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 PoincarĂ©-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.

December 15, 2009 at 4:17 pm |

Is there any reason why such systems will have periodic, instead of almost-periodic solutions? In the integrable case it is trivial, but in general, in a system with two unstable fixed-points, say, would we not expect a generic chaotic behaviour?

That said, I saw the BZ reaction in a chemistry class once. After primed by a semester of The Second Law and Thermodynamic Equilibrium (in a different class, but still fresh on the memory), that was the most counter-intuitive thing.

December 15, 2009 at 10:01 pm |

A good point which I did not think about. It demonstrates how I sometimes

manage to fail to make connections between things I know. I now realized

that the Field-Noyes model is a monotone dynamical system – checking this is a trivial computation. It is discussed in the book ‘Monotone Dynamical Systems’ by Hal Smith. As I mentioned in a previous post, being monotone implies that the dynamics of a given system is no more complicated than a generic system of one dimension less. Since the Field-Noyes model is three-dimensional this means that it is effectively two-dimensional and so there can be no chaos. I suppose that chaos is the usual thing in most dynamical systems and when it is absent there must be a special reason.I feel I would prefer if the world was not like that but it seems that it is.

December 16, 2009 at 8:43 pm |

Thanks for the response.

I took a look at the paper of Hastings and Murray. The proof is rather cute. I guess the monotonicity is also manifest in their proof of the cycling condition among the sub-boxes.

I don’t have too much background in dynamical systems, so another quick question: let us put ourselves in the shoes of Hastings and Murray. Suppose we have a three dimensional dynamical system. We know that it is monotone. We also know that there exists a convex set with smooth boundary (say that’s true. I know they have something slightly weaker) such that two things holds.

(a) The flow on the boundary points inwards. (By Brouwer’s this implies existence of at least one fixed point.)

(b) The fixed point is unique and unstable.

Then is it sufficient to just apply Poincare-Benedixson and say that “if we can show there are no homoclinic trajectories then there must exist an orbit”? Or does that actually require more work. (I am just wondering if there are more abstract/general ways of arriving at Hastings and Murray’s result. Another way of phrasing my question is probably “is it s theorem that Poincare-Benedixson is applicable to three-dimensional monotone systems, or is there a caveat?”)

Thanks in advance for your time.

December 17, 2009 at 8:27 pm

Unfortunately I do not know the answer to this. If I have an idea on it I will let you know.

January 14, 2010 at 11:08 am

Now I went back to the book by Smith. (We do not have it in our library and so I had to go somewhere else to find it.) His Theorem 4.1 is a result of Poincare-Bendixson type for three-dimensional monotone systems which answers your question. It says that a compact limit set which contains no stationary points is a periodic orbit. Later in that chapter he explicitly talks about the Field-Noyes system.

January 10, 2010 at 1:14 pm |

[...] is a kind of oscillator. The Oregonator is nothing other then the Field-Noyes model discussed in a recent post. As mentioned there the Field-Noyes model also exhibits Hopf bifurcations. Hopf bifurcations occur [...]

July 30, 2012 at 8:33 pm |

[...] proof relies on the Brouwer fixed point theorem and is similar to a proof I described in a previous post. Although the Goodwin system is three dimensional the method is not restricted to that case. The [...]