Construction of four dimensional chaotic finance model and its applications

Transcription

1 Volume 8 No. 8, 7-87 ISSN: (on-line version) url: ijpam.eu Construction of four dimensional chaotic finance model and its applications Dharmendra Kumar and Sachin Kumar Department of Mathematics, SGTB Khalsa College, University of Delhi, Delhi - 7, INDIA dharmendrakumar@sgtbkhalsa.du.ac.in Department of Mathematics, Faculty of Sciences, University of Delhi, Delhi - 7, INDIA sachinambariya@gmail.com March 5, 8 Abstract This paper considers the transition from hyperchaotic to chaotic with respect to a parameter in four dimensional autonomous nonlinear finance chaotic system. By two methods the bound- edness of Four dimensional chaotic Dynamical Finance Model is discussed. First by Lyapunov stability theory combined with the comparison principle method and second by optimization method. Furthermore, lyapunov Exponent calculated using Wolf algorithm and shown graphi- cally. Lyapunov dimension of Dynamic Finance Model also discussed. Finally, we have shown ultimate bound for the a four dimensional dynamical finance model as contained in an ellipsoidal set and Numerical simulations presented to show the effectiveness of the proposed scheme. AMS Subject Classification: 37D45, 39A33, 74H65, 65L7 Key Words and Phrases: Ultimate bounds, chaotic finance model, positively invariant set, Lyapunov exponent, optimization Introduction The chaotic dynamics of deterministic systems have been studied extensively, great achievements have been made in the area of nonlinear dynamical systems. The application of the nonlinear chaotic dynamic system in management, economics, and finance has expanded rapidly in the last few years. It was first studied as chaotic dynamics in []. Dynamical Analysis of financial models observed in recent research on financial chaotic dynamics. To 7

2 understand the highly complex dynamics of real economic and financial systems we need to study its global dynamical properties. Chaotic phenomenon in economics was first found in 985 []. A nonlinear differential equation in one variable can generate chaos and an Ordinary Differential Equation in three variables can generate chaos. Chaotic solutions are only accurate for a length of time governed by the errors on initial conditions and the Lyapunov exponent of the system, which quantifies the exponential divergence of trajectories in chaotic systems. However, when considered in the underlying state space, in many cases chaotic solutions relax onto a strange attractor which has a fractal structure and typically a non-integral dimension. Though a chaotic system is bounded, it is not an easy work to estimate and examine its bound. Therefore, an interesting fundamental question is how to estimate the bound of strange attractor. This objective can be achieved using four methods viz, the hyper plane oriented method [3], the iteration theorem and the first order extremum theorem [4], Lyapunov stability theory combined with the comparison principle method [5], and the optimization method [6, 7, 9]. Estimating Bound of an attractor is very difficult task analytically as well as numerically. In recent papers, optimization theory is used for estimating the ultimate bounds of a class of High Dimensional Quadratic Autonomous Dynamical System [3]. In [, 4], author used the Lagrange multiplier method to find two kinds of explicit ultimate bound sets and estimates the Hausdorff dimension of the novel hyperchaotic system. An estimate through the Lyapunov function of the upper bound of a threshold is precisely the threshold itself [5]. Using optimization method and the comparison principle, ultimate bounds and positively invariant sets of the hyperchaotic Lorenz Stenflo system found in [7]. Four dimensional ellipsoidal ultimate bound and two dimensional parabolic bound of Lorenz Haken system discussed in [8]. Ultimate bound and positively invariant set for the Lorenz system and the unified chaotic systems was studied in [9]. The discussion on ellipsoidal ultimate bound for unified chaotic system and two dimensional bound for the Chen system, Lu system, and unified system can be found in []. In [], unification of the Lorenz and the Chen system using the unified system. Partial bound for the Chen system using suitable Lyapunov function [] was discussed. The boundedness of Four dimensional chaotic Dynamical Finance Model is discussed. Two types of methods for the ultimate bounds of the system are derived. First by Lyapunov stability theory combined with the comparison principle method and second by optimization method it was verified by using fmincon solver. Ultimate bound for the a 4-D dynamical finance model to be contained in an ellipsoidal set. Chaotic attractor is called Ellipsoidal Chaos. Motivated by above issues, based on Langrange multiplier method and optimization method we explore the ultimate bound of 4-D dynamical finance model. Section discussed basics of 3-D dynamical finance model. Section 3 investigates extended 4-D dynamical finance model. Lyapunov Stability theory in generalised and particular case given in section 4. Optimization method and its numerical simulation discussed in section 5. Concluding remarks are presented in section 6. 7

