From Fragmented Basins to Synchrony: Coupling and Nonlinearity in Oscillator Arrays

Abstract We quantify how coupling and cubic nonlinearity organize multistability, hysteresis, and synchronization in forced Duffing oscillator arrays. Although all units are identical and driven in phase, the arrays support a diverse set of coexisting periodic responses whose prevalence depends on the balance between stiffness nonlinearity (k 3) and coupling (k c). We develop a unified workflow that combines time–domain simulation, automated basin classification in high–dimensional state spaces, continuation with basin fractions that measure practical stability along branches, and an enhanced frequency sweep that reconstructs solution branches missed by standard path–dependent scans. The results reveal clear regimes: increasing coupling consolidates fragmented basins and promotes synchronized responses, while stronger nonlinearity preserves layered multistability over wider parameter intervals. The enhanced sweep exposes oscillator–wise hysteresis shifts during upward transitions and shows that many mixed–amplitude patterns collapse through intermediate partially synchronized states in which three or four oscillators occupy the high response before full synchronization. Continuation confirms these branches and quantifies their basin share, and time histories uncover long transients consistent with ghost attractors near lost stability. Together these findings provide a coherent map from (k 3 , k c) and system size to global outcomes and offer design guidance for distributed mechanical systems that rely on controlled multistability, including metamaterials, energy harvesting devices, and logic architectures.