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.

Structure of complex-periodic and chaotic media with spiral waves

“…Similar difficulties arise in the consideration of nonchaotic oscillatory dynamics which is nevertheless more complex than a single loop in phase space; for example, in the oscillations that appear in the period doubling cascade to chaos or in the mixedmode oscillations observed in experiments in chemical systems. [9] In this paper we study the spatiotemporal organization of a reacting medium which supports a single spiral wave and where the local rate law exhibits period-2 n or chaotic oscillations. Through an analysis of the dynamics at different spatial points in the medium we show that a number of phenomena arise for n > 0 which are nonexistent in period-1 oscillatory media.…”

Phase diagram and complex patterns in the modeling of the bromate-oxalic acid-Ce-acetone oscillating reaction

“…The first and the last patterns are obvious daughters of the parent patterns 1 3 and 1 1 but the pattern in Figure 7 has three parents, 1 3 ,1 1 , and 1 2 and can not be understood by a Farey tree but by a Farey triangule 54 . At lower [bromate] 0 the tendency for the predominance of high-amplitude oscillations is observed again with the appearance of the sequence of the patterns 1 1 , 2 1 1 1 , 2 1 ,2 1 (3 1 ) 2 , 4 1 , and 5 1 .…”

“…This pattern, together with the pattern 1 2 , can form the daughter 1 1 (1 2 ) 2 which has the firing number 5/8 = (3+2)/(5+3), which is one of the patterns observed. These combination of patterns can be still understood collecting them in a Farey tree 54 .…”

“…These complex oscillations were shown to exist for the BZ reaction by Mase³ko and Swinney 54 . These complex oscillations combine low and high amplitude oscillations in repetitive sequences which are called mixed-mode oscillations (MMO) which can be rationalized using Farey aritmetic.…”

Resonance tongues and patterns in periodically forced reaction-diffusion systems

“…In all cases the different m:n tongues were ordered in a Farey sequence, similar to the Devil's staircase ordering of resonance tongues for two coupled oscillators [30] and for the homogeneous BZ reaction [31,32].…”