3 Dynamic Finance Model In [8], author reported a dynamic model of finance, composed of three first-order differential equations. The model described the time-variation of three state variables: the interest rate, X, the investment demand, Y, and the price index, Z. The factors that influence changes in X mainly come from two aspects: first, contradictions from the investment market, i.e., the surplus between investment and savings, and second, structural adjustment from goods prices. The changing rate of Y is in proportion to the rate of investment, and in proportion to an inversion to the cost of investment and interest rates. Changes in Z, on the one hand, are controlled by a contradiction between supply and demand in commercial markets, and on the other hand, are influenced by inflation rates. By choosing an appropriate coordinate system and setting appropriate dimensions for every state variable, [8] offers the simplified finance model as: ẋ = z + (y a) x ẏ = b y x () ż = x c z where a is the saving amount, b is the cost per investment, and c is the elasticity of demand of commercial markets. It is obvious that all the three constants a, b, and c, are non-negative. The parameters was chosen to be for system () a = 3, b =, c =. () with an initial state (x, y, z ) = (, 3, ). In this case, the system has Lyapunov Exponents: L =.7848, L =.6, L 3 =.333 It can be seen that the largest Lyapunov exponent is positive, indicating that the system has chaotic characteristics. Two L, L are positive Lyapunov exponent, and the third one is negative. Thus, the system is chaotic. The time histories, phase diagrams, and the largest Lyapunov Exponent was used to identify the dynamics of the system. The largest Lyapunov Exponent was calculated using the scheme proposed by Wolf [4]. 3 Construction of Four Dimensional Financial system Recently [] introduced a novel hyperchaotic system with parameters a =.9, b =., c =.5, d =. and k =.7. We apply result [3] with parameter k =. instead of k =.7. Under this change the system is under transition from hyperchaotic to chaotic. 3 73

4 u u z u y z International Journal of Pure and Applied Mathematics x (a) Phase Portrait in x-y plane y (b) Phase Portrait in y-z plane x (c) Phase Portrait in x-z plane x (d) Phase Portrait in x-u plane y (e) Phase Portrait in y-u plane z (f) Phase Portrait in z-u plane Figure : Phase diagrams for the Dynamic Finance Model. In [], the basic properties and complex dynamics of the new system (3), such as waveform, spectrum, Lyapunov exponent, fractal dimensions have been investigated. Bifurcation diagram illustrate how the dynamics of the hyperchaotic system alters with the increasing value of the parameters k and c. The new hyperchaotic finance system which 4 74

5 x u x u International Journal of Pure and Applied Mathematics is defined by two positive Lyapunov exponents can be generated by adding an additional state variable u, which is the average profit margin. The system of four dimensional autonomous differential equations has the following form: z 4 3 y z 3 x 3 (a) Phase Portrait in x-y-z plane (b) Phase Portrait in u-z-x plane u y 3 4 z y 3 4 (c) Phase Portrait in x-u-y plane (d) Phase Portrait in u-z-y plane Figure : Phase diagrams for the Dynamic Finance Model. ẋ = z + (y a) x + u ẏ = b y x (3) ż = x c z u = d x y k u where a, b, c, d, k are the parameters of system (3), and they are positive constants. When parameters a =.9, b =., c =.5, d =. and k =.7, the four Lyapunov exponents of system (3) calculated by Wolf algorithm [4] are: L =.44474, L =.3496, L 3 =.4955, L 4 =.4659 (4) Therefore, the Lyapunov dimension of the hyperchaotic system (3) for k =.7 is given 5 75

6 Table : Simulation results using MATLAB k L L L 3 L 4 Nature Hyperchaotic Hyperchaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic chaotic 6 76

7 by D L = j + L j+ j i= L i = 3 + L + L + L 3 L 4 = So, the hyperchaos in this system (3) is present. Thus the corresponding Lyapunov Exponent is shown in Fig 3..5 Dynamics of Lyapunov exponents Lyapunov exponents Time Figure 3: Lyapunov Exponent of the Hyperchaotic Four Dimensional Dynamic Finance Model for k =.7. Also, Lyapunov dimension of the chaotic system (3) for k =. is given by D L = j + L j+ j i= L i = 3 + L + L + L 3 L 4 =.637 So, the chaos in this system (3) is very obvious. Exponent is shown in Fig 4. Thus the corresponding Lyapunov 4 First Method: Lyapunov stability theory Define the following Lyapunov function as: V = [x + (y + a) + z + u ] (5) 7 77

