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Bifurcation Analysis and Pattern Selection of Solutions for the Modified Swift–Hohenberg Equation
International Journal of Bifurcation and Chaos ( IF 2.2 ) Pub Date : 2020-09-22 , DOI: 10.1142/s0218127420300311
Yuncherl Choi 1 , Taeyoung Ha 2 , Jongmin Han 3
Affiliation  

In this paper, motivated by [Peletier & Rottschäfer, 2004; Peletier & Williams, 2007], we study the dynamical bifurcation of the modified Swift–Hohenberg equation endowed with an evenly periodic condition on the interval [Formula: see text]. As [Formula: see text] crosses over the critical points, the trivial solution bifurcates to an attractor and some new patterns of solutions emerge. We provide detailed descriptions of all possible final patterns of solutions on the overlapped intervals of [Formula: see text], which emerge after a gap collapses to a point. We also compute all critical values of [Formula: see text], [Formula: see text] and [Formula: see text] precisely, which are responsible for bifurcation and pattern formations. We finally provide numerical results that explain the main theorems.

中文翻译:

修正 Swift-Hohenberg 方程的分岔分析和解的模式选择

在本文中,受 [Peletier & Rottschäfer, 2004; Peletier & Williams, 2007],我们研究了修正的 Swift-Hohenberg 方程在区间上具有均匀周期条件的动力学分岔 [公式:见正文]。当 [公式:见正文] 越过临界点时,平凡的解决方案分叉为吸引子,并且出现了一些新的解决方案模式。我们在[公式:见文本]的重叠区间上提供了所有可能的最终解决方案模式的详细描述,这些模式在间隙塌陷到一个点后出现。我们还精确计算了[公式:见文本]、[公式:见文本]和[公式:见文本]的所有临界值,它们负责分叉和模式的形成。我们最终提供了解释主要定理的数值结果。
更新日期:2020-09-22
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