DYNAMIC BIFURCATION OF THE PERIODIC SWIFT-HOHENBERG EQUATION
2
조회
0
다운로드
2
인용
0
참고문헌
키워드
초록
In this paper we study the dynamic bifurcation of the Swift-Hohenberg equation on a periodic cell ${\Omega}=[-L,L]$. It is shown that the equations bifurcates from the trivial solution to an attractor $\mathcal{A}_{\lambda}$ when th control parameter λ crosses the critical value. In the odd periodic case $\mathcal{A}_{\lambda}$ is homeomorphic to $S^1$ and consists of eight singular points and thei connecting orbits. In the periodic case, $\mathcal{A}_{\lambda}$ is homeomorphic to $S^1$, an contains a torus and two circles which consist of singular points.
Abstract
In this paper we study the dynamic bifurcation of the Swift-Hohenberg equation on a periodic cell ${\Omega}=[-L,L]$. It is shown that the equations bifurcates from the trivial solution to an attractor $\mathcal{A}_{\lambda}$ when th control parameter λ crosses the critical value. In the odd periodic case $\mathcal{A}_{\lambda}$ is homeomorphic to $S^1$ and consists of eight singular points and thei connecting orbits. In the periodic case, $\mathcal{A}_{\lambda}$ is homeomorphic to $S^1$, an contains a torus and two circles which consist of singular points.
참고문헌 (16)
- E. Bodenschatz, "Recent Developments in Rayleigh-B´enard Convections", Annual review of fluid mechanics, 32, pp.709~778 (2000), Annual Reviews
- M. C. Cross, "Pattern formation outside of equilibrium", Rev. Mod. Phys, 65, pp.851~111 (1993)
- P. G. Daniels, "Solutions for nonlinear convection in the presence of a lateral boundary", Phys. D, 178(1), pp.83~102 (2003)
- D. Henry, "Geometric Theory of Semilinear Parabolic Equations", Lecture Notes in Mathematics, 840 (1981), Springer
- C.-H. Hsia, "On a Burgers’ type equation", Discrete Contin. Dyn. Syst. Ser. B, 6(5), pp.1121~1139 (2006)
- T. Ma, "Dynamic bifurcation of the Ginzburg-Landau equation", SIAM J. Appl. Dyn. Syst, 3, pp.620~635 (2004)
- T. Ma, "Addendum to the paper “Two-dimensional infinite prandtl number convec- tion: structure of bifurcated solutions", J. Nonlinear Sci, 17(3), pp.199~220 (2007)
- T. Ma, "Bifurcation Theory and Applications" (2005), World Scientific
- T. Ma, "Stability and Bifurcation of Nonlinear Evolution Equations" (2007), Science Press
- T. Ma, "Rayleigh-B´enard convection: dynamics and structure in the physical space", Commun. Math. Sci, 5(3), pp.553~574 (2007)
- T. Ma, "Cahn-Hilliard equations and phase transition dynamics for binary systems", Discrete Contin. Dyn. Syst. Ser. B, 11, pp.741~784 (2009)
- L. A. Peletier, "Pattern selection of solutions of the Swift-Hohenberg equation", Phys. D, 194(1), pp.95~126 (2004)
- L. A. Peletier, "Some Canonical Bifurcations in the Swift-Hohenberg Equation", SIAM J. Appl. Dyn. Syst, 6(1), pp.208~235 (2007)
- J. Swift, "Hydrodynamic fluctuations at the convective instability", Phys. Rev. A, 15, pp.319~328 (1977)
- A.-M.Wazwaz, "New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawa- hara equations", Appl. Math. Comput, 182(2), pp.1642~1650 (2006)
- 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)