Archive for July, 2021

Codimension three bifurcations

July 13, 2021

In our recent work with Lisa Kreusser on Lck we observed that it can happen that to reach a certain goal it is easier to work with a bifurcation of higher codimension although this means working with an object which is intrinsically more complicated. In that specific case we used a Bogdanov-Takens bifurcation (codimension two) instead of a Hopf bifurcation (codimension one). The goal that we reached was to prove the existence of a periodic solution, which turned out to be unstable. We did not prove the existence of a stable periodic solution of the system being considered. I wondered if this strategy could be pushed further, going to a bifurcation of codimension three.

The general idea is as follows. We have a dynamical system \dot x=f(x,\alpha) depending on parameters. We want to compare it with a normal form which is a standard model dynamical system depending on parameters. Here everything is local in a neighbourhood of (0,0), which is a steady state. The number of parameters in the normal form is the codimension. In the end we want to conclude that dynamical features which are present in the normal form are also present in the original system. The normal form for the Hopf bifurcation contains periodic solutions. In fact there are two versions depending on a parameter which can take the value plus or minus one. These are known as supercritical and subcritical. In the supercritical case the periodic solutions are stable, in the subcritical case unstable. The normal form for the Bogdanov-Takens bifurcation also has two cases which may, by analogy with the Hopf case, be called super- and subcritical. The fact we used is that when a Bogdanov-Takens bifurcation exists there is always a Hopf bifurcation close to it. Moreover the super- or subcritical nature of the bifurcation is inherited. What I wanted to investigate is whether the presence of a suitable bifurcation of codimension three could imply the existence of a Bogdanov-Takens bifurcation close to it and whether it might even be the case that both super- and subcritical Bogdanov-Takens bifurcations are obtained. If this were the case then it would indicate an avenue to a statement of the kind that the existence of a generic bifurcation of codimension three of a certain type implies the existence of stable periodic solutions.

The successful comparison of a given system with the normal form relies on certain genericity conditions being satisfied. These are of two types. The first type is a condition on the system defined by setting \alpha=0. It says that the system does not belong to a certain degenerate set defined by the vanishing of particular functions of the derivatives of f(x,0) at x=0. In an abstract way this degenerate set may be thought of a subset of the set of k-jets of functions at the origin. This subset is a real algebraic variety which is a union of smooth submanifolds (strata) of different dimensions which fit together in a nice way. The set defining the bifurcation itself is also a variety of this type and the degenerate set consists of all strata of the bifurcation set except that of highest dimension. The family f(x,\alpha) induces a mapping from the parameter space into the set of k-jets. The second type of genericity condition says that this mapping should be transverse to the bifurcation set. Because of the genericity condition of the first type the image of x=0 lies in the stratum of highest dimension in the bifurcation set, which is a smooth submanifold. The transversality condition says that the sum of the tangent space to this manifold and the image of the linearization at x=0 of the mapping defined by f is the whole space. Writing about these things gives me a strange feeling since they involve concepts which I used in a very different context in my PhD, which I submitted 34 years ago, and not very often since then.

The following refers to a two-dimensional system. In the Bogdanov-Takens case the bifurcation set is defined by the condition that the linearization about (0,0) has a double zero eigenvalue. The genericity condition of the first type involves the 2-jet. The first part BT.0 (cf. this post for the terminology) says that the linearization should not be identically zero. The second part consists of conditions BT.1 and BT.2 involving the second derivatives. The remaining condition BT.3 is the transversality condition (genericity condition of the second type). Now I want to study bifurcations which satisfy the eigenvalue condition for a Bogdanov-Takens bifurcation and the condition BT.0 but which may fail to satisfy BT.1 or BT.2. At one point the literature on the subject appeared to me like a high wall which I did not have the resolve to try to climb. Fortunately Sebastiaan Janssens pointed me to a paper of Kuznetsov (Int. J. Bif. Chaos, 15, 3535) which was a gate through the wall and gave me access to things on the other side which I use in the following discussion. Following Dumortier et al. (Erg. Th. Dyn. Sys. 7, 375) we look at cases where the 2-jet of the right hand side of the equation for \dot y is of the form \alpha x^2+\beta xy. It turns out that we can divide the problem into two cases, in each of which one of the coefficients \alpha and \beta is zero and the other non-zero. We first concentrate on the case \beta=0 which is that studied in the paper just quoted. There it is referred to as the cusp case. It is the case where BT.2 is satisfied and BT.1 fails. As explained in the book of Dumortier et al. (Bifurcations of planar vector fields. Nilpotent singularites and Abelian integrals.) it is useful to further subdivide the case \alpha=0 into three subcases, known as the saddle, focus and elliptic cases. These cases give rise to different normal forms (partially conjectural). A bifurcation diagram for the cusp case can be found in the paper of Dumortier et al. Since we are dealing with a codimension 3 bifurcation the bifurcation diagram should be three-dimensional. It turns out, however, that the bifurcation set has a conical structure near the origin, so that its structure is determined by its intersection with any small sphere near the origin. This gives rise to a two-dimensional object which can be well represented in the plane. Note, however, that passing from the sphere to the plane necessarily involves discarding a ‘point at infinity’. It is this intersection which is represented in Fig. 3 of the paper of Dumortier et al. That this is an accurate representation of the bifurcation diagram in this case is the main content of the paper of Dumortier et al.

In the bifurcation diagram in the paper of Dumortier et al. we see that there are two Bogdanov-Takens points b_1 and b_2 which are joined by a curve of Hopf bifurcations. The points b_1 and b_2 are stated to be codimension 2 and this indicates that they are non-degenerate Bogdanov-Takens points. There is one exceptional point c_2 on the curve of Hopf bifurcations which is stated to be of codimension 2. This indicates that it is a non-degenerte Bautin bifurcation point. This in turn implies that one of the Bogdanov-Takens points is supercritical and the other subcritical. Thus in this case there exist both stable and unstable periodic solutions in each neighbourhood of the bifurcation point. This indicates that the strategy outlined at the beginning of this post can in principle work. Whether it can work in practise in a given system depends on how difficult it is to check the relevant non-degeneracy conditions. I end with a brief comment on the case \alpha=0. The book of Dumortier et al. presents conjectured bifurcation diagrams for all subcases but does not claim complete proofs that they are all correct. In all cases we have two Bogdanov-Takens points joined by a curve of Hopf bifurcations on thich there is precisely one Bautin point. Thus in this sense we have the same configuration as in the case \beta=0. In a discussion of these matters in the paper of Kuznetsov mentioned above (which is from 2005) it is stated that a complete proof is only available in the saddle case. I do not know if there has been any further progress on this since then.