Exotic patterns on quasi-ring networks

Pattern formation in network-coupled dynamical systems plays an important role in many physical, chemical, and biological processes. In this work, we investigate the emergence of spatial patterns in reaction–diffusion dynamics on networks, with a particular focus on quasi-ring topologies. The dynamics are studied using a two-species reaction–diffusion model on a network, where diffusion is represented through the graph Laplacian. The stability of the homogeneous steady state is analyzed using the Master Stability Function framework. By decomposing perturbations into Laplacian eigenmodes, the stability of each spatial mode can be determined from the associated Jacobian spectrum. To understand the nonlinear development of instabilities, weakly nonlinear theory is employed, leading to amplitude equations that describe the growth and saturation of patterns near the instability threshold. Numerical simulations of the Brusselator model are performed on both regular ring networks and perturbed quasi-ring networks. While the symmetric ring supports spatially extended patterns associated with delocalized Fourier modes, small perturbations to the coupling weights break this symmetry and produce localized Laplacian eigenvectors. In the quasi-ring topology, the eigenvectors that become localized are those associated with the short-wave instability, and their interaction with the delocalized modes responsible for the long-wave instability plays a key role in shaping the resulting patterns. This eigenvector localization strongly influences the resulting dynamics and can confine activity to specific regions of the network. As a consequence, the quasi-ring network exhibits a variety of dynamical regimes, including synchronized oscillations, oscillation death, traveling waves, and mixed spatial states in which stationary and oscillatory regions coexist. These results demonstrate how small structural perturbations can significantly alter the spectral properties of a network and lead to complex pattern formation, including chimera-like behavior.