通过正规形约化将共振孤立态刻画为相对平衡
Characterizing resonant solitary states as relative equilibria via normal form reduction
- Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出正规形与对称性方法，将耦合振子网络约化为二维系统，把共振孤立态的频率选择转化为相对平衡搜索，并给出存在性的显式代数条件。
AI中文摘要：
我们研究了耦合振子网络中的孤立态，其中单个振子脱离同步簇并以不同的平均频率演化。我们聚焦于共振孤立态，即脱离振子的平均频率由与簇的某一模态的共振所塑造。在此，我们引入一种基于正规形和对称性的方法来识别此类状态。我们将高维网络动力学约化为一个二维二阶系统，该系统将一个代表孤立振子的旋转子与一个代表簇的主导拉普拉斯模态的谐振子耦合。我们利用正规形变换在该系统中引入对称性。借助这种对称性，我们将频率选择问题重新表述为对正规形对称性的相对平衡的搜索。我们推导出这些相对平衡存在的显式代数条件，从而也推导出共振孤立态及其有效频率存在的显式代数条件。
英文摘要：
We investigate solitary states in coupled oscillator networks, where a single oscillator detaches from a synchronized cluster and evolves at a distinct mean frequency. We focus on resonant solitary states, in which the mean frequency of the detached oscillator is shaped by resonance with a mode of the cluster. Here, we introduce a normal-form and symmetry-based approach to identify such states. We reduce the high-dimensional network dynamics to a two-dimensional second-order system that couples a rotator representing the solitary oscillator to a harmonic oscillator representing the dominant Laplacian mode of the cluster. We use normal-form transformations to introduce symmetry into this system. Using this symmetry, we reformulate the frequency selection problem as the search for relative equilibria of the normal form symmetry. We derive explicit algebraic conditions for the existence of these relative equilibria, and hence for resonant solitary states and their effective frequencies.
