  The Turing instability is a paradigmatic route to patterns formation in
reaction-diffusion systems. Following a diffusion-driven instability,
homogeneous fixed points can become unstable when subject to external
perturbation. As a consequence, the system evolves towards a stationary,
nonhomogeneous attractor. Stable patterns can be also obtained via oscillation
quenching of an initially synchronous state of diffusively coupled oscillators.
In the literature this is known as the oscillation death phenomenon. Here we
show that oscillation death is nothing but a Turing instability for the first
return map associated to the excitable system in its synchronous periodic
state. In particular we obtain a set of closed conditions for identifying the
domain in the parameters space that yields the instability. This is a natural
generalisation of the original Turing relations, to the case where the
homogeneous solution of the examined system is a periodic function of time. The
obtained framework applies to systems embedded in continuum space, as well as
those defined on a network-like support. The predictive ability of the theory
is tested numerically, using different reaction schemes.
