Canards

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.