On the Application of the Multistage Differential Transform Method to the Rabinovich-Fabrikant System

Transcription

1 The African Review of Physics (24) 9:23 69 On the Application of the Multistage Differential Transform Method to the Rabinovich-Fabrikant System O. T. Kolebaje,*, M. O. Ojo, O. L. Ojo and A. J. Omoliki 2 Department of Physics, Adeyemi College of Education, Ondo, Nigeria 2 Department of Physics and Solar Energy, Bowen University, Iwo, Nigeria In this paper, the Differential Transform Method (DTM) and the Multistage Differential Transform Method (MDTM) were applied to obtain solutions to the Rabinovich-Fabrikant system for non-chaotic and chaotic cases. The study shows that the DTM only gives reliable results for t. Comparison between the MDTM and the classical fourth-order Runge-Kutta (RK4) solutions shows that the MDTM performs well with high accuracy. The study shows that MDTM is a powerful and promising tool for solving nonlinear systems of ODEs of chaotic and non-chaotic nature.. Introduction Chaos theory studies the behaviour of dynamical systems that are highly sensitive to initial conditions. Chaos theory is applied in many scientific disciplines including physics, geology, mathematics, biology, computer science, economics [-2], engineering, finance, meteorology, philosophy, and population dynamics. The Rabinovich Fabrikant equations are a set of three coupled ordinary differential equations exhibiting chaotic behaviour for certain values of the parameters. They are named after Mikhail Rabinovich and Anatoly Fabrikant, who described them in 979. The equations are [3]: dx dt = yz +x +γx dy dt = x3z+ x +γy dz dt = 2zα+xy Where, α, γ are constants that control the evolution of the system. For some values of α and γ, the system is chaotic, but for others it tends to a stable periodic orbit. These equations were used by Rabinovich and Fabrikant to model waves in nonequilibrium substances. Reference [4] noted that the Rabinovich-Fabrikant system is difficult to analyse due to the presence of quadratic and cubic terms and that different attractors can be obtained for the same parameters by using different step sizes in the integration. The concept of differential transform method (DTM) was first proposed by [5] to solve linear and nonlinear initial value problems in electric circuit analysis [6]. The method is an iterative semi-analytic scheme that employs Taylor series to approximate solutions of differential equations in the form of polynomials. The merit of the DTM is that the method does not require discretization or perturbation and is easy to implement, while also greatly reducing the size of computational work to be done. The method is very effective and has been applied to the Lorenz system [7], to a nonlinear biochemical model [8], to fractional differential equation [9] and to the Riccati equation []. However, the differential transform method does not give a satisfactory approximation for a large time as observed in [7,] where the convergence region of semianalytic methods is said to be narrow. Since the DTM solutions blow out after a short time a multistaging technique known as the Multistage Differential Transform Method (MDTM) is proposed. The motivation of this paper is to apply the DTM and MDTM to the Rabinovich-Fabrikant system for both chaotic and non-chaotic scenarios and test the accuracy of the method with the well known fourth order Runge-Kutta method. The computations in this paper were carried out with the computer algebraic system MATHEMATICA. * olusolakolebaje28@gmail.com

2 The African Review of Physics (24) 9: Methodology 2.. Differential Transform Method (DTM) If the function, is analytic and differentiable continuously with respect to time and space in the domain of interest, then the function can be expanded in Taylor series about a point = as,=,! 2 The differential transformation of, is defined as =,! 3 Where, known as the -dimensional spectrum function which is the transformation of the function,. The inverse differential transform of is defined as follows The differential transformations of common functions following Eqn. (6) are presented in Table. Table : Differential Transform Table. FUNCTIONAL FORM,,± 2, 4, : ( : (,, 2, TRANSFORMED FORM =!, ±3 4 4 is a constant : ; + : + 3 =,= 4 When =, then Eqns. (2) and (3) are expressed as,, +>!!?,=,! 5 Nonlinear Function EF,G=HF,G E =! H =,!,= 6 7 The inverse transformation of the set of values ( & ' gives approximate solution as ))),= ( ( 8 Where, + is the order of approximation. Therefore, the exact solution of the equation is given by,= lim ))), ( ( Multistage Differential Transform Method (MDTM) An efficient way of ensuring the validity of solutions to differential equations for large value of t is by multi-staging the solution procedure to be employed. Let J,KL be the interval over which the solutions to the differential equation () is to be determined. The solution interval J,KL is divided into E subintervals + =,2,,E of equal step size given by h =K E with the interval end points ( =+h. Initially, the DTM scheme is applied to obtain the approximate solutions of,p and Q of Eqn. () over the interval J, R L by using initial the condition,p and Q respectively. For obtaining the approximate solution of () over the next interval J R, L, we take R,P R and Q R as the initial condition. Generally the scheme is repeated for any + with the right endpoints :=R,P :=R and Q :=R at the previous

