Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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DYNAMICAL BIFURCATION OF THE ONE DIMENSIONAL MODIFIED SWIFT-HOHENBERG EQUATION

  • Received : 2014.08.26
  • Published : 2015.07.31
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Abstract

In this paper, we study the dynamical bifurcation of the modified Swift-Hohenberg equation on a periodic interval as the system control parameter crosses through a critical number. This critical number depends on the period. We show that there happens the pitchfork bifurcation under the spatially even periodic condition. We also prove that in the general periodic condition the equation bifurcates to an attractor which is homeomorphic to a circle and consists of steady states solutions.

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References

  1. M. Bestehorn and H. Haken, Transient patterns of the convection instability: A model-calculation, Z. Phys. B 57 (1984), 329-333. https://doi.org/10.1007/BF01470424
  2. Y. Choi and J. Han, Dynamical bifurcation of the damped Kuramoto-Sivashinsky equation, J. Math. Anal. Appl. 421 (2015), 383-398. https://doi.org/10.1016/j.jmaa.2014.07.009
  3. Y. Choi, J. Han, and C.-H. Hsia, Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 7, 1933-1957. https://doi.org/10.3934/dcdsb.2015.20.1933
  4. Y. Choi, J. Han, and J. Park, Dynamical bifurcation of the generalized Swift-Hohenberg equation, Int. J. Bifurcat. Chaos 25 (2015), no. 8.
  5. M. Cross and P. Hohenbrg, Pattern formation outside equilibrium, Rev. Mod. Phys. 65 (1993), 581-1112.
  6. A. Doleman, B. Sandstede, A. Scheel, and G. Schneider, Propagation of hexagonal patterns near onset, Euro. J. Appl. Math. 14 (2003), no. 9, 85-110.
  7. N. Duan andW. Gao, Optimal control of a modified Swift-Hohenberg equation, Electron. J. Differential Equations 2012 (2012), no. 155, 1-12.
  8. H. Gao and Q. Xiao, Bifurcation analysis of the 1D and 2D generalized Swift-Hohenberg equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), no. 3, 619-643. https://doi.org/10.1142/S0218127410025922
  9. J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics, Rev. Mod. Phys. 71 (1999), 396-403. https://doi.org/10.1103/RevModPhys.71.S396
  10. J. Han and C.-H. Hsia, Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 7, 2431-2449. https://doi.org/10.3934/dcdsb.2012.17.2431
  11. J. Han and M. Yari, Dynamic bifurcation of the periodic Swift-Hohenberg equation, Bull. Korean Math. Soc. 49 (2012), no. 5, 923-937. https://doi.org/10.4134/BKMS.2012.49.5.923
  12. M. Hilali, S. Metens, P. Borckmans, and G. Dewwl, Pattern selection in the generalized Swift-Hohenberg model, Phys. Rev. E 51 (1995), 2046-2052. https://doi.org/10.1103/PhysRevE.51.2046
  13. T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, 2005.
  14. L. A. Peletier and V. Rottschafer, Pattern selection of solutions of the Swift-Hohenberg equations, Phys. D 194 (2004), no. 1-2, 95-126. https://doi.org/10.1016/j.physd.2004.01.043
  15. L. Peletier and J. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst. 6 (2007), no. 1, 208-235. https://doi.org/10.1137/050647232
  16. M. Polat, Global attractor for a modified Swift-Hohenberg equation, Comput. Math. Appl. 57 (2009), no. 1, 62-66. https://doi.org/10.1016/j.camwa.2008.09.028
  17. L. Song, Y. Zhang, and T. Ma, Global attractor for a modified Swift-Hohenberg equation in $H^k$ spaces, Nonlinear Anal. 72 (2010), 183-191. https://doi.org/10.1016/j.na.2009.06.103
  18. J. Swift and P. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15 (1977), 319. https://doi.org/10.1103/PhysRevA.15.319
  19. Q. Xiao and H. Gao, Bifurcation analysis of the Swift-Hohenberg equation with quintic nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), no. 9, 2927-2937. https://doi.org/10.1142/S0218127409024542
  20. M. Yari, Attractor bifurcation and final patterns of the N-dimensional and generalized Swift-Hohenberg equations, Discrete Contin. Dyn. Syst. Ser. B 7 (2007), no. 2, 441-456. https://doi.org/10.3934/dcdsb.2007.7.441
  21. X. Zhao, B. Liu, P. Zhang, W. Zhang, and F. Liu, Fourier spectral method for the modified Swift-Hohenberg equation, Adv. Difference Eqns. 2013 (2013), no. 156, 1-19. https://doi.org/10.1186/1687-1847-2013-1

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