, Firm supply and demand…… [4] Day, R., Irregular Growth Cycles, American Economic Review, 72, 406-414, 1982.

Chaos in Economics • Topics on chaotic economics: • Investigating real economic data: to find evidence of chaos • Analyzing nonlinear dynamics of some economic behaviors • Explaining the intrinsic mechanism and reasons of economic behaviors • Predicting economic behavior • Modeling and analyzing economic behavior

Chaos in Economics Istanbul stock exchange [5]: ISE system has very high chaotic phenomena • Phase space reconstruction: The embedding dimension of ISE time series is very high, • and the strange attractor dimension is 0.15. Time series of ISE index 3D phase space of ISE time series [5] Muge Iseri,Hikmet Caglar, Nazan Caglar . A model proposal for the chaotic structure of Istanbul stock exchange . Chaos, Solitons and Fractals 36 (2008) 1392–1398

Chaos in Economics The $C/$US exchange rate [6]: chaotic structure Time series of daily exchange rate data 14/02/1973---29/03/2003 Lyapunov exponent of the time series • Chaos also exists in daily data for the Swedish Krona against Deutsche Mark, • the ECU, the US Dollar and the Yen exchange rates.[7] [6] R. Weston, The chaotic structure of the $C/$US exchange rate, International Business & Economics Research Journal, 2007, 6:19-28. [7] Mikael Bask.A positive Lyapunov exponent in Swedish exchange rate? Chaos, Solitons and Fractals 14(2002) 1295-1304.

Chaos in Economics Economic prediction[8]: • Several economic time series are tested by using a deterministic predictive • technique is introduced, which is based on the embedding theorem by Takens • and the recently developed wavelet networks • Based on phase space reconstruction technique, the predicted values correspond • quite well with the actual values. Chinese microeconomic time series National financial expenditure Gross output value of industry [8] LG Cao, YG Hong, HZ Zhao and SH Deng, Predicting economic time series using a nonlinear deterministic technique, Computational Economics, 1996, 9:149-178.

Chaos in Economics Economic Modeling: • Lots of economic models are presented to study the rich nolinear dynamical behavior. Such as: cobweb price adjustment processes, optimal growth models, overlapping generations models, Keynesian business cycle models, Kaldor and Goodwin growth cycle models, demand models with adaptive preferences, models of productivity growth, duopoly models, and others.. • Researchers analyze the chaotic properties of these models: • Equilibrium, Lyapunov exponents, bifurcation diagram……

The Analysis of two Economic Systems • The Cournot duopoly Kopel economic Model • A Business cycle model

2010 年系统科学与复杂网络研讨会学术报告 Chaos Modelling and Applications in Financial Engineering 混沌动力学系统建模及在金融工程领域中的应用陈增强教授南开大学