Shilnikov’s theorems on bifurcation from a homoclinic orbit

If is a stationary solution of a dynamical system a homoclinic orbit based at is the image of a solution which converges to both as and . Homoclinic orbits are typically unstable. In other words, if the given dynamical system is embedded into a family depending smoothly on a parameter then generically the qualitative behaviour of solutions for is different from that of the solutions of the original system. The homoclinic loop breaks. Consider first the case of a two-dimensional dynamical system and a hyperbolic stationary solution. Then it can be shown (Andronov-Leontovich theorem) that generically there exist periodic solutions near . Information can also be obtained about the stability of these periodic solutions. One of the genericity assumptions is that the sum of the two eigenvalues of the linearization is non-zero. In higher dimensions generalizations of these phenomena were studied in detail by Leonid Shilnikov in the 1960’s. Here I present some of the ideas involved. My main source for this is chapter 6 of the book ‘Elements of applied bifurcation theory’ by Y. Kuznetzov. Consider a homoclinic orbit based at a hyperbolic stationary solution where all eigenvalues of the linearization are real and of multiplicity one. Suppose that exactly one of them is positive. Label them in decreasing order , , . The eigenvalue is referred to as the leading eigenvalue. Let . It is called the saddle quantity. Under some genericity assumptions, which include it can be shown that a periodic orbit bifurcates from the homoclinic orbit. When the periodic orbit is stable while for positive it is of saddle type. When all eigenvalues are real higher dimensional systems may be reduced to the three-dimensional case by a kind of centre manifold construction.

Kuznetsov’s book discusses applications of the above general constructions to the example of travelling wave solutions of the FitzHugh-Nagumo system mentioned in a previous post. Different types of bifurcations can be used to classify different patterns of firing of neurons. This was already done by Hodgkin and is discussed in the book ‘Dynamical Systems in Neuroscience’ by E. Ishikevich, of which the first chapter is available on the Internet. The bifurcations which play a central role are different from those discussed above. One of them, called SNIC bifurcation by Ishikevich, arises from a homoclinic orbit based at a non-hyperbolic stationary solution and is discussed (under a different name) in Chapter 7 of Kuznetzov. There is a distinction between Class 1 and Class 2 excitability which reminded me of type I and type II critical collapse (see below) due to the similarity of the corresponding diagrams. I doubt if there is a deeper connection – the quantities plotted in the two cases are of quite a different nature.

The type of bifurcations studied by Shilnikov have been implicated in phenomena observed in a field of study in general relativity known as critical collapse. This concerns the gravitational collapse of matter. A typical scenario is as follows. A small amount of matter (‘case of small data’) disperses to infinity since its gravitational field is too weak to hold it together. A very large amount of matter collapses to a black hole since the effect of the gravitational field dominates. What happens for data of intermediate size? This question was investigated intensively using numerical methods by M. Choptuik in the case a self-gravitating scalar field. Since then many authors have investigated many related cases. The results are almost entirely numerical and it would be nice to have some theorems. Choptuik found that there is a transition between the small data and large data behaviour where there is a unique attractor solution giving rise to universal properties of the evolution. There is still no proof of the existence of ‘the Choptuik solution’ but there is strong numerical evidence pointing to an object of this kind. It is discretely self-similar (DSS). Equivalently, it is periodic when expressed in dimensionless variables. With other choices of matter model attractors are encountered which are continuously self-similar (CSS). In terms of dimensionless variables these become stationary solutions. These objects are called critical solutions. Both of these correspond to what is called type II critical collapse. The mass of the black hole formed approaches zero as the threshold between collapse and dispersion is approached. In type I collapse it tends to a non-zero value and the critical solution is stationary in the original variables. For more information see the review article of Gundlach and Martín-García.

There are examples where the existence of CSS critical solutions has been proved, although it has not been proved that they are attractors. In the case of DSS critical solutions, on the other hand, there is not a single existence theorem available. How might a theorem of this kind be proved? One possible idea would be to obtain a DSS solution from a CSS solution by bifurcation in a one-parameter family. The simplest scenario would be a Hopf bifurcation but no example of this kind has yet been found. A more complicated possibility is a bifurcation of the Shilnikov type described above. In work of Aichelburg, Bizon and Tabor (gr-qc/0512136) the authors gave numerical evidence for the occurrence of this type of bifurcation in a model of critical collapse where the matter model is a wave map with values in the three-sphere. Perhaps this could serve as a basis for an existence theorem. There are however major obstacles. It is, for instance, not clear how a rigorous formulation as a dynamical system should be set up. Moreover it is clear that whatever this dynamical system is it must be infinite-dimensional. In the most optimistic case some kind of centre manifold or Lyapunov-Schmidt reduction might be used to get back to a finite-dimensional (even low-dimensional) system. Another problem, and perhaps the most serious one, is that this is a non-local bifurcation which cannot be identified by local calculations at the stationary point.

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