Emergent dynamics of the Kuramoto model with adaptive coupling on undirected networks

We study the emergent dynamics for the Kuramoto model with adaptive and local couplings. With conditions satisfied by network topology, sufficient frameworks for the complete synchronization and phase-locking estimates are established in terms of initial configurations and system parameters. For a homogeneous ensemble with Hebbian adaptive coupling, we demonstrate that complete phase synchronization occurs exponentially on connected symmetric networks for initial phase confined in a quarter circle. When initial phase diameters exceed $\frac{\pi }{2}$, synchronization is achieved under stricter scrambling undirected networks and admissible coupling strength. Moreover, complete frequency synchronization is guaranteed unconditionally. For a homogeneous ensemble with anti-Hebbian adaptive coupling, we prove that the complete phase synchronization emerges on connected symmetric network when initial configurations are located on the same semicircle. For a heterogeneous ensemble with Hebbian adaptive coupling, we establish the emergence of phase-locked state under two frameworks: scrambling undirected networks with initial phase diameter below $\frac{\pi }{2}$ and admissible coupling strength, and connected undirected networks with restricted initial phase configuration. Both ensure complete frequency synchronization and convergence to an equilibrium. Moreover, a practical phase synchronization is proved on connected symmetric network with anti-Hebbian adaptive coupling when initial configurations are located on the same semicircle. Finally, numerical simulations are provided to demonstrate our theoretical results.