Cyclic periodic synchronization in a ring-star network of piecewise-linear discontinuous maps

In this paper, we explored the emergence of cyclic periodic synchronization patterns in networks of coupled piecewise-linear discontinuous (PLD) maps. The maps, characterized by the discontinuities in their dynamics, are capable of generating rich nonlinear behavior, including multistability, sensitivity to initial conditions, and complex bifurcation structures. Notably, this PLD map is a discrete version of the differential Chay neuron model, widely used in computational neuroscience, and can generate a similar bifurcation scenario of the inter-spike interval observed in neural experiments. We first analyzed the individual map dynamics through bifurcation diagrams, Lyapunov exponent spectrum, and border-collision bifurcations. In particular, the system exhibits period-adding structures whose boundaries are defined by border-collision bifurcations, providing a clear route to complex synchronization dynamics. Then, we investigated the behavior of these maps in structured coupled networks, focusing on ring, and hybrid ring-star topologies. In these coupled systems, we observed a wide range of spatiotemporal patterns, including the full synchronized state, unsynchronized state, traveling waves, clustered state, and most importantly, cyclic period-3 synchronization pattern, cyclic period-4 synchronization pattern, and cyclic period-5 synchronization pattern. The emergence and stability of these states are strongly influenced by the coupling strength parameters, εr and εs, which play an important role in governing cluster competition and cluster collision, thereby determining the persistence or breakdown of cyclic periodic synchronization patterns. Regime maps in both the parameter and coupling strength revealed the boundaries of these distinct behaviors and demonstrated how interface width and basin boundary influence pattern transitions.