  Oscillators have two main limitations: their synchronization properties are
limited (i.e they have a finite synchronization region) and they have no memory
of past interactions (i.e. they return to their intrinsic frequency whenever
the entraining signal disappears). We previously proposed a general mechanism
to transform an oscillator into an adaptive frequency oscillator which adapts
its parameters to learn the frequency of any input signal. The synchronization
region then becomes infinite and the oscillator retains the entrainment
frequency when the driving signal disappears. While this mechanism has been
successfully used in various applications, such as robot control or observer
design for active prosthesis, a formal understanding of its properties is still
missing. In this paper, we study the adaptation mechanism in the case of
strongly coupled phase oscillators and show that non-trivial slow-fast dynamics
is at the origin of the adaptation. We show the existence of a layered
structure of stable and unstable invariant slow manifolds and demonstrate how
the input signal forces the dynamics to jump between these manifolds at regular
intervals, leading to exponential convergence of the frequency adaptation. We
extend the idea to a network of oscillators with amplitude adaptation and show
that the slow invariant manifolds structure persists. Numerical simulations
validate our analysis and extend the discussion to more complex cases.
