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Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition
| 1. | Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, 1 Hoegi-Dong, Dongdaemun-Gu, Seoul, 130-701, South Korea |
| 2. | Department of Mathematics, National Taiwan University, Taipei, 10617 |
References:
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I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equations, Rev. Mod. Phys., 74 (2002), 99-143.
doi: 10.1103/RevModPhys.74.99. |
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M. C. Cross and P. C. Hohenberg, Pattern formation outside of equillibrium, Rev. Mod. Phys., 65 (1993), 851-1112.
doi: 10.1103/RevModPhys.65.851. |
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S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equationm, SIAM J. Appl. Dyn. Sys., 4 (2005), 1-31. |
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J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics, Rev. Mod. Phys., 71 (1999), 396-403.
doi: 10.1103/RevModPhys.71.S396. |
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J. Han and M. Yari, Dynamic bifurcation of the one-dimensional periodic Swift-Hohenberg equation, Bull. Korean Math. Soc., to appear. |
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D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
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T. Ma and S. Wang, "Bifurcation Theory and Applications," World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. |
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T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences," Springer, in press. |
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L. A. Peletier and, V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D, 194 (2004), 95-126.
doi: 10.1016/j.physd.2004.01.043. |
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L. A. Peletier and W. C. Troy, "Spatial Patterns: Higher Order Models in Physics and Mecahnics," Progress in Nonlinear Differential Equations and their Applications, 45, Birkhäuser Boston, Inc., Boston, MA, 2001. |
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L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Sys., 6 (2007), 208-235. |
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J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.
doi: 10.1103/PhysRevA.15.319. |
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M. Yari, Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations, Dis. Cont. Dyn. Sys. Ser. B, 7 (2007), 441-456. |
show all references
References:
| [1] |
I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equations, Rev. Mod. Phys., 74 (2002), 99-143.
doi: 10.1103/RevModPhys.74.99. |
| [2] |
M. C. Cross and P. C. Hohenberg, Pattern formation outside of equillibrium, Rev. Mod. Phys., 65 (1993), 851-1112.
doi: 10.1103/RevModPhys.65.851. |
| [3] |
S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equationm, SIAM J. Appl. Dyn. Sys., 4 (2005), 1-31. |
| [4] |
J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics, Rev. Mod. Phys., 71 (1999), 396-403.
doi: 10.1103/RevModPhys.71.S396. |
| [5] |
J. Han and M. Yari, Dynamic bifurcation of the one-dimensional periodic Swift-Hohenberg equation, Bull. Korean Math. Soc., to appear. |
| [6] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
| [7] |
T. Ma and S. Wang, "Bifurcation Theory and Applications," World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. |
| [8] |
T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences," Springer, in press. |
| [9] |
L. A. Peletier and, V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D, 194 (2004), 95-126.
doi: 10.1016/j.physd.2004.01.043. |
| [10] |
L. A. Peletier and W. C. Troy, "Spatial Patterns: Higher Order Models in Physics and Mecahnics," Progress in Nonlinear Differential Equations and their Applications, 45, Birkhäuser Boston, Inc., Boston, MA, 2001. |
| [11] |
L. A. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Sys., 6 (2007), 208-235. |
| [12] |
J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.
doi: 10.1103/PhysRevA.15.319. |
| [13] |
M. Yari, Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations, Dis. Cont. Dyn. Sys. Ser. B, 7 (2007), 441-456. |
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