Research PapersNo Access

Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

Department of Physics, North University of China, Taiyuan 030051, China

E-mail Address: dongchengwei@tsinghua.org.cn

https://doi.org/10.1142/S0217984918501555Cited by:10 (Source: Crossref)

Abstract

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter $ν$ $ν$ = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

Keywords:

References

1. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Hachette, UK, 2014). Google Scholar
2. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, New York, 1997). Google Scholar
3. R. Artuso, E. Aurell and P. Cvitanovic, Nonlinearity 3 (1990) 325. Web of Science, ADS, Google Scholar
4. R. Artuso, E. Aurell and P. Cvitanovic, Nonlinearity 3 (1990) 361. Web of Science, ADS, Google Scholar
5. P. Cvitanović, R. Artuso, R. Mainieri, G. Tanner, G. Vattay, N. Whelan and A. Wirzba, Chaos: Classical and Quantum (Niels Bohr Institute, Copenhagen, 2012). Google Scholar
6. F. Christiansen, P. Cvitanović and V. Putkaradze, Nonlinearity 10 (1997) 55. Web of Science, ADS, Google Scholar
7. S. M. Zoldi and H. S. Greenside, Phys. Rev. E 57 (1998) R2511. Web of Science, ADS, Google Scholar
8. M. Kurulay, A. Secer and M. A. Akinlar, Appl. Math. Inform. Sci. 7 (2013) 267. Web of Science, Google Scholar
9. P. Rebelo, Int. J. Numer. Meth. Eng. 27 (2011) 874. Web of Science, Google Scholar
10. F. Khellat and N. Vasegh, Commun. Nonlinear Sci. 19 (2014) 3011. Web of Science, Google Scholar
11. X. Ding, H. Chaté, P. Cvitanović, E. Siminos and K. A. Takeuchi, Phys. Rev. Lett. 117 (2016) 024101. Web of Science, ADS, Google Scholar
12. E. L. Rempel, A. C. Chian, E. E. Macau and R. R. Rosa, Chaos 14 (2004) 545. Web of Science, ADS, Google Scholar
13. J. Nickel, Chaos Soliton. Fract. 33 (2007) 1376. Web of Science, ADS, Google Scholar
14. V. A. Galaktionov, E. Mitidieri and S. I. Pohozaev, Dokl. Math. 77 (2008) 238. Web of Science, Google Scholar
15. M. Gameiro and J. P. Lessard, SIAM J. Appl. Dyn. Syst. 16 (2017) 687. Web of Science, Google Scholar
16. J. L. Figueras and R. D. L. Llave, SIAM J. Appl. Dyn. Syst. 16 (2017) 834. Web of Science, Google Scholar
17. N. B. Budanur and P. Cvitanović, J. Stat. Phys. 167 (2017) 636. Web of Science, ADS, Google Scholar
18. C. Dong and Y. Lan, Commun. Nonlinear Sci. 19 (2014) 2140. Web of Science, ADS, Google Scholar
19. S. A. Kashchenko, Theor. Math. Phys. 192 (2017) 958. Web of Science, Google Scholar
20. K. W. Ong, Discrete Contin. Dyn. B 21 (2017) 1225. Web of Science, Google Scholar
21. Y. Saiki, M. Yamada, A. C. L. Chian, R. A. Miranda and E. L. Rempel, Chaos 25 (2015) 103123. Web of Science, ADS, Google Scholar
22. Y. Lan and P. Cvitanović, Phys. Rev. E 69 (2004) 016217. Web of Science, ADS, Google Scholar
23. Y. Kuramoto, Prog. Theor. Phys. Suppl. 64 (1978) 346. ADS, Google Scholar
24. G. I. Sivashinsky, SIAM J. Appl. Math. 39 (1980) 67. Web of Science, Google Scholar
25. G. I. Sivashinsky and D. M. Michelson, Prog. Theor. Phys. Suppl. 63 (1980) 2112. Google Scholar
26. D. J. Benney, J. Math. Phys. 45 (1966) 150. Web of Science, Google Scholar
27. B. I. Cohen, J. A. Krommes, W. M. Tang and M. N. Rosenbluth, Nucl. Fusion 16 (1972) 971. Web of Science, ADS, Google Scholar
28. R. E. Quey, S. M. Mahajan, P. H. Rutherford and W. M. Tang, Phys. Rev. Lett. 34 (1974) 391. Web of Science, Google Scholar
29. A. P. Hooper and R. Grimshaw, Phys. Fluids 28 (1985) 37. Web of Science, ADS, Google Scholar
30. Q. Chen, J. D. Meiss and I. C. Percival, Physica D 23 (1987) 143. Web of Science, ADS, Google Scholar
31. B. Mestel and I. Percival, Physica D 24 (1987) 172. Web of Science, ADS, Google Scholar
32. A. A. Olyaei, C. Wu and W. Kinsner, Phys. Rev. E 96 (2017) 012207. Web of Science, ADS, Google Scholar
33. C. Dong, P. Wang, M. Du, T. Uzer and Y. Lan, Mod. Phys. Lett. B 30 (2016) 1650183. Link, Web of Science, ADS, Google Scholar
34. W. H. Press, S. A. Teukolsky, W. T. Veterling and B. P. Flannery, Numerical Recipes in Fortran 77. The Art of Scientific Computing (Cambridge University Press, New York, 1992). Google Scholar
35. B. L. Hao and W. M. Zheng, Applied Symbolic Dynamics and Chaos (World Scientific, Singapore, 1998). Link, Google Scholar
36. E. N. Lorenz, J. Atmos. Sci. 20 (1963) 130. Web of Science, ADS, Google Scholar
37. D. Viswanath, Nonlinearity 16 (2003) 1035. Web of Science, ADS, Google Scholar