Emergent behaviors of the continuum thermodynamic Kuramoto model in a large coupling regime

We study the emergent dynamics of the continuum thermodynamic Kuramoto model which arises from the continuum limit of the lattice thermodynamic Kuramoto (TK) model [17]. The continuum TK model governs the time-evolution of the Kuramoto phase field in a temperature field, and the solution to the lattice TK model corresponds to the simple function-valued solution to the continuum TK model on a compact spatial region. Asymptotic emergent estimates for the continuum TK model consist of two sequential processes (temperature homogenization and phase-locking). First, we show that the temperature field relaxes to a positive constant temperature exponentially fast pointwise depending on the nature of the communication weight function. In contrast, the emergent dynamics of phase field exhibits a more rich phenomena. For the phase field in a constant natural frequency field, the phase field concentrates to either one-point cluster or bi-polar cluster, whereas in a nonconstant natural frequency field, the phase field exhibits a phased-locked state asymptotically. We also provide several numerical simulations and compare them with analytical results.