Critical parameters of the synchronisation's stability for coupled maps in regular graphs

Coupled Map Lattice (CML) models are suitable to study spatially extended behaviours, such as wave-like patterns, spatio-temporal chaos, and synchronisation. In CMLs, synchronisation emerges when all the maps have their state variables with equal magnitude, forming a spatially uniform pattern that evolves in time. In this talk, I will show some closed-form expressions that we derived for the critical values of parameters that control the synchronisation-manifold's 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). These parameters include the coupling strength, maximum Lyapunov exponent, and link density, which reveal interesting implications to the synchronisation properties of CMLs. Our derivations are based on the Laplacian matrix eigenvalues. In particular, we show that all regular graphs can be classified into two classes (according to a topological condition and the stability of the synchronisation manifold), and derive closed-form expressions for the smallest non-zero eigenvalue and largest eigenvalue of 2 types of cyclic graphs: $k$-cycles (i.e., regular lattices of even degree $k$, which can be embedded in $T^k$ tori) and $k$-M{\"o}bius ladders, which we introduce to generalise the M{\"o}bius ladder of degree $k = 3$ so that we can extend our analyses to the thermodynamic limit. Our results highlight differences in the synchronisation manifold's stability of these cyclic graphs, even for identical node degrees, and for the finite and infinite limit-size.