Approximate solutions of the hyperchaotic Rössler system by using the Bessel collocation scheme

Abstract

The purpose of this study is to give a Bessel polynomial approximation for the solutions of the hyperchaotic R¨ossler system.For this purpose, the Bessel collocation method applied to different problems is developed for the mentioned system. This method isbased on taking the truncated Bessel expansions of the functions in the hyperchaotic R¨ossler systems. The suggested secheme convertsthe problem into a system of nonlinear algebraic equations by means of the matrix operations and collocation points, The accuracy andefficiency of the proposed approach are demonstrated by numerical applications and performed with the help of a computer code writtenin Maple. Also, comparison between our method and the differential transformation method is made with the accuracy of solutions

Keywords

• Hyperchaotic R¨ossler system,Bessel collocation method,approximate solution,Nonlinear differential equation systems,Bessel functions of first kind,Collocation points

Approximate solutions of the hyperchaotic R ¨ossler system by using the Bessel collocation scheme

Öz

References

1. O. Abdulaziz, N.F.M. Noor, I. Hashim, M.S.M. Noorani, Further accuracy tests on Adomian decomposition method for chaotic systems, Chaos, Solitons & Fractals 36 (2008) 1405–1411.
2. M.M. Al-Sawalha, M.S.M. Noorani, I. Hashim, On accuracy of Adomian decomposition method for hyperchaotic R¨ossler system, Chaos, Solitons & Fractals, 40 (2009) 1801–1807.
3. I. Hashim, M.S.M. Noorani, R. Ahmad, S.A. Bakar, E.S.I. Ismail, A.M. Zakaria, Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos, Solitons & Fractals, 28 (2006) 1149–1158.
4. M.S.M. Noorani, I. Hashim, R. Ahmad, S.A. Bakar, E.S.I. Ismail, A.M. Zakaria, Comparing numerical methods for the solutions of the Chen system, Chaos, Solitons & Fractals, 32 (2007) 1296–1304.
5. S.M. Goh, M.S.M. Noorani, I. Hashim, Efficacy of variational iteration method for chaotic Genesio system-Classical and multistage approach, Chaos, Solitons & Fractals, 40 (2009) 2152-2159.
6. S.M. Goh , M.S.M. Noorani, I. Hashim, A new application of variational iteration method for the chaotic R¨ossler system, Chaos, Solitons & Fractals, 42 (2009) 1604–1610.
7. S.M. Goh, M.S.M. Noorani, I. Hashim, Prescribing a multistage analytical method to a prey-predator dynamical system, Phys. Lett. A, 373 (2008) 107–110.
8. S.M. Goh, M.S.M. Noorani, I. Hashim, M.M. Al-Sawalha, Variational iteration method as a reliable treatment for the hyperchaotic R¨ossler system, Int. J. Nonlinear Sci. Numer. Simul., 10 (2009) 363–371.

9. S.M. Goh, A.I.M. Ismail, M.S.M. Noorani, I. Hashim, Dynamics of the Hantavirus infection through variational iteration method, Non Anal: Real World Appl.,10 (2009) 2171–2176.
10. F.M. Allan, Construction of analytic solution to chaotic dynamical systems using the Homotopy analysis method, Chaos, Solitons & Fractals, 39 (2009) 1744–1752.
11. M.S.H. Chowdhury, I. Hashim, Application of multistage homotopy-perturbation method for the solutions of the Chen system, Nonlinear Anal: Real World Appl, 10 (2009) 381–391.
12. M.M. Al-Sawalha, M.S.M. Noorani, Application of the differential transformation method for the solution of the hyperchaotic R¨ossler system, Commun Nonlinear Sci Numer Simul, 14 (2009) 1509–1514.
13. J. H. Park, Chaos synchronization between two different chaotic dynamical systems, Chaos, Solitons and Fractals 27(2006) 549–554.
14. T. Plienpanich, P. Niamsup and Y. Lenbury, Controllability and stability of the perturbed chen chaotic dynamical system, Appl. Math. Comput. 171 (2005) 927–947.
15. M. T. Yassen, The optimal control of Chen chaotic dynamical system, Appl. Math. Comput. 131 (2002) 171–180.
16. M. T. Yassen, Chaos control of Chen chaotic dynamical system, Chaos, Solitons and Fractals 15 (2003) 271–283.
17. O.E. R¨ossler, An equation for continuous chaos, Phys. Lett.,57A (1976) 397–398.
18. S¸. Y¨uzbas¸ı, A numerical approach for solving the high-order linear singular differential-difference equations, Comput. Math. Appl., 62 (5) (2011) 2289–2303.
19. S¸. Y¨uzbas¸ı, N. S¸ahin, M. Sezer, A Bessel collocation method for numerical solution of generalized pantograph equations, Numerical Methods for Partial Differential Equations,28(4)(2012) 1105-1123.
20. S¸ Y¨uzbas¸ı, A numerical approach for solving a class of the nonlinear Lane–Emden type equations arising in astrophysics, Math. Meth. Appl. Sci. 34 (2011) 2218–2230.
21. S¸. Y¨uzbas¸ı, N. S¸ahin, M. Sezer, A numerical approach for solving linear differential equation systems, J. Adv. Res. Diff. Equ. 3(3): (2011) 8-32.
22. A. Aky¨uz-Das¸cıo˘glu, H. C¸ . Yaslan, An approximation method for solution of nonlinear integral equations, Appl. Math. Comput. 174 (2006) 619-629.