Effect of biharmonic coupling in frequency-weighted Kuramoto oscillators
arXiv:2607.27642
The paper studies how combining frequency‑weighted and biharmonic coupling in globally coupled Kuramoto phase oscillators alters their collective dynamics, revealing multiple stable states and abrupt, hysteretic transitions.
Abstract
We investigate the collective dynamics of globally coupled phase oscillators with frequency-weighted biharmonic coupling. Although frequency-weighted and biharmonic extensions of the Kuramoto model have each been studied independently, their combined effect has remained unexplored. By combining numerical branch continuation with analytical stability analysis, we construct the complete phase diagram in the two-parameter coupling plane. The competition between the first- and second-harmonic interactions gives rise to three distinct collective states: the incoherent state (IC), the antipodal multibranch state (APMS), and the symmetry-broken multibranch state (SBMS). These states are separated by three bistable regions and a narrow tristable region in which all three states coexist as stable attractors. The transitions from the incoherent state to both ordered states are discontinuous and hysteretic, reflecting the explosive synchronization induced by frequency weighting, while the transition between the APMS and SBMS is also abrupt. Analytically, we derive the linear instability thresholds of the incoherent state together with the existence and local stability conditions for the SBMS, and show that the theoretical predictions are in good agreement with extensive numerical simulations.
11 pages, 7 Figures