An investigation is undertaken of coupled reaction–diffusion systems in one spatial dimension that are able to support, in different regions of their parameter space, either an isolated spike solution, or stable localized patterns with an arbitrary number of peaks. The distinction between the two cases is characterized through the behavior of the far field, where there is either an oscillatory or a monotonic decay. This transition is illustrated with two examples: a generalized Schnakenberg system that arises in cellular-level morphogenesis and a continuum model of urban crime spread. In each, it is found that localized patterns connected via a so-called homoclinic snaking curve in parameter space transition into a single spike solution as a second parameter is varied, via a change in topology of the snake into a series of disconnected branches. The transition is caused by a so-called Belyakov–Devaney transition between complex and real spatial eigenvalues of the far field of the primary pulse. A codimension-two problem is studied in detail where a non-transverse homoclinic orbit undergoes this transition. A Shilnikov-style analysis is undertaken which reveals the asymptotics of how the infinite family of folds of multi-pulse orbits are all destroyed at the same parameter value. The results are shown to be consistent with numerical experiments on the examples.

对在一个空间维度上的耦合反应扩散系统进行了研究，该系统能够在其参数空间的不同区域中支持孤立的尖峰解或具有任意数量峰的稳定局部模式。两种情况之间的区别是通过远场的行为来表征的，在远场中，存在振荡或单调衰减。通过两个示例说明了这种过渡：在细胞水平的形态发生中出现的广义Schnakenberg系统和城市犯罪蔓延的连续模型。在每种情况下，都发现在第二个参数改变时，通过蛇形拓扑结构转变为一系列不连续的分支，在第二个参数变化的情况下，通过所谓的同斜弯曲曲线在参数空间中连接的局部模式会转变为单个峰值解决方案。过渡是由一次脉冲远场的复数和实际空间特征值之间的所谓Belyakov-Devaney过渡引起的。当非横向同宿轨道经历这种过渡时，将详细研究一个二维问题。进行了希尔尔科夫式分析，它揭示了在相同的参数值下多脉冲轨道的无限大折叠家族如何被破坏的渐近现象。结果显示与实施例的数值实验一致。进行了希尔尔科夫式分析，它揭示了在相同的参数值下多脉冲轨道的无限大折叠家族如何被破坏的渐近现象。结果显示与实施例的数值实验一致。进行了希尔尔科夫式分析，它揭示了在相同的参数值下多脉冲轨道的无限大折叠家族如何被破坏的渐近现象。结果显示与实施例的数值实验一致。