2010 年系统科学与复杂网络研讨会学术报告 chaos modelling and applications in...

20102010 年系统科学与复杂网络研讨会学术报告年系统科学与复杂网络研讨会学术报告

Chaos Modelling and Applications in Financial EngineeringChaos 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 Chaos Modelling and Applications in Financial Engineeringin Financial Engineering

ZENGQIANG CHEN

Department of Automation

Nankai University

Email: [email protected]

感谢感谢

上海理工大学许晓鸣校长上海理工大学许晓鸣校长

香港城市大学陈关荣教授香港城市大学陈关荣教授

的热情邀请的热情邀请

和上海系统工程研究院的支持和上海系统工程研究院的支持

OutlineOutline

Introduction to ChaosIntroduction to Chaos

Topological Horseshoe TheoryTopological Horseshoe Theory

Chaos in EconomicsChaos in Economics

The Analysis of two Economic SystemsThe Analysis of two Economic Systems


What is Chaos?

Chaos exists in nonlinear dynamical systems


Basic properties of Chaos

sensitive dependence on initial conditions

.506

.506127

1961, Lorenz’s experiment of weather prediction


The trajectory is bounded and never repeats

Self-similar


Unpredictability

Chaos is aperiodic 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.


Classical Chaotic attractors:Lorenz attractor


Rössler attractor

Classical Chaotic attractors:


Chen attractor

Classical Chaotic attractors:


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

OutlineOutline






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: J. Kennedy and J.A. Yorke’s work [2]

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.

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).


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 with multiple expanding direction Combines with computer numerical simulations


Topological horseshoe: Yang Xiao-Song’s work [3]

Definition: f-connected family

[3] X. S. Yang, and Y. Tang, “Horseshoe in piecewise continuous maps,” Chaos Solitons & Fractals, vol. 19, pp. 841–845, Apr. 2004.

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 .

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 Horseshoe TheoryTopological Horseshoe Theory Theorem:

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 entropy=logm

more applicable, can be applied to many systems provides a geometrical method to find the topological horseshoe


Important Comment

Topological horseshoe theorem ~ the computer-assisted computation

Continuous time system Topological horseshoe theorem

Poincaré Map

(1)

(2) Every statement about existence of horseshoe can tolerate some fixed bounded errors , because of inevitable of errors in computer computation


Steps for applying the Theorem :

I. Construct Poincaré cross-section and the proper Poincaré mapII. Find an invariant set, such that the Poincaré map is semi-conjugate to a m-shift map.

-- Continuous case

-- Discrete caseFind a proper map which is semi-conjugate to a m-shift map.

拓扑马蹄理论简介

● 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

研究背景

● 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

(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

(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 Yang 提出的拓扑马蹄引理 [7]

[7] Yang X S, Tang Y. Chaos Solitons & Fractals, 2004, 19(4): 841~845

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

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。

拓扑马蹄理论简介连续系统中寻找拓扑马蹄的步骤

（ 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。

2D

mD

1D

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

OutlineOutline






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.

[4] Day, R., Irregular Growth Cycles, American Economic Review, 72, 406-414, 1982.

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……


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


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


The $C/$US exchange rate [6]: chaotic structure

Chaos also exists in daily data for the Swedish Krona against Deutsche Mark, the ECU, the US Dollar and the Yen exchange rates.[7]

Time series of daily exchange rate data14/02/1973---29/03/2003

Lyapunov exponent of the time series

[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.


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 seriesNational 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.


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……

OutlineOutline






I. The Cournot duopoly Kopel economic Model

II. 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...

Documents