Oscillatory and isochronous rate equations possibly describing chemical reactions

Abstract

We study a simple mathematical model that can be interpreted as a description of the kinetics of the following four reactions involving the two chemicals U and W: (i) U + U double right arrow U with rate alpha, (ii) U + W double right arrow U with rate beta, (iii) W + W double right arrow U with rate gamma and (iv) W + W double right arrow W + W + W with rate delta + 2 gamma. The model can be generally solved by quadratures, and in the special case beta = 2 alpha, explicitly in terms of elementary functions. We focus on the case characterized by the two inequalities gamma beta(2) > alpha delta(2) and 2 beta gamma > delta(2), and we show that in this case the solutions vanish asymptotically at large times. But if a constant decay with rate theta of chemical U is added, then a nonvanishing equilibrium configuration arises. Moreover, for arbitrary strictly positive initial conditions, the solutions remain bounded. They either tend asymptotically (in the remote future) to this nonvanishing equilibrium configuration, or are periodic, or tend to a limit cycle. Indeed, we find that this system goes through a standard supercritical Hopf bifurcation at an appropriate value of the parameters. Another interesting case arises when, in addition to the original reaction, a negative constant term is added to the equation corresponding to chemical U, corresponding to siphoning out a constant amount of chemical U per unit time, independent of its concentration. A very remarkable feature of this (possibly not very realistic) model is the following: in the special case beta = 2 alpha, we again find an explicit solution in terms of elementary functions, which oscillates at a fixed frequency, independent of the initial condition. In other words, it is an isochronous system. If beta not equal 2 alpha, however, no periodic orbits exist, implying that the nature of the bifurcation at beta = 2 alpha is rather peculiar.