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Localized patterns in planar bistable weakly coupled lattice systems
Nonlinearity ( IF 1.7 ) Pub Date : 2020-06-02 , DOI: 10.1088/1361-6544/ab7d1e
Jason J Bramburger 1, 2 , Bjrn Sandstede 1
Affiliation  

Localized planar patterns in spatially extended bistable systems are known to exist along intricate bifurcation diagrams, which are commonly referred to as snaking curves. Their analysis is challenging as techniques such as spatial dynamics that have been used to explain snaking in one space dimension no longer work in the planar case. Here, we consider bistable systems posed on square lattices and provide an analytical explanation of snaking near the anti-continuum limit using Lyapunov–Schmidt reduction. We also establish stability results for localized patterns, discuss bifurcations to asymmetric states, and provide further numerical evidence that the shape of snaking curves changes drastically as the coefficient that reflects the strength of the spatial coupling crosses a finite threshold.

中文翻译:

平面双稳态弱耦合晶格系统中的局部模式

已知空间扩展双稳态系统中的局部平面图案沿着复杂的分叉图存在,这通常被称为蛇形曲线。他们的分析具有挑战性,因为用于解释一个空间维度中的蛇行的空间动力学等技术不再适用于平面情况。在这里,我们考虑在方形晶格上构成的双稳态系统,并使用 Lyapunov-Schmidt 约简对在反连续谱附近的蛇行进行分析解释。我们还建立了局部模式的稳定性结果,讨论了不对称状态的分岔,并提供了进一步的数值证据,表明随着反映空间耦合强度的系数越过有限阈值,蛇形曲线的形状发生了剧烈变化。
更新日期:2020-06-02
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