張 光 云

(重慶工商大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，重慶 400067)

基于動(dòng)力系統(tǒng)理論的一類金融混沌系統(tǒng)的定性分析

張 光 云

(重慶工商大學(xué) 數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，重慶 400067)

根據(jù)動(dòng)力系統(tǒng)的基本理論與方法，針對(duì)一類三維金融動(dòng)力系統(tǒng)，研究了該類金融動(dòng)力系統(tǒng)的平衡點(diǎn)及其附近軌線的性態(tài)、解的最終界、全局吸引域、不變集等；最后，給出了相應(yīng)的計(jì)算機(jī)仿真.這有助于加深人們對(duì)各種金融政策的理解，該混沌系統(tǒng)有望應(yīng)用于控制工程、圖像加密、混沌電路設(shè)計(jì)等領(lǐng)域中.

混沌系統(tǒng)；穩(wěn)定性；奇點(diǎn)；計(jì)算機(jī)仿真；拓?fù)浣Y(jié)構(gòu)

20世紀(jì)以來，非線性系統(tǒng)科學(xué)得到了進(jìn)一步的研究和發(fā)展，自從Lorenz系統(tǒng)被發(fā)現(xiàn)以來，其他一些混沌系統(tǒng)相繼被發(fā)現(xiàn)和研究，如R?ssler系統(tǒng)，Chua電路系統(tǒng)、Chen系統(tǒng)、Lü系統(tǒng)、超混沌MCK電路系統(tǒng)、廣義Lorenz系統(tǒng)、超混沌R?ssler系統(tǒng)、超混沌Chen系統(tǒng)等，這些系統(tǒng)的動(dòng)力學(xué)特性，如分岔、控制和同步等也被廣泛地研究[1-14].

本文依據(jù)動(dòng)力系統(tǒng)的基本理論與方法對(duì)一類金融動(dòng)力系統(tǒng)的動(dòng)力學(xué)特性進(jìn)行了研究. 這有助于加深人們對(duì)各種金融政策的理解，該混沌系統(tǒng)有望應(yīng)用于控制工程、圖像加密等領(lǐng)域中.

1 數(shù)學(xué)模型及其主要結(jié)果

一個(gè)三維金融混沌系統(tǒng)的數(shù)學(xué)模型為[15]

(1)

其中，變量x,y和z分別代表利率、投資需要和價(jià)格指數(shù)；a>k>0,b>0,c>0是系統(tǒng)的正參數(shù)，分別代表節(jié)省成本、單位投資成本和市場(chǎng)需要的彈性數(shù)；kx表示平均利潤(rùn)率.當(dāng)a=0.6,b=0.2,c=0.9,k=0.5時(shí)，系統(tǒng)(1)的混沌吸引子見圖1.混沌系統(tǒng)(1)各個(gè)變量x,y和z隨時(shí)間演化的圖形見圖2.

圖1 系統(tǒng)(1)的混沌吸引子Fig.1 Chaotic attractor of system (1)

圖2 系統(tǒng)(1)的各個(gè)狀態(tài)變量隨時(shí)間演化圖形Fig.2 The diagram of all state variables changing with time

1.1 對(duì)稱性和不變性

系統(tǒng)(1)具有對(duì)稱性，即在坐標(biāo)變換(x,y,z)→(-x,y,-z)下，系統(tǒng)(1)保持不變.y軸為系統(tǒng)(1)的一個(gè)不變集，并且從y軸上任何點(diǎn)出發(fā)的軌線當(dāng)t→+∞時(shí)都趨于點(diǎn)(0,0,0).

