Collective Phase Reorganization and Cluster Synchronization in Networks of Coupled Gumowski--Mira Maps

Abstract

We investigate the collective dynamics of networks composed of diffusively coupled Gumowski-Mira maps and analyze how modifications in the intrinsic dynamics of the local oscillator reorganize the emergent phase structure of the network. The coupling strength and the local control parameter are treated as bifurcation parameters, and the resulting collective states are quantified using the largest Lyapunov exponent, a synchronization error measure, cluster-count statistics, and collective phase-classification diagrams. Two representative regimes of the local dynamics are examined. In the first regime, the network exhibits a smooth and highly organized collective parameter space, featuring a synchronization wedge embedded within an extended region of periodic cluster states. In the second regime, the same coupling architecture yields a fragmented phase organization, comprising disconnected synchronization islands, incoherent domains, and enhanced chaotic-cluster states. These results indicate that variations in the intrinsic dynamics of the individual Gumowski-Mira oscillator do not simply shift synchronization thresholds but can fundamentally restructure the topology of the collective phase space. Our findings thus establish a direct relationship between local nonlinear dynamics and the emergent organization of collective phases in coupled discrete-time networks.