ultimate numerical bound estimation of chaotic dynamical finance model

Ultimate Numerical Bound Estimation of ChaoticDynamical Finance Model

Dharmendra Kumar1 Sachin Kumar2

1SGTB Khalsa CollegeDepartment of Mathematics

University of Delhi

2Department of MathematicsUniversity of Delhi

M3HPCST - 2015

Dharmendra Kumar, Sachin Kumar Ultimate Numerical Bound Estimation of Chaotic Dynamical Finance ModelM3HPCST - 2015 1 / 17

Outline

1 Dynamical SystemReal Life Examples with chaotic system

2 Main Problem under discussion Ultimate Bound

3 Four Methods to estimate ultimate bound of chaotic systems.

4 Dynamic Finance ModelDynamical behaviour of the systemPhase diagrams for the Dynamic Finance ModelLyapunov Exponent for Dynamic Finance ModelLyapunov Dimension

5 Langrange Multiplier Method

6 Optimization MethodMain Optimization ProblemNumerical Calcualtions for BoundEllipsoidal Bound

7 Future Work


Dynamical System

Dynamical System

Dynamics is the study of change

A Dynamical System is just a recipe for saying how a system ofvariables interacts and changes with time.


Dynamical System Real Life Examples with chaotic system

Real Life Examples with chaotic system

Some real life examples are as follows:

The solar system (Poincare in 1888)

Complex behaviour in Hamiltonian mechanics(Birkhoff, Kolmogorov, Arnold & Moser in 1920 - 1960)

The Weather Forcasting (Lorenz in 1963)

Turbulence in fluids (Libchaber)

Solar activity (Parker)

Population growth or Logistic Map(May in 1970)

lots and lots of other systems


Main Problem under discussion Ultimate Bound

Main Problem

Ultimate Bound

The estimate of the ultimate bound for a chaotic system is of greatimportance for chaos control, chaos synchronization, Hausdorff dimensionand the Lyapunov dimension of chaotic attractors

There are a few studies which have discussed the solution bounds andinvariant sets of the chaotic systems, such as

The Lorenz system in 2001

The hyperchaotic Lorenz–Haken system in 2009

Lorenz–Stenflo system in 2015

Stochastic cellular neural networks with delays in 2011

One tumor growth model in 2013

A class of HDQA chaotic systems in 2011


Four Methods to estimate ultimate bound of chaotic systems.

How to estimate the bound of strange attractor?

Task is difficult

Chaotic systems are bounded, the bounds have important application inchaos control and synchronization; But it is a difficult task to estimate theultimate bounds of them.

There are mainly four methods to estimate the bounds of chaotic systemsin current literatures. Four popular methods are following:

The hyper plane oriented method

The iteration theorem and the first order extremum theorem,

Lyapunov stability theory combined with the comparison principlemethod

The optimization method.


Dynamic Finance Model Dynamical behaviour of the system

Dynamic Finance Model

Ma JH & Chen YS in 2002 reported a dynamic model of finance,composed of three first-order differential equations. The model describesthe time-variation of three state variables

x = z + (y − a)x

y = 1− by − x2 (1)

z = −x − cz

x: interest rate.

y: investment demand.

z: price index.

where a is the saving amount, b is the cost per investment, c is theelasticity of demand. Dynamical behaviour of the system:

Symmetry and Invariance

Dissipativity and Existence of Attractor


Dynamic Finance Model Phase diagrams for the Dynamic Finance Model

Phase diagrams for the Dynamic Finance Model.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.51.5

2

2.5

3

3.5

4

4.5

5

5.5

x

y

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−1.5

−1

−0.5

0

0.5

1

1.5

x

z

1.5 2 2.5 3 3.5 4 4.5 5 5.5−1.5

−1

−0.5

0

0.5

1

1.5

y

z

−2

−1

0

1

2

0

2

4

6

−3

−2

−1

0

1

2

3

zy

x

Figure: Phase diagrams for the Dynamic Finance Model


Dynamic Finance Model Lyapunov Exponent for Dynamic Finance Model

Lyapunov Exponent

Numerical Values of Lyapunov Exponent

L1 = 0.7848, L2 = 0.2260, L3 = −1.3332

Why Finance Model is chaotic?

Two L1, L2 are positive Lyapunov exponent, and the third one is negative.Thus, the system is chaotic. The time histories, phase diagrams, and thelargest Lyapunov Exponent were used to identify the dynamics of thesystem. The largest Lyapunov Exponent were calculated using the schemeproposed by Wolf in 1985.


