Dynamics of adaptive network with connection delay and external stimulus

We consider an adaptive network of identical Kuramoto oscillators with connection delay and external stimulus. We show that the adaptive network exhibits a rich variety of dynamical states, including antipodal, solitary, double antipodal, splay, multi-cluster, synchronized and force entrained states in the phase diagrams. In particular, we elucidate the delay induced double antipodal state in the adaptive network in the absence of the phase lag parameter. Further, we observe the solitary state in the globally coupled adaptive network for the first time in the literature. We corroborate the dynamical state and elucidate their dynamical transitions using the Kuramoto order parameters and the strength of incoherences. We also show that the network exhibits a rich variety of multistable states as a function of the system parameters. We also elucidate that the distance dependent delay facilitates the manifestation of similar dynamical states as that of the constant delay. Further, we show that the stability condition for the forced entrained state turns out to be the existence condition for the same, which agrees well with the simulation results.