Jumping off a critical manifold

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.


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