稳定嵌合体态：一种几何奇异摄动方法
Stable Chimera States: A Geometric Singular Perturbation Approach
AI总结：
本研究运用平均场技术与几何奇异摄动理论，分析含协同演化耦合强度的多层Kuramoto振子网络，推导稳定嵌合体态的充要条件，还生成呼吸型嵌合体、张弛振荡与鸭轨周期等模式。
摘要：
过去几十年间，嵌合体态（chimera states）因其出人意料的对称性破缺时空特性——在特定条件下同时表现出同步与非同步行为——受到了广泛关注。尽管这类反常状态已在多种物理和拓扑构型中取得了相关先导性成果，但仍有部分结构和机制有待揭示。本研究采用平均场技术，分析了由两类异质Kuramoto相位振子种群构成、且耦合强度协同演化的多层网络。此外，我们引入网络单元动力学与连接单元的自适应耦合强度之间的时间尺度分离，运用几何奇异摄动理论（Geometric Singular Perturbation Theory, GSPT），从快慢动力学视角更深入地洞察系统行为。由此，我们推导了存在协同演化层间耦合强度时，产生稳定嵌合体态的充要条件。另外，在上述约束下，通过选择合适的自适应律，能够生成有趣的模式，例如持续呼吸型嵌合体态。随后，我们分析了具有协同演化层内耦合强度的平均场系统的几何特性，并证明了依赖于关联网络的稳定嵌合体态的存在性。最后，由于系统的几何特性，我们在已确定的条件下通过数值方法生成了同样与呼吸型嵌合体相关的张弛振荡和鸭轨（canard）周期。
英文摘要：
Over the past decades chimera states have attracted considerable attention given their unexpected symmetry-breaking spatio-temporal nature, simultaneously exhibiting synchronous and incoherent behaviours under specific conditions. Despite relevant precursory results of such unforeseen states for diverse physical and topological configurations, there remain structures and mechanisms yet to be unveiled. In this work, using mean-field techniques, we analyze a multilayer network composed by two populations of heterogeneous Kuramoto phase oscillators with coevolutive coupling strengths. Moreover, we employ Geometric Singular Perturbation Theory (GSPT) with the inclusion of a time-scale separation between the dynamics of the network elements and the adaptive coupling strength connecting them, gaining a better insight into the behaviour of the system from a fast-slow dynamics perspective. Consequently, we derive the necessary and sufficient condition to produce stable chimera states when considering a co-evolutionary intercoupling strength. Additionally, under the aforementioned constraint and with a suitable adaptive law election, it is possible to generate intriguing patterns, such as persistent breathing chimera states. Thereafter, we analyze the geometric properties of the mean-field system with a co-evolutionary intracoupling strength and demonstrate the production of stable chimera states which depend on the associated network. Finally, relaxation oscillations and canard cycles, also related to breathing chimeras, are numerically produced under identified conditions due to the geometry of our system.
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