Research PaperNo Access

Performance of a hybrid computational scheme on traveling waves and its dynamic transition for Gilson–Pickering equation

Armutlu Vocational School, Yalova University, Armutlu, 77500 Yalova, Turkey

Corresponding author.

Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Rangpo, East-Sikkim 737136, India

E-mail Address: asit_saha123@rediffmail.com

Search for more papers by this author

, and

Sharanjeet Dhawan

Department of Mathematics, Central University of Haryana, 123029 Haryana, India

E-mail Address: dhawan311@gmail.com

Search for more papers by this author

https://doi.org/10.1142/S0129183119500281Cited by:13 (Source: Crossref)

Abstract

In this work, we developed a hybrid scheme for Gilson–Pickering equation by extracting advantageous features of collocation method and B-splines. Collocation method involves satisfying a differential equation to some tolerance at finite number of points and has low computational cost. The idea is to extract the superior accuracy and smoothness of B-spline functions with the low computational cost of collocation scheme. In the vision of a superficial periodic force, we obtain the quasiperiodic and chaotic behaviors of the forced Gilson–Pickering equation by perusing phase portrait technique, time series and Poincaré section. The sequel of frequency ( $ω$ ) of the superficial periodic force is very prime in case of transition from weakly chaotic feature to quasiperiodic feature of the forced Gilson–Pickering equation.

Keywords:

PACS: 02.60.Cb, 02.70.Jn, 47.11.Fg, 47.35.Fg

You currently do not have access to the full text article.
Recommend the journal to your library today!

References

1. P. A. Clarkson, E. L. Mansfield and T. J. Priestley, Math. Comput. Model. 25, 195 (1997). Web of Science, Google Scholar
2. C. Gilson and A. Pickering, J. Phys. A, Math. Gen. 28, 2871 (1995). ADS, Google Scholar
3. A. Chen, W. Huang and S. Tang, Nonlinear Anal. Real World Appl. 10, 2659 (2009). Web of Science, Google Scholar
4. A. Irshad and S. T. Mohyud-Din, Stud. Nonlinear Sci. 3, 24 (2012). Google Scholar
5. X. Fan, S. Yang and D. Zhao, Int. J. Nonlinear Sci. 8, 368 (2009). Google Scholar
6. G. Ebadi, A. H. Kara, M. D. Petkovic and A. Biswas, Waves Random Complex Media 21, 378 (2011). Web of Science, ADS, Google Scholar
7. B. Tian, Y.-T. Goa and W. Hong, Comput. Math. Appl. 44, 525 (2002). Web of Science, Google Scholar
8. Z.-Y. Yan and H.-Q. Zhang, J. Phys. A, Math. Gen. 34, 1785 (2001). ADS, Google Scholar
9. Y.-T. Gao and B. Tian, Phys. Lett. A 349, 314 (2006). Web of Science, ADS, Google Scholar
10. A. S. T. Nguetcho, J. Li and J. M. Bilbault, Commun. Nonlinear Sci. Numer. Simul. 19, 2590 (2014). Web of Science, ADS, Google Scholar
11. B. Jiang, Y. Lu, J. Zhang and Q. Bi, Appl. Math. Comput. 228, 220 (2014). Web of Science, Google Scholar
12. A. Saha, Commun. Nonlinear Sci. Numer. Simul. 17, 3539 (2012). Web of Science, ADS, Google Scholar
13. H. Liu and F. Yan, Appl. Math. Comput. 228, 384 (2014). Web of Science, Google Scholar
14. F. K. Abdullaev, Phys. Rep. 179, 1 (1989). Web of Science, ADS, Google Scholar
15. R. Grimshaw and X. Tian, Proc. R. Soc. A 445, 1 (1994). ADS, Google Scholar
16. D. J. Zheng, W. J. Yeh and O. G. Symko, Phys. Lett. A 140, 225 (1989). Web of Science, ADS, Google Scholar
17. K. B. Blyuss, Rep. Math. Phys. 49, 29 (2012). Web of Science, ADS, Google Scholar
18. Douvagai, Y. Salathiel, G. Betchewe, S. Y. Doka and K. T. Crepin, Opt. Int. J. Light Electron Opt. 140, 735 (2017). Google Scholar
19. P. M. Prenter, Splines and Variational Methods (John Wiley, New York, 1975). Google Scholar
20. C. de Boor, A Practicle Guide to Splines (Springer-Verlag, New York, 2001). Google Scholar
21. M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos and Patterns (Springer-Verlag, Berlin, 2003). Google Scholar
22. A. Saha, Nonlinear Dyn. 87, 2193 (2017). Web of Science, Google Scholar
23. A. Saha, Comput. Math. Appl. 73, 1879 (2017). Web of Science, Google Scholar