  Adaptive coupling, where the coupling is dynamical and depends on the
behaviour of the oscillators in a complex system, is one of the most crucial
factors to control the dynamics and streamline various processes in complex
networks. In this paper, we have demonstrated the occurrence of multi-stable
states in a system of identical phase oscillators that are dynamically coupled.
We find that the multi-stable state is comprised of a two cluster
synchronization state where the clusters are in anti-phase relationship with
each other and a desynchronization state. We also find that the phase
relationship between the oscillators is asymptotically stable irrespective of
whether there is synchronization or desynchronization in the system. The time
scale of the coupling affects the size of the clusters in the two cluster
state. We also investigate the effect of both the coupling asymmetry and
plasticity asymmetry on the multi-stable states. In the absence of coupling
asymmetry, increasing the plasticity asymmetry causes the system to go from a
two clustered state to a desynchronization state and then to a two clustered
state. Further, the coupling asymmetry, if present, also affects this
transition. We also analytically find the occurrence of the above mentioned
multi-stable-desynchronization-multi-stable state transition. A brief
discussion on the phase evolution of nonidentical oscillators is also provided.
Our analytical results are in good agreement with our numerical observations.
