A simple system with two different timescales

While struggling with the proof of a result for a model for photosynthesis (which I intend to report on in more detail at a later date) I decided to apply the following principle which I heard about as an undergraduate and is apparently due to Paul Halmos: if there is a mathematical problem you cannot solve then there is also a simpler problem you cannot solve. With this in mind I developed a model problem for what I wanted to do which I still could not solve. In the meantime, with the help of some key input from Juan Velázquez, I can solve the model problem and I will report on that here. It seems that the right conceptual framework for this is that of systems with two different timescales. Consider two functions x(t) and y(t) which satisfy \dot x=x and \dot y=x-y^5. The fact that the power here is exactly five is not so important. This was the power that came up in the problem I was originally interested in and I wanted to rule out any misleading simplifications which might have arisen by replacing it by the power two, for instance. The first equation can be solved explicitly in the form x(t)=\alpha e^t.This reduces the original system to the scalar, but non-autonomous, equation \dot y=\alpha e^t-y^5. The question of interest is whether this equation has solutions which tend to infinity as t\to\infty and if so how many of these are there. A guess at the asymptotics of a solution of this kind which is formally consistent is y=\alpha^{\frac15}e^{\frac15 t}+\ldots. It is possible to take this further by looking for formal power series solutions where y-\alpha^{\frac15}e^{\frac15 t} is a linear combination of integer powers of e^{\frac15 t}. It turns out that there is a solution of this type in the sense of formal power series. In other words, substituting the expression in the equations and comparing coefficients gives a consistent answer. The coefficients are determined uniquely. This means that if there is more than one solution with this type of asymptotics these coefficients cannot distinguish between them and hence cannot be used to parametrize them. In fact there is a one-parameter family of solutions having this type of asymptotics and the difference between any two of these is of order \exp [-Ce^{\frac45 t}] for a constant C. This means that while these solutions decay on a timescale t the difference between them decays on a timescale which is exponentially faster. I am reminded of the term ‘asymptotics beyond all orders’ which I have heard occasionally but I do not know exactly what that means.

How can these results be proved? First introduce a quantity w=1-\frac{y^5}{x} since this can be expected to decay very fast. The evolution equation for y can be rewritten as an evolution equation for w of the form \dot w+5\alpha^{\frac45}e^{\frac45 t}w=q, where the quantity q will eventually be small. This equation can be solved by variation of constants to give an integral equation for w. It contains a parameter \eta_0 which distinguishes the different solutions. The problem of solving the integral equation can be reformulated as a fixed point problem for a mapping \Phi. The key step is to show that, for \alpha sufficiently large and \eta_0 sufficiently small, \Phi maps a suitable set of functions with a certain decay property to itself. The fixed point can then be obtained using the Arzela-Ascoli theorem. The simple system considered here can presumably be treated by simpler methods. My motivation for discussing it here that the technique of proof is of much wider applicability and the conclusions obtained are a model for other problems.

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One Response to “A simple system with two different timescales”

  1. Modelling the Calvin cycle | Hydrobates Says:

    [...] research on these systems myself. Some technical difficulties which arose (which I wrote about in a previous post) led me to consult Juan Velázquez and he joined me in this project. Now we have written a paper on [...]

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