Archive for September, 2018

Canards

September 22, 2018

This post follows on from the last one. For the mathematics I want to describe here my main source is the book ‘Multiple Time Scale Dynamics’ by Christian Kuehn. In the last post I discussed the transitions from slow to fast dynamics in a relaxation oscillator. A point where this takes place is a fold point. More insight into the transformations which can be used to analyse the dynamics near a point of this kind can be obtained using geometric singular perturbation theory and this is described in Chapter 7 of Kuehn’s book. The point can be blown up using quasihomogeneous directional blow-ups similar to those which I used in my work with Pia Brechmann on the Selkov oscillator, described here. The main conceptual difference in comparison to our work is that in the case of the fold point there is a small parameter \epsilon involved and it is also rescaled. In this context it is necessary to cover a neighbourhood of the fold point by three charts, in each of which there is a highly non-trivial dynamics. With a suitable analysis of these dynamics it is possible to get information about the transition map from a section before the fold point to one after it. Here the cube roots already seen in the previous post again come up. With this in hand it becomes relatively easy to prove the existence of a relaxation oscillation in the van der Pol system and also that it is stable and hyperbolic. In particular, the existence statement follows from the fact that the return map, obtained by following a solution from a section back to itself around the singular cycle for \epsilon=0 is a contraction. There are other ways of getting the existence result but they rely on special features, in particular the fact that the system is two-dimensional. The proof using GSPT is more powerful since it can be more easily adapted to other situations, such as higher dimensions and it gives more detailed results, such as the expansion for the period. For instance in the book it is explained how this works for a case with one fast and two slow variables.

I have not yet mentioned the concept in the title of this post.(I did once mention briefly it in a recent post.) A canard, apart from being the French word for a duck is an idea in dynamical systems which has intrigued me for a long time but which I understood very little about. With the help of Chapter 8 of Kuehn’s book I have now been able to change this. What I will not do here is to try to explain the origin of the word canard in this context. It has led to a considerable number of humorous contributions of varying quality and I do not want to add to that literature here. I recall that at a fold point a non-degeneracy condition f(0,0,0)\ne 0 holds. Here f(x,y,\epsilon) is the right hand side of the evolution equation for the fast variable. This means that the slow flow does not stand still at the fold point. A canard is obtained if we assume that f(0,0,0)=0 at the fold point while two other non-degeneracy assumptions are made. In this case the fold point is called a fold singularity. This word is used in the sense that sometimes a steady state of a dynamical system is referred to as a singularity. In this case the fold point is steady state of the slow flow. The first non-degeneracy assumption is that  f_y(0,0,0)\ne 0. This means that the steady state is hyperbolic. For the other condition the setting has to be extended by introducing an additional parameter \lambda. Then we have a function f(x,y,\epsilon,\lambda) and it is assumed that f_\lambda(0,0,0,0)\ne 0. In the simplest situation, such as the van der Pol oscillator, the slow dynamics takes place on a critical manifold which is normally hyperbolic and stable. The curious think about  a canard is that there the slow dynamics can follow an unstable critical manifold for a relatively long time before jumping off. More precisely it can remain within distance \epsilon to a repelling part of the slow manifold for a time which is of order one on the slow time scale. Information can be obtained on the dynamics of this type of situation by doing blow-ups. A surprising feature of this type of point is that it is associated with the production of small periodic solutions in a scenario called a singular Hopf bifurcation. Some intuition for this can be obtained by thinking about a periodic solution which starts near the fold singularity, moves a short distance along an unstable branch of the slow manifold (canard), jumps to the stable branch and then returns to its starting point along that branch. A simple example where a canard occurs is the van der Pol system with constant forcing, in other words a system obtained by modifying the basic van der Pol system by introducing an additive constant at an appropriate place on the right hand side.

Jumping off a critical manifold

September 21, 2018

In a previous post I discussed the concept of relaxation oscillations and the classical example, the van der Pol oscillator for large values of the parameter \mu. The periodic orbit consists of two phases of fast motion and two of slow motion.The slow motion is on the critical manifold and in the transition from slow to fast the solution jumps off the critical manifold. This is of course only a heuristic description of what happens when the parameter \epsilon=\mu^{-2} is zero. What we would like to understand is what happens for \epsilon small and positive. Then the non-differentiability related to the jumping off must be smoothed out. To get an expression for the period of the oscillation in leading order (i.e. order zero in \epsilon) it suffices to compute the time taken to complete one of the slow phases. Since the critical manifold is one-dimensional this can be reduced to computing an integral and in the case of the van der Pol oscillator the integral can be computed explicitly. The fast phases make no contribution – in this limit they are instantaneous. In order to get higher order corrections we need to be able to control what happens near the corners where the motion changes from slow to fast. What does the next correction look like? A naive guess would be that it might be an integer power of \epsilon or that at worst this might be corrected by some expression involving \log\epsilon. In reality that corner is really singular and it produces more exotic phenomena. To be specific, it produces integer powers of \epsilon^{1/3}. How can we understand the origin of such terms? The jumping-off point is what is called a fold point since it is analogous to a fold bifurcation. Suppose we have a system \dot y=g(x,y) which undergoes a generic fold bifurcation at the origin. We can extend the system  by adding the trivial equation \dot x=0 for the parameter x. Now we modify this by replacing the equation for x by the equation \dot x=\epsilon f(x,y,\epsilon) with f(0,0,0)\ne 0. Thus for \epsilon=0 we have something which looks like a fold bifurcation for the fast subsystem. For \epsilon non-zero the quantity x, which was constant and could be called a parameter becomes dynamical and starts to move slowly. For this kind of situation there is an approximate  normal form. The system is topologically equivalent to a simple system, up to higher order corrections.

