Stability and Unstability of Chaos in Stretch-Twist-Fold Flow (STF)

Abstract:

In this study, the Adomian's decomposition method (ADM) is applied to the STF system. This method has been tested on the STF system which is a three-dimensional system of ODE with quadratic nonlinearities. A computer based Matlab program has been developed in order to solve the STF system. Chaotic and non-chaotic behavior of the system has been analyzed graphically and finally a comparison as well as accuracy between two systems has been made in two-step sizes with detail.

• Periodical: Advanced Materials Research (Volumes 631-632)
• Pages: 1436-1440
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.631-632.1436
• Online since: January 2013
• Authors: Ud Din Khan Shahab, Yong Lu Shu, Ud Din Khan Salah
• Keywords: Adomian Decomposition Method, Chaos, Harmonic Mean, Modified Fifth-Order Runge-Kutta Method, STF System

References

[1] G. Adomian, Nonlinear stochastic system theory and application to physics, Dordrech: Kluwer Academic, (1989).

[2] G. Adomian, Solving frontier problems of physics: the decomposition method, Boston: Kluwer Academic, (1994).

[3] G. Adomian, Analytic solution of nonlinear integral equations of Hammerstein type, Appl. Math. Lettl. 11 (1998)127–130.

DOI: 10.1016/s0893-9659(98)00045-7

[4] D. J . Evans and K. R. Raslan, The Adomian decomposition method for solving delay differential equations, Int. J. Comput. Math. 82 (2005)49–54.

[5] I. Hashim , M . S . M . Noorani, R. Ahmad, S. A. Bakar, E . S . Ismail and A. M. Zakaria, Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos, Solitons and Fractals 28 (2006)1149–1158.

DOI: 10.1016/j.chaos.2005.08.135

[6] I. Hashim, Adomian decomposition method for solving BVPs for fourth- orderintegral- differential equations, J. Comp. Appl . Math. 193(2) (2006)658–664.

DOI: 10.1016/j.cam.2005.05.034

[7] I. Hashim, Commentson"A new algorithm for solving classical Blasius equation "by L. Wang, Appl. Math. Comput. 176(2) (2006)700–703.

DOI: 10.1016/j.amc.2005.10.016

[8] H. Jafariand V. D aftardar-Gejji, Revised Adomian decomposition method for solving systems of ordinary and fractional differential equations, Appl. Math. Comput. doi: 10. 1016/j. amc. 2005. 12. 049.

[9] D. Kaya and S.M. El-Sayed, Numerical soliton -like solutions of the potential Kadomtsev-Petviaashvili equation by the decomposition method, Phys. Letts. A. 320 (2003)192–199.

DOI: 10.1016/j.physleta.2003.11.021

[10] N. Shawagfeh and D. Kaya, Comparing numerical methods for the solutions of ordinary differential equations, Appl. Math. Lett. 17 (2004)323–328.

DOI: 10.1016/s0893-9659(04)90070-5

[11] M. S. M. Noorani, I. Hashim, R. Ahmad ,S. A. Bakar, E. S. Ismail and A. M. Zakaria, Comparing numerical methods for the solutions of the Chen system, Toappearin Chaos , Solitons and Fractals doi: 10. 1016/j. chaos. 2005. 12. 036.

DOI: 10.1016/j.chaos.2005.12.036

[12] P. Vadasz, S. Olek, Convergence and accuracy of Adomian's decomposition method for the solution of Lorenz equation, Int .J. Heat Mass Transfer 43 (2000)1715–1734.

DOI: 10.1016/s0017-9310(99)00260-4

[13] Noorhelyna Razali, solving Lorenz system by using runge-kutta method, European journal of scientific research, Vol. 32 No. 2 (2009) 241-251.