3 The African Review of Physics (24) 9:23 7 interval being used as the initial condition for the interval J :=R, : L. 3. Application By using the fundamental operations of differential transformation method (Eqns. (5)-(9) and Table ), the transformed form of the Rabinovich-Fabrikant system is given below as +! S!?R = T U = T + T VS W S ==W X + S = W +! T!?R = 3S U = + S S VS W S ==W X +! T!?R = 24U = W + T 2 S VT W U ==W X = W The system of equations is solved using the initial condition =., P= and Q=.5. For α=. and γ=.87 we have a chaotic system, and 4 =.5, =.55 correspond to a non-chaotic system. The recurrence relation (Eqn. ()) was evaluated with the aid of Mathematica to obtain the solution up to the -term approximation for the time range [, 3] with a time step size.. MDTM is implemented by dividing the solution interval J, 3L into 5 subintervals + =, 2,, 5 of equal step size given by h=.2. P = [ \ ] ^ _.52433` a R Q= [ \ ].89985^+.326 _ ` a R The accuracy of the DTM and MDTM is investigated by comparing their solutions to the RK4 solution for the parameters α=.5, γ=.55, where the system is non-chaotic with the initial conditions =., P= and Q=.5. The RK4 with time steps =. and =. with the number of significant digits set to 6 is used. Tables 2 and 3 present the absolute errors between the -term DTM solutions and the -term MDTM solutions for α=.5, γ=.55 and RK4 solutions with time steps =. and =., respectively. In Tables 2 and 3, we can observe that the DTM only gives valid result for. The MDTM solutions on the time step =. for the nonchaotic case agree with RK4 solutions on the time step =. to at least 8 decimal places. Also, the MDTM solutions on the time step =. agree with the RK4 solutions on a smaller time step =. to at least 7 decimal places. Hence, for the non-chaotic Rabinovich-Fabrikant system we observe that the MDTM solutions even agree with the RK4 solutions with a larger time step. The x P, Q, P Q, and P Q phase portraits for non-chaotic case are obtained using the -term MDTM solutions which are shown in Figs. to Results and Discussion The non-chaotic case of the Rabinovich-Fabrikant system was solved using the DTM and MDTM implemented on Mathematica, the -term approximate DTM solution was obtained as = [.9897 \ ] ^ _ ` a R

4 The African Review of Physics (24) 9:23 72 Table 2: Absolute differences between -term DTM and -term MDTM with RK4 solutions =. for 4 =.5, =.55. efg h.hi klm h.hi gefg h.hi klm h.hi n o p q o p q E- 9.58E E E- 9.58E E E-5.8E E E E- 3.83E-.5.66E+.66E E E E E E E E+3.53E E E E E E E-.96E E E E+8.6E+9.52E E E E E+.46E+ 2.3E E-2 5.4E E E E+2.7E E E E E E+3 5.3E-2 2.6E E E E E+4.354E-2.6E- 6.25E E+3.333E+5.787E E E-.26E- Table 3: Absolute differences between -term DTM and -term MDTM with RK4 solutions =. for 4 =.5, =.55. efg h.hi klm h.hhi gefg h.hi klm h.hhi n o p q o p q E E E E E E E-5.8E E-4.8E-9.297E-9.52E E+.66E E E E E E E E E E-9.57E E E E E E E E E+8.6E E- 2.58E-8 6.E E E+.46E+ 4.52E-.922E-8.5E E E E E-9.6E-8 4.E E E E E-9.567E E E E E+4.928E-9.3E E E+3.333E+5.787E E-.2E E-