What does all this have to do with the cube roots in the expansion?The remarkable fact is that the normal form (i.e. the leading part of the approximate normal form) can be reduced by means of a rescaling of x, y and t to a system where the parameter \epsilon is eliminated. The price to be payed for this is that the domain in which this can be done is very small for small \epsilon. The magical rescaling is to replace (x,y,t) by (\epsilon^{1/3}X,\epsilon^{2/3}Y,\epsilon^{2/3}T). The model equation which comes out is (up to signs) \frac{dX}{dY}=X^2+Y. This is a Riccati equation whose solutions can be analysed further by classical methods. There are solutions with three different types of behaviour and for one of these three types there is precisely one solution, which is the solution relevant for the relaxation oscillation. It describes the correct smoothing of the corner.

Visit to Aachen

September 13, 2018

In the past few days I was in Aachen. I had only been there once before, which was for a job interview. It was just before Christmas. I only remember that it was dark, that there was a big Christmas market which I had trouble navigating through and that the building of the RWTH where the job was was large and imposing. This time the visit was more relaxed. I was there for a small meeting organized by Sebastian Walcher centred around the topic of singular perturbation theory and with an emphasis on models coming from biology.

I visited the cathedral of which a major part is older than all comparable buildings in northern Europe (eighth century). Even for someone with such rudimentary knowledge of architecture as myself it did appear very different from other cathedrals, with an oriental look. If I had been asked beforehand to imagine what the church of Charlemagne might look like I would never have guessed anything like the reality.

I gave a talk about my work with Juliette Hell about the existence of periodic solutions of the MAP kinase cascade. The emphasis was not only on the proof of this one result but also on explaining some general techniques with the help of this particular example. A theme running through this meeting was how, given a system of ODE depending on parameters, it is possible to find a small parameter \epsilon such that the limit \epsilon\to 0, while singular, nevertheless has some nice properties. In particular the aim is to obtain a limiting system with less equations and less parameters. Then we would like to lift some features of the dynamics of the limiting system to the full system. One talk, which I would like to mention, was by Satya Samal of the Joint Research Centre for Computational Biomedicine on tropical geometry. This is a term which I have often heard without understanding what it means intrinsically or how it can be applied to the study of chemical reaction networks. A lot of what was in the introduction of the talk can also be found in a paper by Radulescu et al. (Math. Modelling of Natural Phenomena, 10, 124). There was one thing in the talk which I found enlightening, which is not in the paper and which I have not seen elsewhere. I will try to explain it now. In this tropical business an algebra is introduced with ‘funny’ definitions of addition and multiplication for real numbers. It seems that often the formal properties of these operations are explained and then the calculations start. This is perhaps satisfactory for people with a strong affinity for algebra. I would like something more, an intuitive explanation of where these things come from. Let us consider multiplication first. If I start with two real numbers a and b, multiply them with \log\epsilon and exponentiate, giving the powers of \epsilon with these exponents, then multiply the results and finally take the logarithm and divide by \log\epsilon I get the sum a+b. This should explain why the tropical equivalent of the product is the sum. Now let us consider the sum. I start with numbers a and b, take powers of a quantity \epsilon with these exponents as before, then sum the results and take the logarithm. Thus we get \log (\epsilon^a+\epsilon^b). Now we need an extra step, namely to take the limit \epsilon\to 0. If b>a then the second term is negligible with respect to the first for \epsilon small and the limit is a\log\epsilon. If a>b then the limit is b\log\epsilon. Thus after dividing by \log\epsilon I get \min\{a,b\}. This quantity is the tropical equivalent of the sum. I expect that this discussion might need some improvement but it does show on some level where the ‘funny’ operations come from. It also has the following intuitive interpretation related to asymptotic expansions. If I have two functions whose leading order contributions in the limit \epsilon\to 0 are given by the powers a and b then the leading order of the product is the power given by the tropical product of a and b while the leading order of the sum is the power given by the tropical sum of a and b.

The technique of matched asymptotic expansions has always seemed to me like black magic or, to use a different metaphor, tight-rope walking without a safety net. At some point I guessed that there might be a connection between (at least some instances of) this technique and geometric singular perturbation theory. I asked Walcher about this and he made an interesting suggestion. He proposed that what is being calculated in leading order is the point at which a solution of a slow-fast system hits the critical manifold (asymptotically) thus providing an initial condition for the slow system. I must try to investigate this idea more carefully.