  Networks of coupled dynamical units give rise to collective dynamics such as
the synchronization of oscillators or neurons in the brain. The ability of the
network to adapt coupling strengths between units in accordance with their
activity arises naturally in a variety of contexts, including neural plasticity
in the brain, and adds an additional layer of complexity: the dynamics on the
nodes influence the dynamics of the network and vice versa. We study a minimal
model of Kuramoto phase oscillators including a general adaptive learning rule
with three parameters (strength of adaptivity, adaptivity offset, adaptivity
shift), mimicking learning paradigms based on spike-time-dependent-plasticity
(STDP). Importantly, the strength of adaptivity allows to tune the system away
away from the limit of the classical Kuramoto model, corresponding to
stationary coupling strengths and no adaptation, and thus, to systematically
study the impact of adaptivity on the collective dynamics. We carry out a
detailed bifurcation analysis for the minimal model consisting of N = 2
oscillators. The non-adaptive Kuramoto model exhibits very simple dynamic
behavior, drift or frequency-locking; but once the strength of adaptivity
exceeds a critical threshold non-trivial bifurcation structures unravel: A
symmetric adaptation rule results in multi-stability and bifurcation scenarios,
and an asymmetric adaptation rule generates even more intriguing and rich
dynamics, including a period doubling-cascade to chaos as well as oscillations
displaying features of both librations and rotations simultaneously. Generally,
adaptation improves the synchronizability of the oscillators. Finally, we also
numerically investigate a larger system consisting of N = 50 oscillators and
compare the resulting dynamics with the case of N = 2 oscillators.