Dynamic Finance Model Lyapunov Dimension

Lyapunov Dimension for finance chaotic system

The Lyapunov dimension of the chaotic system (1) is given by

DL = j +1

|Lj+1|

j∑i=1

Li = 2 +L1 + L2|L3|

= 2.7582 (2)

0 200 400 600 800 1000−2

−1.5

−1

−0.5

0

0.5

1Dynamics of Lyapunov exponents

Time

Lya

pu

no

v e

xp

on

en

ts

Figure: Lyapunov Exponent of the Dynamic Finance Model.

So, the chaos in the system is very obvious.Dharmendra Kumar, Sachin Kumar Ultimate Numerical Bound Estimation of Chaotic Dynamical Finance ModelM3HPCST - 2015 10 / 17

Langrange Multiplier Method

Langrange Multiplier Method

Consider the following problem which is constructed using Lyapunovfunction theory

Max V (x , y , z) =1

2(x2 + y2 + (z − a)2) (3)

Subject to constraint:

Γ : 3(x − 1

2)2 +

1

10(y − 5)2 + (z − 3)2 =

11

4(4)

Estimate of ultimate bound for (1) is

Ω =

(x , y , z)|1

2(x2 + y2 + (z − 3)2) ≤ 52.5971


Optimization Method

Optimization Method using matrix analysis

Theorem

Suppose that there exits a real symmetric matrix P > 0 and a vectorµ ∈ R3 such that

Q = ATP + PA + 2(BT1 PuT ,BT

2 PuT , ...,BTn PuT )T < 0

and∑

xiXT (BT

i P + PBi )X = 0 for any X = (x1, x2, ..., xn)T ∈ R3 andu = (u1, u2, ..., un) = 2µTP then (1) is bounded and has the followingultimate bound set also called positively invarient set:

Ω =X ∈ R3|(X + µ)TP(X + µ) ≤ Rmax

(5)

Rmax to be calculated.


Optimization Method Main Optimization Problem

Main Optimization Problem

Rmax is a real number to be determined by following OptimizationProblem:Objective Function:

Max (X + µ)TP(X + µ) (6)

subject to

XTQX + 2(µTPA + CTP)X + 2CTPµ = 0. (7)


Optimization Method Numerical Calcualtions for Bound

Numerical Calcualtions for Bound

Bounds are numerically calculated and observed for different parameter. Inthis paper, we have shown the ellipsoidal boundedness of the Financialdynamic Model.

Parameters Values Initial Condition (x0, y0, z0) Rmax

p11 = 1, p33 = 2 (2, 3, 2) 281560.49

p11 = 0.1, p33 = 0.2 (2, 3, 2) 129247.49

p11 = 0.001, p33 = 0.001 (2, 3, 2) 56796

p11 = 0.001, p33 = 0.002 (2, 3, 2) 79519.34

p11 = 1000, p33 = 2 (2, 3, 2) 1.2E48

p11 = 1, p33 = 2000 (2, 3, 2) 1.2E12

p11 = 0.001, p33 = 0.001 (2, 3, 2) 56796

p11 = 1, p33 = 2 (0, 0, 0) 165775.15

p11 = 1, p33 = 1 (2, 3, 2) 639152.0

Table: Simulation results using MATLAB


Optimization Method Ellipsoidal Bound

Ellipsoidal Bound

Attractor formed using finance chaotic system is enclosed with an ellipsoidwhich is shown in the following fig:

−4

−2

0

2

4

0

2

4

6−4

−2

0

2

4

6

x(t)y(t)

z(t

)

−10

−5

0

5

10

−10

−5

0

5

10−5

0

5

10

15

y(t)z(t)

x(t

)

Figure: The bound of the chaotic attractor of system with a = 3, b = 0.1 andc = 1, p11 = 1, p33 = 2.


Future Work

Future Work

New Problems in this topic

At present, ultimate bound estimations of many chaotic systems are still adifficult mathematical problem, such as the well-known

Chen system,

Lu system and

multi-scroll chaotic attractors.

How to find explicit ultimate bound sets for these systems is still anunsolved question.


Summary

For Further Reading I

S.H. Strogatz.Nonlinear Dynamics And Chaos.Studies in Nonlinearity, 2000.

Wang P, Li D, Wu X, L J, Yu X.Ultimate bound estimation of a class of high dimensional quadraticautonomous dynamical systems.Int. J. Bifurc. Chaos Appl. Sci. Eng., 21(9):2679–2694, 2011.

Leonov GA.Bound for attractors and the existence of homoclinic orbit in Lorenzsystem.J. Appl. Math. Mech., 65:19–32, 2001.


ultimate numerical bound estimation of chaotic dynamical finance model

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