8 .5 Dynamics of Lyapunov exponents Lyapunov exponents Time Figure 4: Lyapunov Exponent of the Chaotic Four Dimensional Dynamic Finance Model for k =.. dv dt = xẋ + (y + a)ẏ + zż + u u = x(x(y a) + u + z) + (a + y) ( by x + ) + z( cz x) + u( dxy ku) = 3ax by cz ku + ux + y aby duxy + a For a >, b >, c >, V (x, y, z) is positive definite and the quadratic principal part of V = dv is negative definite. V dt =, i.e., the surface Γ : 3ax + by + cz + ku + duxy + aby ux y = a (6) Because of the term d u x y, Lyapunov function V (x, y, z, u) is not positive definite. Hence we will simplify our calculations by assuming one of the variables as suitable constant. In order to solve the 4D problem analytically in particular u =, then the Lyapunov function is given by: V = (x + (y + a) + z ) (7) dv dt = xẋ + (y + a)ẏ + zż = x(x(y a) + w + z) + (y + a) ( by x + ) + z( cz x) = a + 3ax 3ax y aby by cz ( = 3a x ( b y ) ab ) cz + 4a b + 7ab + b 4b 8 78

9 For a >, b >, c >, V (x, y, z) is positive definite and the quadratic principal part of V = dv is negative definite. V dt =, i.e., the surface ( Γ : 3a x ( + b y ) ab ) + cz = 4a b + 7ab + b 4b Γ : ( ) x ( ( y ( + a)) b r 3a ) ( r + z b ) ( r c ) = (8) where 4a b + 7ab + r = 4b is ellipsoid in 3D space for certain a, b, c. Since the system (3) is bounded, then Lyapunov function can reach its maximum value MaxV on (). Because, V (X) = Max V (x, y, z) with respect to constraint () contains the solution of system (3). By solving the following conditional extremum problem, one can get the ultimate bound for the system (3): Maximize: V (x, y, z) = (x + (y + a) + z ) (9) Subject to constraint: Γ : ( ) x ( ( y ( + a)) b r 3a ) ( r + z b ) ( r c ) = () In particularly, when the system parameters in (3) are constants, it is easy to obtain ultimate bound. For example, when a =.9, b =., c =.5, d =. and k =. we have Max V (x, y, z) = ( x + (y +.8) + z ) () Subject to constraint: ( Γ :.7 x ) +.(y.6) +.5z = () The typical solution for constructed optimization problem can be obtained by Langrange multiplier method analytically, explicit ultimate bound for the system is: { Ω = X R 3 ( x + (y +.8) + z ) }

10 5 Second Method:Optimization Method 5. Priliminaries Consider the following autonomous system [3]: Ẋ = f(x) (3) where f : R n R n sufficiently smooth, where X = (x, x...x n ), t is the time, X(t, t, X ) is the solution which satisfies X(t, t, X ) = X. Suppose Ω R n is a compact set. Define the distance between X(t, t, X ) and Ω as: ρ(x(t, t, X ), Ω) = inf Y Ω X(t, t, X ) Y where Ω ɛ = {X ρ(x, Ω) < ɛ}. Clearly Ω Ω ɛ. Definition. Suppose Ω R n is compact set, if for any X R n /Ω satisfying lim t ρ(x, Ω) = that is for any ɛ there exits T > t satisfying X(t, t, X ) Ω ɛ, then the set Ω is called the positively invariant set for (3). As an application of Theorem this section aims to estimate the ultimate bound of a chaotic system. Theorem. [3] Suppose that there exits a real symmetric matrix P > and a vector µ R 3 such that and Q = A T P + P A + (B T P u T, B T P u T,..., B T n P u T ) T < xi X T (B T i P + P B i )X = (4) for any X = (x, x,..., x n ) T R 3 and u = (u, u,..., u n ) = µ T P then is bounded and has the following ultimate bound set also called positively invarient set: Ω = { X R 3 (X + µ) T P (X + µ) R max } (5) where R max is a real number to be determined by subject to Max (X + µ) T P (X + µ) (6) X T QX + (µ T P A + C T P )X + C T P µ =. (7) 8

11 The initial values of system (3) are appointed as (,,.5,.5). According to numerical as well as detailed theoretical analysis, it has been confirmed that the new system (3) with parameters a =.9, b =., c =.5, d =. and k =.7 displays sophisticated and abundant hyperchaotic dynamical behaviors. The strange attractors and phase portraits of the hyperchaotic finance system (3) are shown in figure, and they are also the new reverse butterfly shape attractors. a x A = b c ; X = x x 3 ; k x 4 / B = ; B = d/ 5. Numerical Simulation / ; B 3 = ; d/ B 4 = ; C = ; Let P = (p ij ) 4 4 for i, j =,, 3, 4, µ = (µ, µ, µ 3, µ 4 ). Since 5 x i X T (Bi T P + P B i )X = (8) i= holds for any x i R with i =,, 3, 4, let p = p 4 = p 4 = dp 44, p = d( d)p 44 (9) since, P is positive symmetric definite then we have dp 44 dp 44 P = d( d)p 44 p 33 ; dp 44 p 44 Eigenvalues of P are λ = p 33, λ = (d )dp 44, λ 3 = (dp 44 + p 44 ) 5d d + p 44, λ 4 = (dp 44 + p 44 + ) 5d d + p 44 () 8

