Memory-aware event-triggered bipartite synchronisation of signed fractional RC–Schmitt oscillator networks

We develop a memory-aware event-triggered framework for structurally balanced networks of Schmitt-trigger relaxation oscillators whose storage branch is represented by a constant-phase element. The hybrid circuit model yields Caputo dynamics of order α∈(0,1] and, under weak signed coupling, admits a circuit-informed reduction to a signed Sakaguchi–Kuramoto network with α-dependent coupling gain and phase lag. A gauge transformation converts the balanced signed graph into its absolute graph. Exact frequency locking is established for the zero-lag model on a cohesive set, while nonzero phase lag and effective-frequency mismatch lead to explicit practical synchronisation bounds. For identical effective frequencies, the gauge disagreement decays exponentially and produces two π-separated phase groups in the original coordinates. The communication layer uses a reset transmission error together with an event-local fractional-memory term, a minimum dwell guard, and an anti-staleness timeout. A positive inter-event-time bound rules out Zeno behaviour, and a sample-and-hold perturbation analysis shows that disagreement-energy decay is preserved outside an explicit tube. Phase-level numerical experiments illustrate frequency entrainment, cohesive bipartite locking, event-count sensitivity, and node-wise triggering times.