  We study low-dimensional dynamics in a Kuramoto model with inertia and
Hebbian learning. In this model, the coupling strength between oscillators
depends on the phase differences between the oscillators and changes according
to a Hebbian learning rule. We analyze the special case of two coupled
oscillators, which yields a five-dimensional dynamical system that decouples
into a two-dimensional longitudinal system and a three-dimensional transverse
system. We readily write an exact solution of the longitudinal system, and we
then focus our attention on the transverse system. We classify the stability of
the transverse system's equilibrium points using linear stability analysis. We
show that the transverse system is dissipative and that all of its trajectories
are eventually confined to a bounded region. We compute Lyapunov exponents to
infer the transverse system's possible limiting behaviors, and we demarcate the
parameter regions of three qualitatively different behaviors. Using insights
from our analysis of the low-dimensional dynamics, we study the original
high-dimensional system in a situation in which we draw the intrinsic
frequencies of the oscillators from Gaussian distributions with different
variances.
