Cyclic periodic synchronization in networks of piecewise nonlinear discontinuous maps

Vismaya V S and Sishu Shankar Muni

Chaos, Solitons & Fractals, 2026, vol. 208, issue P1

Abstract: We investigate cyclic periodic synchronization (CPS) in a ring–star network of piecewise nonlinear discontinuous (PND) maps. While CPS has previously been reported in piecewise linear discontinuous (PLD) systems, the linear structure of such maps restricts the diversity of emergent spatiotemporal behaviours. In this work, we extend the PLD framework by introducing quadratic nonlinearities on either side of the switching manifold. Analytical expressions for border–collision bifurcation curves associated with fixed points, period–2, and higher–order periodic orbits are derived and validated through two–parameter period diagrams. When coupled in a ring–star network, the PND maps exhibit a wide spectrum of spatiotemporal patterns, including CPS, traveling waves, unsynchronized states, clustered states, and two novel collective behaviours: cluster period–2 states and cyclic periodic cluster patterns. The mechanism for CPS patterns arises from the competition and collision of synchronized clusters, which reorganize periodically to produce robust cyclic patterns. We further examine nonlocally coupled ring–star networks, demonstrating that cyclic periodic synchronization persists even under nonlocal interactions. Two-parameter regime maps in the coupling strength plane reveal how local ring coupling, global hub coupling, and coupling range collectively govern transitions among synchronized, clustered, traveling-wave, and CPS patterns. Apart from the quadratic PND map, the framework is generalized to five additional nonlinear discontinuous maps: sinusoidal, tangential, hyperbolic tangent, rational, and exponential, confirming the robustness and persistence of cyclic period-3 synchronization across diverse nonlinear formulations.

Keywords: Ring–star network; Cyclic periodic synchronization; Piecewise nonlinear discontinuous map; Cluster competition; Cluster collision (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:208:y:2026:i:p1:s0960077926002638

DOI: 10.1016/j.chaos.2026.118122

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