Lyapunov exponents and the Belousov-Zhabotinsky oscillator: an interactive computational approach

Abstract

The Belousov-Zhabotinsky (BZ) chemical oscillator is the most studied oscillator. It has been modelled on the basis of single-step mechanisms which have been continuously refined since the seminal manuscript by Field, Koros and Noyes in 1972. This manuscript reports on a unique way of modelling the global dynamics of the oscillator by assuming that the BZ oscillator has shown chaotic behaviour. The unique mathematical definition of chaos is very stringent, and, in this manuscript, we attempt to trace this unique exotic behaviour by the use of 'onto' maps of the interval onto itself which are known to exhaustively show a universal sequence of states that has all the hallmarks of chaotic behaviour. A series of one-humped maps of the interval display, through iterations and subsequent symbolic dynamics, a universal sequence of steps that commence with period-doubling, culminating in chaotic behaviour at some accumulation point of an appropriate bifurcation parameter. We put this theory to the test for the BZ oscillator in this manuscript by selecting a unique continuous map of the interval. This was then decomposed by an iterative treatment. Metric entropy and subsequent arbiter of chaotic behaviour was determined by evaluation of Lyapunov exponents which were then compared to observed BZ oscillator states. Our proposed map satisfactorily modelled the global dynamics of the BZ oscillator; predicted period-doubling, and a regime after a critical bifurcation parameter, where chaotic sequences were dense. We also produce, in the Addendum, an iterative MatLab procedure that any reader can utilize to reveal the type of states and behaviour reported here. KEYWORDS BZ Oscillator, chemical chaos, one-dimensional maps, maps of the interval, symbolic dynamics, Lyapunov characteristic exponent.