From heteroclinic loops to homoclinic snaking in reversible systems: rigorous forcing through computer-assisted proofs
Jan Bouwe van den Berg, Gabriel William Duchesne and Jean-Philippe Lessard
同宿蛇行是在许多模式形成系统中观察到的普遍现象。尽管基于模式前沿解的识别已经发展出了强制理论，但在非微扰区域证明其存在性仍然很困难。由于问题的非线性特性，这些异宿解本身也难以分析。本文使用计算机辅助证明方法，在时间可逆系统中寻找平衡点与周期轨道之间参数化的异宿连接环。这导致我们在Swift-Hohenberg和Gray-Scott问题中都证明了同宿蛇行现象的存在。我们的结果表明，非线性动力系统中连续族连接轨道的计算机辅助证明是理解全局动力学及其参数依赖性的有力工具。
Homoclinic snaking is a widespread phenomenon observed in many pattern-forming systems. Demonstrating its occurrence in non-perturbative regimes has proven difficult, although a forcing theory has been developed based on the identification of patterned front solutions. These heteroclinic solutions are themselves challenging to analyze due to the nonlinear nature of the problem. In this paper, we use computer-assisted proofs to find parameterized loops of heteroclinic connections between equilibr...