1.2 耗散性和吸引子

記系統(tǒng)(1)的向量場(chǎng)為

F(x,y,

則對(duì)于系統(tǒng)(1)，有

1.3 奇點(diǎn)及其附近軌線的性態(tài)

ii) 當(dāng)c-b-(a-k)bc≥0時(shí)，系統(tǒng)(1)有3個(gè)平衡點(diǎn)：

當(dāng)c-b-(a-k)bc=0,01時(shí)，平衡點(diǎn)S0為漸近穩(wěn)定的平衡點(diǎn)；當(dāng)c-b-(a-k)bc>0時(shí)，由中心流形定理可以知道，當(dāng)平衡點(diǎn)P1,P2在正參數(shù)a,b,c,k滿足條件bc4+b2c3-2(a-k)b2c2+[2(a-k)b-2-3b2]c+3b=0時(shí)，系統(tǒng)(1)可能出現(xiàn)分岔,因此平衡點(diǎn)P1,P2附近軌線的拓?fù)浣Y(jié)構(gòu)類型變得非常復(fù)雜.

1.4 解的最終界

定理1 設(shè)X(t)=(x(t),y(t),z(t))為系統(tǒng)(1)的任意一個(gè)解,對(duì)任意的參數(shù)a>k>0,b>0,c>0,令

R2=

證明 定義Lyapunov函數(shù)

V(X)=x2+y2+z2

對(duì)此函數(shù)沿著系統(tǒng)(1)正半軌線求導(dǎo)數(shù)：

2x(z+xy-ax+kx)+2y(1-by-x2)+

2z(-x-xz)=-2(a-k)x2-2by2-2cz2+2y

Γ

由條件極值問題的求法可以得到：

容易證明Δ為系統(tǒng)(1)的一個(gè)最終有界集與正向不變集.證畢.

1.5 全局吸引域

定理2 設(shè)X(t)=(x(t),y(t),z(t))為系統(tǒng)(1)的任意一個(gè)解，則對(duì)任意的a>k>0,b>0,c>0,令

V(X)=V(x,y,z)=x2+y2+z2

θ=min(a-k,c,b)>0,

當(dāng)V(X(t))>L0,V(X(t0))>L0,系統(tǒng)(1)軌線存在下列估計(jì)式:

V(X(t))-L0≤(V(X(t0))-L0)e-θ(t-t0)

從而

≤L0}

為系統(tǒng)(1)的一個(gè)全局指數(shù)吸引集.

證明 定義Lyapunov函數(shù)

V(X)=V(x,y,z)=x2+y2+z2

當(dāng)V(X(t))>L0,V(X(t0))>L0,沿著系統(tǒng)(1)正半軌線求導(dǎo)有：

2x(z+xy-ax+kx)+2y(1-by-x2)+2z(-x-xz)=-2(a-k)x2-2by2-2cz2+2y≤-(a-k)x2-by2-cz2-(a-k)x2-by2-cz2+2y≤

-θx2-θy2-θz2-by2+2y≤

(2)

對(duì)不等式(2)兩邊積分有:

V(X(t))-L0≤(V(X(t0))-L0)e-θ(t-t0)

(3)

令t→+∞,對(duì)不等式(3)兩邊取上極限:

≤L0

2 數(shù)值仿真

圖3 系統(tǒng)的軌線最終界估計(jì)圖示Fig.3 The diagram of ultimate bound estimate of the trajectory of the system

3 結(jié) 論

基于動(dòng)力系統(tǒng)的理論與方法，定量和定性地分析了一類金融動(dòng)力系統(tǒng)的動(dòng)力學(xué)行為，包括奇點(diǎn)及其附近軌線的拓?fù)漕愋?、吸引子、最終界、全局吸引集、不變集等. 這有助于加深人們對(duì)各種金融政策的理解，該混沌系統(tǒng)有望應(yīng)用于控制工程、圖像加密等領(lǐng)域中.

