In a previous post I mentioned the Field-Noyes model which is a three-dimensional dynamical system which gives a description of the Belousov-Zhabotinski oscillatory chemical reaction. I want to discuss it here in the context of methods to show the existence or non-existence of periodic solutions of a dynamical system. For two-dimensional systems there are special possibilities due to the availability of Poincaré-Bendixson theory. Here I want to concentrate on what can be done in higher dimensions.
A task which is often simpler than that of finding periodic solutions is that of finding stationary solutions. This reduces to algebra and is in many cases tractable. Another algebraic task is to linearize about the stationary solutions and compute the corresponding eigenvalues. This gives information about the stability of the stationary solutions. Suppose we have a dynamical system with a compact invariant set where the stationary solutions, if any, are all such that no solution can converge to them at late times. Then the -limit set of a non-stationary solution must contain a periodic solution or must be associated with some more complicated kind of attractor. Under the weaker condition that each stationary point is unstable it could be hoped that a similar conclusion will hold for generic solutions. (This might fail in the presence of heteroclinic cycles.) In the Field-Noyes model there are precisely two stationary solutions, one of which is always unstable. The stability of the other depends on the values of the parameters in the system, which are related to reaction rates. When the second is unstable we are in the situation described above. So are there periodic solutions corresponding to the oscillations observed experimentally?
One way of proving the existence of periodic solutions is to use a bifurcation analysis. If a dynamical system depends on parameters and a stationary solution loses stability at some point of parameter space it may be possible to do a local analysis to determine the behaviour of the system for parameter values just beyond the bifurcation. The simplest case of this is a Hopf bifurcation where a pair of eigenvalues passes through the imaginary axis in a sufficiently non-degenerate way. In this case the existence of periodic solutions in a certain (a priori small) region of parameter space follows. This kind of analysis can be done in the case of the Field-Noyes model. This is described in some detail in Murray’s book ‘Mathematical Biology’, where it is also indicated that the experimantally observed oscillations cannot plausibly be thought of as corresponding to parameter values belonging to this small region. There is an interesting further analysis due to Hastings and Murray (SIAM J. Appl. Math. 28, 678). They show that almost all solutions are oscillatory by the following procedure. It is shown that all solutions enter a certain box in state space at late times. This box is the union of eight smaller boxes. It is proved that, apart from some rare exceptions, a given solution must visit each box repeatedly in a certain order at late times. This shows that generic solutions are oscillatory and gives some information about the qualitative properties of the oscillations. In addition it is shown that there is at least one periodic solution for given values of the parameters. This is proved using the Brouwer fixed point theorem.