Higher dimensional bifurcations

Here I return to a subject which has been mentioned in this blog on several occasions, bifurcation theory. The general set-up is that we have a dynamical system of the form \dot x=f(x,\mu) where x\in R^m denotes the unknowns and \mu\in R^k stands for parameters. A central aim of the theory is to find conditions under which the system is topologically equivalent to a simple model system. In other words it only differs from the model system by an invertible continuous change of parameters. It is natural to start in the case where m=1 and k=1. By choosing the coordinates appropriately we can focus on the case x=0 and \mu=0. The subject can be explored by going to successively more complicated cases. If f(0,0)=0 and f'(0,0)\ne 0 we have the case that there is no bifurcation. It follows from the implicit function theorem that for parameters close to zero there is exactly one stationary solution close to zero and it has the same character (source or sink) as the stationary solution for \mu=0. The case where f(0,0)=0, f'(0,0)=0 and f''(0,0)\ne 0 is the fold bifurcation, which was discussed in a previous post. In this case these conditions together with the condition f_\mu\ne 0 imply topological equivalence to a standard system. The case where f(0,0)=0, f'(0,0)=0, f''(0,0)=0 and f'''(0,0)\ne 0 is the cusp bifurcation. Here topological equivalence to a standard case does not hold in a context with only one parameter. It can be obtained by passing to the case k=2 and requiring that a suitable combination of derivatives with respect to \mu_1 and \mu_2 does not vanish. I was able to use this to prove the existence of more than one stable stationary solution in a model for the competition between Th1 and Th2 dominance in the immune system, cf. a previous post on this subject. The system of interest in that case is of dimension four and the fact I could obtain results using a system of dimension one resulted from exploiting symmetry properties. Later I was able to prove the existence of more (four) stable stationary solutions of that system using other methods.

It is possible to talk about fold and cusp bifurcations in higher dimensional systems. This is possible in the case that the linearization of the system at the bifurcation point (which is necessarily singular) has a zero eigenvalue with a corresponding eigenspace of dimension one and no other eigenvalues with real part zero. Then the reduction theorem tells us that near the stationary point the dynamical system is topologically equivalent to the product of a standard saddle with the restriction of the system to a one-dimensional centre manifold. This centre manifold is by definition tangent to the eigenspace of the linearization corresponding to the zero eigenvalue of the linearization at the bifurcation point. It is now rather clear what has to be done in order to analyse this type of situation. It is necessary to determine an approximation of sufficiently high order to the centre manifold and to carry out a qualitative analysis of the dynamics on the centre manifold. In practice this leads to cumbersome calculations and so it is worth thinking carefully about how they can best be organized. A method of doing both calculations together in a way which makes them as simple as possible is described in Section 8.7 of Kuznetsov’s book on bifurcation theory. A number of bifurcations, including the fold and the cusp, are treated in detail there. One way of understanding why the example from immunology I mentioned above was relatively easy to handle is that in that case the centre manifold could be written down explicitly. I did not look at the problem in that way at the time but with hindsight it seems to be an explanation why certain things could be done.

In the case of higher dimensional systems the quantities which should vanish or not in order to get a certain type of bifurcation are replaced by more complicated expressions. In the fold or the cusp f''(0,0) is replaced by [L_i({\partial^2 f^i}/{\partial x_j\partial x_k})R^jR^k](0,0) where L and R are left and right eigenvectors of the linearization corresponding to the eigenvalue zero. Naively one might hope that for the cusp f'''(0,0) would be replaced by [L_i{(\partial^3 f^i}/{\partial x_j\partial x_k\partial x_l})R^jR^kR^l](0,0) but unfortunately, as explained in Kuznetsov’s book, this is not the case. There is a extra correction term which involves the second derivatives and which is somewhat inconvenient to calculate. We should be happy that a topological normal form can be obtained at all in these cases. Going more deeply into the landscape of bifurcations reveals cases where this is not possible. An example is the fold-Hopf bifurcation where there is one zero eigenvalue and one pair of non-zero imaginary eigenvalues. There it is possible to get a truncated normal form which is a standard form for the terms of the lowest orders. It is, however, in general the case that adding higher order terms to this gives topologically inequivalent systems. A simple kind of mechanism behind this is the breaking of a heteroclinic orbit. It is also possible that things can happen which are much nore complicated and not completely understood. There is an extended discussion of this in Kuznetsov’s book.


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