  Dynamical systems on networks with adaptive couplings appear naturally in
real-world systems such as power grid networks, social networks as well as
neuronal networks. We investigate a paradigmatic system of adaptively coupled
phase oscillators inspired by neuronal networks with synaptic plasticity. One
important behaviour of such systems reveals splitting of the network into
clusters of oscillators with the same frequencies, where different clusters
correspond to different frequencies. Starting from one-cluster solutions we
provide existence criteria for multi-cluster solutions and present their
explicit form. The phases of the oscillators within one cluster can be
organized in different patterns: antipodal, double antipodal, and splay type.
Interestingly, multi-clusters are shown to exist where different clusters
exhibit different patterns. For instance, an antipodal cluster can coexist with
a splay cluster. We also provide stability conditions for one- and
multi-cluster solutions. These conditions, in particular, reveal a high level
of multistability.
