Final patterns and bifurcation analysis of the odd-periodic Swift–Hohenberg equation with respect to the period

ABSTRACT

We study the bifurcation analysis of the Swift–Hohenberg equation (SHE) with the odd-periodic condition as period parameter λ moves. Motivated by Peletier and Rottschäfer [Pattern selection of solutions of the Swift–Hohenberg equations. Phys D. 2004;194:95–126] and Peletier and Williams [Some canonical bifurcations in the Swift–Hohenberg equation. SIAM J Appl Dyn Syst. 2007;6:208–235], with the complete proof based on center manifold reduction, we show how the periodic SHE bifurcates from the trivial solution to an attractor when λ passes a critical number, and mainly when a gap collapsed to a point, and an overlapped interval emerges. Peletier and Williams provided the local behavior about nontrivial solutions of the SHE depending on critical lengths or the overlapped interval based on $L^2$-norm. In this paper, by dropping the symmetric condition given in the paper by Peletier and Williams, we extend their results and find the explicit stationary solution of the SHE. We also present several numerical results explaining our results.

Keywords:

• Swift–Hohenberg equation
• attractor bifurcation
• center manifold

2010 Mathematics Subject Classifications:

• 35G25
• 37G35

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Taeyoung Ha  http://orcid.org/0000-0002-9440-1918

Additional information

Funding

Y. Choi was supported by the Research Grant of Kwangwoon University in 2017. And T. Ha was supported by the National Institute for Mathematical Sciences(NIMS) grant funded by the Korean government (No. B181300000, No. B19610000).