Codimension 3+ bifurcations?

Question

I'm about to begin a study of "Elements of Applied Bifurcation Theory" by Kuznetsov. In going through the chapter headings, I noticed that his exposition stops with codimension 2 bifurcations. Indeed, after googling "codimension 3 bifurcations", there was a scattering of research papers - however, it seemed like there was no large body of theory for codimension 3+ bifurcations (as there is for codim 1 and 2). Why is this? Is the complexity of analysis too difficult and not rewarding enough to develop? Are codimension 3+ bifurcations not actually found naturally/not interesting mathematically?

Answer 1

In my view, you're right to observe that 'standard' bifurcation theory does not really go beyond codimension 2. Kuznetsov's book is a canonical reference on this subject; it's also interesting to look at Scholarpedia entries edited/created by him (see for example the article on the Bogdanov-Takens bifurcation).

There are a couple of reasons why there is no extensive general theory of higher codimension bifurcations. From a practical point of view, as you increase the bifurcation codimension, you increase the number of conditions that have to be fulfilled at the bifurcation point. In other words, the point becomes more and more 'special', and therefore more and more rare. So, in practice, there will be very few specific systems where a bifurcation of relatively high ($\geq 3$) codimension plays a pivotal role in understanding the dynamics of those systems. Also, the higher a codimension of a bifurcation point, the more 'degenerate' it becomes. That is, the existence of such a point will be very sensitive on the inclusion (or exclusion) of highly nonlinear terms in the dynamical system (for example, high degree monomials in the dependent variable). Since (again) from the practical point of view, dynamical systems are generally kept as simple as possible to effectively describe the natural phenomena in question, adding extraneous nonlinear terms (which give rise to the bifurcation of high codimension) goes against 'good practice', or Occam's razor, what you will. Personally, I would be quite suspicious of a high codimension bifurcation in a specific dynamical system, treating it as a sign of bad modelling, rather than something intrinsically interesting.

That being said, from the theoretical point of view, there is no a priori reason to assume that high codimension bifurcations will not occur in a general dynamical system $\dot{x} = f(x)$. I would argue that a general theory of bifurcations does exist, and is called centre manifold theory. Centre manifold theory provides a clear, geometric insight in the local dynamics; bifurcations of high codimension are studied by locally expanding the centre manifold to a sufficiently high order. The procedure, however tedious to carry out in practice, is clear from the abstract point of view.

Lastly, I would say that, if it's clear in principle how to deal with bifurcations of any codimension, the question is how far one should go in painstakingly cataloguing all possible bifurcations of increasing order. It's not like the classification of finite simple groups, where there is a clear incentive to understand all (a finite number of) possible options. Rather, the higher the bifurcation codimension gets, the more tedious the calculations become, the less relevant (practically applicable) it gets, and frankly, the less interesting it gets for potential readers/fellow scientists.