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DYNAMICAL BIFURCATION OF THE ONE DIMENSIONAL MODIFIED SWIFT-HOHENBERG EQUATION

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대한수학회 (The Korean Mathematical Society)|2015.07|대한수학회보 (Bulletin of the KMS)|제52권 제4호|pp. 1241-1252

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참고문헌

키워드

modified Swift-Hohenberg equationdynamic bifurcationcenter manifold function
주제분류수학

초록

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.

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.

참고문헌 (21)

  1. M. Bestehorn, "Transient patterns of the convection instability: A model-calculation", Z. Phys. B, 57, pp.329~333 (1984)
  2. Y. Choi, "Dynamical bifurcation of the damped Kuramoto-Sivashinsky equa-tion", J. Math. Anal. Appl, 421, pp.383~398 (2015)
  3. Y. Choi, "Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period", Discrete Contin. Dyn. Syst. Ser. B, 20(7), pp.1933~1957 (2015)
  4. Y. Choi, "Dynamical bifurcation of the generalized Swift-Hohenberg equation", Int. J. Bifurcat. Chaos, 25(8) (2015)
  5. M. Cross, "Pattern formation outside equilibrium", Rev. Mod. Phys, 65, pp.581~1112 (1993)
  6. A. Doleman, "Propagation of hexagonal pat-terns near onset", Euro. J. Appl. Math, 14(9), pp.85~110 (2003)
  7. N. Duan, "Optimal control of a modified Swift-Hohenberg equation", Electron. J. Differential Equations, 2012(155), pp.1~12 (2012)
  8. H. Gao, "Bifurcation analysis of the 1D and 2D generalized Swift-Hohenberg equation", Internat. J. Bifur. Chaos Appl. Sci. Engrg, 20(3), pp.619~643 (2010)
  9. J. P. Gollub, "Pattern formation in nonequilibrium physics", Rev. Mod. Phys, 71, pp.396~403 (1999)
  10. J. Han, "Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition", Discrete Contin. Dyn. Syst. Ser. B, 17(7), pp.2431~2449 (2012)
  11. 한종민, "DYNAMIC BIFURCATION OF THE PERIODIC SWIFT-HOHENBERG EQUATION", 대한수학회보, 49(5), pp.923~937 (2012), 대한수학회
  12. M. Hilali, "Pattern selection in the generalized Swift-Hohenberg model", Phys. Rev. E, 51, pp.2046~2052 (1995)
  13. T. Ma, "Bifurcation Theory and Applications" (2005), World Scientific
  14. L. A. Peletier, "Pattern selection of solutions of the Swift-Hohenberg equations", Phys. D, 194(1-2), pp.95~126 (2004)
  15. L. Peletier, "Some canonical bifurcations in the Swift-Hohenberg equa-tion", SIAM J. Appl. Dyn. Syst, 6(1), pp.208~235 (2007)
  16. M. Polat, "Global attractor for a modified Swift-Hohenberg equation", Comput. Math. Appl, 57(1), pp.62~66 (2009)
  17. L. Song, "Global attractor for a modified Swift-Hohenberg equation in Hk spaces", Nonlinear Anal, 72, pp.183~191 (2010)
  18. J. Swift, "Hydrodynamic fluctuations at the convective instability", Phys. Rev. A, 15, pp.319~ (1977)
  19. Q. Xiao, "Bifurcation analysis of the Swift-Hohenberg equation with quintic nonlinearity", Internat. J. Bifur. Chaos Appl. Sci. Engrg, 19(9), pp.2927~2937 (2009)
  20. M. Yari, "Attractor bifurcation and final patterns of the N-dimensional and generalized Swift-Hohenberg equations", Discrete Contin. Dyn. Syst. Ser. B, 7(2), pp.441~456 (2007)
  21. X. Zhao, "Fourier spectral method for the modified Swift-Hohenberg equation", Adv. Difference Eqns, 2013(156), pp.1~19 (2013)

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