Mixed-mode oscillations in an electrochemical system. I. A Farey sequence which does not occur on a torus

We analyze measurements of an oscillatory current in an electrochemical process in which copper dissolves into phosphoric acid from a rotating-disk electrode. The focus is on a set of states in which each member consists of a different combination of large and small oscillations (mixed-mode oscillations). This set of mixed-mode oscillations is shown to constitute a Farey sequence, i.e., a periodic sequence for which a one-to-one correspondence exists with an ordered sequence of rational numbers. Plots of a measured quantity known as the "firing number" are presented which reveal a structure that is similar to a "devil's staircase." The states surrounding the mixed-mode oscillations are analyzed by examining one-dimensional maps, surfaces of section, and phase portraits constructed from experimental data. This analysis shows that the Farey sequence of these mixed-mode oscillations is of a different nature than the Farey sequences associated with phase locking on a torus.