Circle maps and the devil's staircase in a periodically perturbed oregonator

Markman and Bar-Eli has studied a periodically forced Oregonator numerically and found a parameter range with the following properties: (1) Only periodic solutions are found in frequency-locked steps, each with a certain pattern of large and small oscillations (2) Between any two steps there is a step with the period being the sum of the two periods and the concatenation of the two patterns (3) Certain scaling properties as the period tends to infinity. We show that such behavior occurs if the dynamics of the system is governed by a family of diffeomorphisms of a circle with a Devil's staircase. Using invariant manifold theory we argue that an invariant circle must indeed exist when, as in the present case, the unforced system is close to a saddle-loop bifurcation. Generalizations of the results are briefly discussed.