5 The African Review of Physics (24) 9: X.5 Fig.: X- Phase portrait using -term MDTM on =. for 4 =.5 and = Fig.2: - Phase portrait using -term MDTM on =. for 4 =.5 and = Fig.3: X- Phase portrait using -term MDTM on =. for 4 =.5 and = Fig.4: X-- Phase portrait using -term MDTM on =. for 4 =.5 and =.55. The terms approximate DTM solution for the chaotic Rabinovich-Fabrikant system was obtained as = [ \ ] ^ _ ` a R + P= [ \ ] ^ _.8878` a R + Q= [ \ ].7489^ _ ` a R + The differences of the -term DTM solutions and -term MDTM solutions for α=., γ=.87 and RK4 solutions with time steps =. and =. are given in Tables 4 and 5, respectively. In Tables 4 and 5, we can also observe that the DTM only gives valid result for. The MDTM solutions on the time step =. for the chaotic case agree with the RK4 solutions on both time steps =. and =. to at least 5 decimal places. The P, Q,P Q, and P Q phase portraits for the chaotic Rabinovich-Fabrikant system are obtained using -term MDTM solutions and shown in Figs. 5-8.

6 The African Review of Physics (24) 9:23 74 Table 4: Absolute differences between -term DTM and -term MDTM with RK4 solutions =. for 4 =., =.87. efg h.hi klm h.hi gefg h.hi klm h.hi n o p q o p q E E E- 2.47E E E E-4.48E E E- 3.77E E-.5.67E E+ 5.62E- 7.22E E- 3.78E E E E E- 4.99E E E E+6.786E E-.54E-.839E E+8.E E+7.224E-9.553E E E+.7E+.38E E-.947E-.583E E E E E- 7.77E- 6.26E E E E+ 3.7E E- 3.57E E E E+ 6.44E-2 9.7E E- 3..7E+5.872E E+.762E E-6.779E-8 Table 5: Absolute differences between -term DTM and -term MDTM with RK4 solutions =. for 4 =., =.87. efg h.hi klm h.hhi gefg h.hi klm h.hhi n o p q o p q E E E- 6.59E E E E-4.49E E-5.93E-9.63E E-.5.67E E+ 5.62E-.2E E E E E E+2 4.6E E-9.25E E E+6.786E+5.4E E-.58E E+8.E E E E- 8.7E E+.7E+.38E E E E E E E E E E E E E E-7.7E E E E E+ 8.E E E E+5.872E E+ 6.83E E E-6

7 The African Review of Physics (24) 9: X X. 5 Fig.5: X- Phase portrait using -term MDTM on =. for 4 =. and = Fig.8: X-- Phase portrait using -term MDTM on =. for 4 =. and = Conclusion In this work, the DTM and MDTM were applied to the solutions of the Rabinovich-Fabrikant system for the chaotic and non-chaotic case. Comparisons were made between the DTM, MDTM and the fourth-order Runge-Kutta (RK4) method. For the chaotic and non-chaotic case, we observe that the MDTM solutions were consistent with the RK4 solutions. Hence, the MDTM is a simple, efficient and accurate method of solving the Rabinovich- Fabrikant system and other nonlinear systems. Fig.6: - Phase portrait using -term MDTM on =. for 4 =. and = Fig.7: X- Phase portrait using -term MDTM on =. for 4 =. and = X References [] C. Kyrtsou and C. Vorlow, Complex Dynamics in Macroeconomics: A Novel Approach, New Trends in Macroeconomics (Springer-Verlag, Berlin, 25). [2] C. Kyrtsou and W. Labys, J. Macroeconomics 28, 256 (26). [3] M. I. Rabinovich and A. L. Fabrikant, Sov. Phys. JETP 5, 3 (979). [4] M. F. Danca and G. Chen, Int. J. Bifurcation and Chaos 4(), 349 (24). [5] J. K. hou, Differential Transformation and its Applications for Electrical Circuits (Huazhong University Press, Wuhan, China, 986). [6] C. K. Chen and S. H. Ho, Applied Mathematics and Computation 79, 73 (996). [7] M. M. Al-Sawalha and M. S. M. Noorani, Chin. Phys. Lett. 25(4), 27 (28). [8] Abdul-Monim Batiha and Belal Batiha, Advance Studies in Biology 3(8), 355 (2). [9] A. Arikoglu and I. Ozkol, Chaos, Solitons and Fractals 34, 473 (27).

8 The African Review of Physics (24) 9:23 76 [] J. Biazar and M. Eslami, Int. J. Nonlinear Science 9(4), 444 (2). [] O. T. Kolebaje and E. O. Oyewande, Int. J. Basic and Applied Sciences, Science Publishing Corporation (3), 32 (22). Received: January, 24 Accepted: 3 June, 24