论文标题
常规图中耦合图的同步稳定性的关键参数
Critical parameters of the synchronisation's stability for coupled maps in regular graphs
论文作者
论文摘要
耦合的地图晶格（CML）模型特别适合研究空间扩展行为，例如波浪状模式，时空混乱和同步。当所有地图都具有相等大小的状态变量时，CMLS中的完全同步出现，形成了随时间演变的空间均匀模式。 Here, we derive critical values for the parameters -- coupling strength, maximum Lyapunov exponent, and link density -- that control the synchronisation-manifold's linear stability of diffusively-coupled, identical, chaotic maps in generic regular graphs (i.e., graphs with uniform node degrees) and class-specific cyclic graphs (i.e., periodic lattices with cyclical node permutation对称）。我们的派生基于拉普拉斯矩阵特征值，在其中，我们给出了常规图的最小非零特征值和最大的特征值的封闭形式表达式，并表明这些图可以根据拓扑条件分为两个集合（从稳定性分析中得出）。我们还为两类环状图制作了派生：$ k $ -cycles（即，可以嵌入$ t^k $ tori的定期晶格，甚至可以嵌入$ t^k $ tori）和$ k $-Möbiusladders，我们在此处介绍这些内容，以概括为$ k = 3 $ k = 3 $的Möbiusladder ladder。我们的结果突出了有限尺寸和无限尺寸限制的同步歧管稳定性（即使对于相同的节点度）的稳定性的差异。
Coupled Map Lattice (CML) models are particularly suitable to study spatially extended behaviours, such as wave-like patterns, spatio-temporal chaos, and synchronisation. Complete synchronisation in CMLs emerges when all maps have their state variables with equal magnitude, forming a spatially-uniform pattern that evolves in time. Here, we derive critical values for the parameters -- coupling strength, maximum Lyapunov exponent, and link density -- that control the synchronisation-manifold's linear stability of diffusively-coupled, identical, chaotic maps in generic regular graphs (i.e., graphs with uniform node degrees) and class-specific cyclic graphs (i.e., periodic lattices with cyclical node permutation symmetries). Our derivations are based on the Laplacian matrix eigenvalues, where we give closed-form expressions for the smallest non-zero eigenvalue and largest eigenvalue of regular graphs and show that these graphs can be classified into two sets according to a topological condition (derived from the stability analysis). We also make derivations for two classes of cyclic graph: $k$-cycles (i.e., regular lattices of even degree $k$, which can be embedded in $T^k$ tori) and $k$-Möbius ladders, which we introduce here to generalise the Möbius ladder of degree $k = 3$. Our results highlight differences in the synchronisation manifold's stability of these graphs -- even for identical node degrees -- in the finite size and infinite size limit.
