发表机构
Institut für Theoretische Chemie, Universität Ulm（乌尔姆大学理论化学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究局域非线性谐振器网络中的尖点灾变，提出极点组织机制，揭示尖点对产生条件与分类规则，并扩展至Kerr网络。
AI 中文摘要
一个具有单个局域三次非线性的谐振器，当它与一个原本线性的网络耦合时，可以获得额外的尖点对。在一次谐波阶次下，尖点轨迹是驱动点响应函数 $G(\Omega)$ 在非线性坐标处的相位等值线。我们将该等值线分解为有源极点及其背景的贡献。对于轻阻尼、孤立模态极点，在缓慢变化、近似实数的背景上，当该极点在非线性坐标处的参与度超过由阻尼半宽和该背景设定的闭式阈值时，尖点对即产生。在由主共振上方极点产生的弱混合硬化分支上，产生耦合随阻尼的平方根增长（固定模式间隔），并随模式间隔的平方根增长（固定阻尼）。完整尖点轨迹的曲率决定了局部折叠集是打开间隙（喙型）还是创建孤立环（唇型）。孤立极点近似将该分类器简化为背景响应规则，其有效性范围在100个随机网络上进行了量化。同一集合数值测试了极点阶梯，而多谐波延拓在原始方程中跟随整个重连序列——连接带、间隙、孤立透镜、无——在出生轨迹的折叠处，该处分类器保护最弱。该构造还扩展到耦合模式Kerr网络，其中驱动点响应设定固有折叠几何，驱动端口传递响应设定其可达性。
英文摘要
A resonator with one localized cubic nonlinearity can acquire additional cusp pairs when it is coupled to an otherwise linear network. At first-harmonic order, the cusp locus is a phase contour of the driving-point receptance $G(Ω)$ at the nonlinear coordinate. We resolve that contour into the contribution of an active pole and its background. For a lightly damped, isolated modal pole on a slowly varying, nearly real background, cusp-pair birth occurs once the participation of that pole at the nonlinear coordinate exceeds a closed-form threshold set by its damping half-width and that background. On the weak-mixing hardening branch generated by a pole above the primary resonance, the birth coupling then grows as the square root of the damping at fixed mode separation, and of the mode separation at fixed damping. The curvature of the full cusp contour determines whether the local fold set opens a gap (beaks type) or creates an isolated loop (lips type). The isolated-pole approximation reduces this classifier to a background-receptance rule whose range of validity is quantified on 100 random networks. The same ensemble tests the pole ladder numerically, while multi-harmonic continuation follows the whole reconnection sequence---connected band, gap, isolated lens, none---in the original equations at the fold of the birth locus, where the classifier is least protected. The construction also extends to coupled-mode Kerr networks, where the driving-point response sets the intrinsic fold geometry and the drive-port transfer response sets its accessibility.
