  Analytically tracking patterns emerging from a small amplitude Turing
instability to large amplitude remains a challenge as no general theory exists.
In this paper, we consider a three component reaction-diffusion system with one
of its components singularly perturbed, this component is known as the fast
variable. We develop an analytical theory describing the periodic patterns
emerging from a Turing instability using geometric singular perturbation
theory. We show analytically that after the initial Turing instability,
spatially periodic patterns evolve into a small amplitude spike in the fast
variable whose amplitude grows as one moves away from onset. This is followed
by a secondary transition where the spike in the fast variable widens, its
periodic pattern develops two sharp transitions between two flat states and the
amplitudes of the other variables grow. The final type of transition we uncover
analytically is where the flat states of the fast variable develop structure in
the periodic pattern. The analysis is illustrated and motivated by a numerical
investigation. We conclude with a preliminary numerical investigation where we
uncover more complicated periodic patterns and snaking-like behaviour that are
driven by the three transitions analysed in this paper. This paper provides a
crucial step towards understanding how periodic patterns transition from a
Turing instability to large amplitude.

Dynamical Systems