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Research Papers

Bifurcation in the Swift–Hohenberg Equation

[+] Author and Article Information
Qingkun Xiao

College of Sciences,
Nanjing Agricultural University,
Nanjing 210095, China
e-mail: xiaoqk@njau.edu.cn

Hongjun Gao

Institute of Mathematics,
School of Mathematical Sciences,
Nanjing Normal University,
Nanjing 210023, China
e-mail: gaohj@njnu.edu.cn

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 22, 2014; final manuscript received August 26, 2015; published online October 23, 2015. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 11(3), 031002 (Oct 23, 2015) (7 pages) Paper No: CND-14-1052; doi: 10.1115/1.4031489 History: Received February 22, 2014; Revised August 26, 2015

This paper is concerned with the asymptotic behavior of the solutions u(x, t) of the Swift–Hohenberg equation with quintic polynomial on the cylindrical domain Q=(0,L)×R+. With the control parameter α in the Swift–Hohenberg equation and the length L of the domain regarded as bifurcation parameters, branches of nontrivial solutions bifurcating from the trivial solution at certain points are shown. Local behavior of these branches is also investigated. With the help of a center manifold analysis, two types of structures in the bifurcation diagrams are presented when the bifurcation points are close, and their stabilities are analyzed.

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References

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Xiao, Q. K. , and Gao, H. J. , 2009, “ Bifurcation Analysis of the Swift–Hohenberg Equation With Quintic Nonlinearity,” Int. J. Bifurcation Chaos, 19(9), pp. 2927–2937. [CrossRef]
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Figures

Grahic Jump Location
Fig. 1

The asymptotic bifurcation diagram: n = 2 and m = 1

Grahic Jump Location
Fig. 2

The asymptotic bifurcation diagram: n = 3 and m = 2

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