Complex periodic oscillations and Farey arithmetic in the Belousov–Zhabotinskii reaction

Abstract

Our experiments on the manganese-catalyzed Belousov–Zhabotinskii reaction in a stirred flow reactor reveal many sequences of distinct multipeaked periodic states. In the parameter ranges studied the waveform for each periodic state consists of an admixture of small and large amplitude oscillations. No chaos is discernible, and in many cases the transitions from one periodic state to another occur without any observable hysteresis. Two types of sequences were studied in detail, one with waveforms consisting of concatenations of two basic patterns and another with waveforms consisting of concatenations of three basic patterns. The sequences of states with two patterns are described well by Farey arithmetic, which provides rational approximations of irrational numbers. These states can be characterized by a firing number, the ratio of the number of small amplitude oscillations to the total number of oscillations per period. For our data this ratio is a monotone stepwise-increasing function of flow rate, and the steps have a fractal dimension. The relationship between the observed sequence and the Farey arithmetic and the observation of a fractal dimension for the steps in the firing number suggest that the states formed by concatenating two patterns can be interpreted in terms of frequency locking on a 2 torus in phase space. The sequences of states with three basic patterns are described by a generalized Farey arithmetic that provides rational approximations for pairs of irrational numbers that are mutually irrational; this suggests that these states can be interpreted in terms of frequency locking on a 3 torus. A piecewise-linear two-dimensional map is shown to yield a phase diagram in qualitative accord with the measured phase diagram for these sequences.