12 Determinant of P is (d ) d p 33 p 3 44 which is positive and positive eigenvalues, shows that matrix P is real symmetric matrix. M = [M T ] T where M T is given by ( µ p 33 a(dµ p 44 + dµ 4 p 44 )) M T = (( d)dp 44 + b( + d)dµ p 44 ) ( cp 33 µ 3 + dp 44 µ + dp 44 µ 4 ) ( k (dp 44 µ + p 44 µ 4 ) + dp 44 µ 4 + dp 44 µ 4 ) Q = dp 44 ((d )µ a) (d )dp 44µ dp 44 p 33 d(a + k )p 44 (d )dp 44µ b(d )dp 44 dp 44 p 33 cp 33 dp 44 d(a + k )p 44 dp 44 (d k)p 44 for simplificaion, let µ = adp 44 ; M = [ ] (d )dp 44 ( + ac)dp 33 p 44 (d k) p 33 p 44 Q = p 33 adp 44 dp 44 p 33 d( a k)p 44 b( d)dp 44 dp 44 p 33 cp 33 dp 44 d( a k)p 44 dp 44 (k d)p 44 Now from Theorem, we obtain the following result: Theorem 3. Let a >, b >, c >, p ii >, satisfying comparing and balancing the following set: p = p 4 = p 4 = dp 44, p = d( d)p 44 () Ω = { X R 4 d p 44 x + ( d) d p 44 x + p 33 x 3 + p 44 x 4 + (d x a d x 3 + x 4 ) p 44 p 33 + d p 44 x x 4 + p 44 p 33 + a d p 44 p 33 R max } is the ultimate bound region of system (3), where X = (x, x, x 3, x 4 ) T and the value of R max can be obtained by computing the following maximum optimization question: Max V = d p 44 x + ( d) d p 44 x +p 33 x 3 + p 44 x 4 + subject to constraint: ( d x a d x 3 + x 4 ) p 44 p 33 + d p 44 x x 4 + p 44 p 33 + a d p 44 p 33 () a d p 44 x + b (d ) d p 44 x c p 33 x 3 + (d k) p 44 x 4 + d ( a c + ) p 33 p 44 x 3 + ( a k + ) d p 44 x x 4 + ( d k ) p 33 p 44 x 4 + ( d) d p 44 x x x 3 (p 33 d p 44 ) + d p 44 x 3 x 4 = (3) 8

13 Then Ω is the ultimate bound of system (3) where R max can be derived from the following optimization problem: (a) Ultimate Bound for p 33 =, p 44 = (b) Ultimate Bound for p 33 =, p 44 = (c) Ultimate Bound for p 33 =, p 44 = (d) Ultimate Bound for p 33 =, p 44 = 3 Figure 5: Ultimate Bound from second method for different values of p 33, p 44 It is often difficult to solve the optimization problem (3) analytically. However, it is very easy to solve the optimization problem numerically for the given system parameters. From Table, we have the best bound in case, we can call ultimate ellipsoidal bound. These parameters p 33 =, p 44 = are significant for further study. Four dimensional finance dynamical system can be used for further study for these parameters. 3 83

14 Table : Simulation to find the ultimate bound for System 6 Conclusion S.No. Parameter Value R max p 33 =, p 44 = 3.3 p 33 =, p 44 = p 33 =, p 44 = p 33 =, p 44 = 3.44 This paper has investigated the ultimate bound and positively invariant set for a Financial dynamic model. To the best of our knowledge, we discussed first time two type of bounds in a paper. We have shown graphically Lyapunov Exponent for two values of parameters and it has been proved that positive value of Lyapunov Exponent shows chaotic nature of the dynamic finance model. Ultimate bound for the a 4-D dynamical finance model to be contained in an ellipsoidal set. Chaotic attractor is called Ellipsoidal Chaos. Furthermore, ultimate bounds are numerically calculated and observed for different parameter. We have shown the ellipsoidal boundedness of the Financial dynamic Model. Research on the ultimate bound for dynamics system is very important both in control theory and synchronization. Numerical simulations show the effectiveness and advantage of our methods. Acknowledgements First author thank Department of Mathematics, SGTB Khalsa College, University of Delhi, INDIA. References [] E.N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci. (963), 3 4. [] W.J. Baumol and R.E. Quandt, Chaos Models and their Implications for Forecasting, Eastern Economic Journal, (985), 3 5. [3] G.A. Leonov, Bound for attractors and the existence of homoclinic orbit in Lorenz system, J. Appl. Math. Mech. 65 (), 9 3. [4] F.C. Zhang, Y.I. Shu and H.I. Yang Hl, Bounds for a new chaotic system and its application in chaos synchronization, Comm. Nonlinear Sci. Numer. Simulat. 6(3) (),

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