[1] 王興元.廣義M-J集的分形機(jī)理[M].大連:大連理工大學(xué)出版社,2002

WANG X Y.The Generalized Fractal Mechanism of M-J Set[M].Dalian:Dalian University of Technology Press,2002

[2] 王興元.混沌系統(tǒng)的同步及在保密通信中的應(yīng)用[M].北京:科學(xué)出版社,2011

WANG X Y.Synchronization of Chaotic Systems and Its Application in Secret Communication[M].Beijing:Science Press,2011

[3] ZHANG F C,MU C L,ZHOU S M,et al.New Results of the Ultimate Bound on the Trajectories of the Family of the Lorenz Systems[J].Discrete and Continuous Dynamical Systems Series B,2015,20(4): 1261-1276

[4] ZHANG F C,MU C L,LIX W.On the Boundness of Some Solutions of the Lü System[J].International Journal of Bifurcation and Chaos,2012,22(1):1-5

[5] ZHANG F C,ZHANG G Y.Further Results on Ultimate Bound on the Trajectories of the Lorenz System[J].Qualitative Theory of Dynamical Systems,2016,15(1): 221-235

[6] LIN D,ZHANG F C,LIU J M.Symbolic Dynamics-based Error Analysis on Chaos Synchronization Via Noisy Channels[J].Physical Review E,2014,90:1-7

[7] CHOWDHURY M S H,HASHIM I.Application of Multistage Homotopy-Perturbation Method for the Solutions of the Chen System[J].Nonlinear Analysis:Real World Applications,2009,10 (1):381-391[8] ELABBASY E M, El-DESSOKY M M.Synchronization of Van Der Pol Oscillator and Chen Chaotic Dynamical System[J].Chaos,Solitons and Fractals,2008,36:1425-1435

[9] WU X F,CHEN G R,CAI J P.Chaos Synchronization of the Master-Slave Generalized Lorenz Systems Via Linear State Error Feedback Control[J].Physica D,2007,229:52-80[10] WU X Y, GUAN Z H,WU Z P,et al.Chaos Synchronization Between Chen System and Genesio System[J].Physics Letters A,2007,364(6):484-487

[11] STARKOV K E,STARKOV J K K.Localization of Periodic Orbits of the R?Ssler System under Variation of Its Parameters[J].Chaos,Solitons and Fractals,2007,33(5):1445-1449

[12] PARK J H,LEE S M,KWON O M.Dynamic Controller Design for Exponential Synchronization of Chen Chaotic System[J].Physics Letters A,2007,367(4-5):271-275

[13] HEGAZI A,AGIZA H N,EL-DESSOKY M M.Controlling Chaotic Behaviour for Spin Generator and Rossler Dynamical Systems with Feedback Control[J].Chaos,Solitons and Fractals,2001,12:631-658

[14] HEGAZI A S,AGIZA H N,El-DESSOKY M M.Synchronization and Adaptive Synchronization of Nuclear Spin Generator System[J].Chaos,Solitons and Fractals,2001,12(6):1091-1099

[15] CAI G L,YU H J,LI Y X. Localization of Compact Invariant Sets of a New Nonlinear Finance Chaotic System[J].Nonlinear Dynamics,2012,69:2269-2275

責(zé)任編輯：李翠薇

Qualitative Analysis of a Class of Financial Chaotic System Based on Dynamical System Theory

ZHANG Guang-yun

(School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China)

Based on the basic theory and method of dynamical system, according to a class of three-dimensional financial dynamic systems, this paper studies the equilibrium point of this class of financial dynamic system, its nearby trajectory property, the final boundedness of the solution, global attractive sets, invariant set and so on, and ultimately gives corresponding computer simulation. This research is conducive to deepening the understanding of all kinds of financial policies, and this chaotic system is expected to be applied to control engineering, image encryption, chaotic circuit design and so on.

chaotic system; stability; singular point; computer simulation; topological structure

2016-09-14；

2016-10-23.

張光云( 1983-),女,山東臨沂人,助教,碩士,從事外國(guó)語言學(xué)及應(yīng)用語言學(xué)、常微分方程研究．

10.16055/j.issn.1672-058X.2017.0002.009

O241.84;O29;O242.1

A

1672-058X(2017)02-0037-04