We investigate stationary, spatially localized patterns in lattice dynamical systems that exhibit bistability. The profiles associated with these patterns have a long plateau where the pattern resembles one of the bistable states, while the profile is close to the second bistable state outside this plateau. We show that the existence branches of such patterns generically form either an infinite stack of closed loops (isolas) or intertwined s-shaped curves (snaking). We then use bifurcation theory near the anti-continuum limit, where the coupling between edges in the lattice vanishes, to prove existence of isolas and snaking in a bistable discrete real Ginzburg–Landau equation. We also provide numerical evidence for the existence of snaking diagrams for planar localized patches on square and hexagonal lattices and outline a strategy to analyse them rigorously.

中文翻译：

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晶格动力系统中的空间局部结构

我们调查显示双稳态的晶格动力学系统中的固定，空间局部的模式。与这些图案相关联的轮廓具有较长的平台，其中该图案类似于双稳态之一，而轮廓则在该平台之外接近第二双稳态。我们表明，此类模式的存在分支通常形成无限的闭环堆栈（等值线）或交错的S形曲线（蛇形）。然后，我们在反连续极限附近使用分叉理论，在该理论中，晶格边缘之间的耦合消失，以证明在双稳态离散真实Ginzburg-Landau方程中存在isolas和蛇行。