## The Bogdanov-Takens bifurcation

Consider a system of ODE of the form $\dot x=f(x,\alpha)$ with parameters $\alpha$. A steady state is a solution of $f(x_0,\alpha_0)=0$ and it is a bifurcation point if $J=D_x f(x_0,\alpha_0)$ has at least one eigenvalue on the imaginary axis. A common procedure in bifurcation theory is to start with those cases which are least degenerate. Thus we look at those cases with the fewest eigenvalues on the imaginary axis. If there is only one such eigenvalue, which is then necessarily zero, then I call this a spectral fold point. I use the word ‘spectral’ to indicate that this is a condition which only involves the structure of eigenvalues or eigenvectors. The full definition of a fold point also includes conditions which do not only involve the linearized equation $\dot y=Jy$. If the only imaginary eigenvalues are a non-zero complex conjugate pair then this is a spectral Hopf point. If the only imaginary eigenvalues are two zero eigenvalues and the kernel is only one-dimensional (so that $J$ has a non-trivial Jordan block for the eigenvalue zero) then we have a spectral Bogdanov-Takens point.

When we have picked a class of bifurcations on the basis of the eigenvalues of $J$ the aim is to show that it can be reduced to a standard form by transformations of the unknowns and the parameters. This is often done in two steps. The first is to reduce it to an approximate normal form, which still contains error terms which are higher order in the Taylor expansion about the bifurcation point than the terms retained. The second is to transform further to eliminate the error terms and reduce the system to normal form. Normal forms for the Bogdanov-Takens bifurcation were derived independently by Bogdanov and Takens in the early 1970’s. In this post I follow the presentation in the book of Kuznetsov, which in turn is based on Bogdanov’s approach. In the notation of Kuznetsov the BT bifurcation is defined by four conditions, denoted by BT.0, BT.1, BT.2 and BT.3. The condition BT.0 is the spectral condition we have already discussed. To be able to have a BT bifurcation the number of unknowns must be at least two. The following discussion concerns the two-dimensional case. This is the essential case since higher dimensions can then be treated by using centre manifold theory to reduce them to two dimensions. The BT bifurcation is codimension two which means that in a suitable sense the set of dynamical systems exhibiting this bifurcation is a subset of codimension two in the set of all dynamical system. Another way of saying this is that in order to find BT bifurcations which persist under small perturbations it is necessary to have at least two parameters. For these reasons we consider the case where there are two variables $x$ and two variables $\alpha$. The conditions BT.1, BT.2 and BT.3 are formulated in terms of the derivatives of $f$ of order up to two at the bifurcation point. We choose coordinates so that the bifurcation point is at the origin.

By a linear transformation from $x$ to new variables $y$ the system can be put into the form $\dot y=J_0y+R(y,\alpha)$ where $J_0$ is a Jordan block and the remainder term $R$ is of order two in $y$ and of order one in $\alpha$. After this a sequence of transformations are carried out leading to new unknowns $\eta$, new parameters $\beta$ and a new time coordinate. This eventually leads to the equations $\dot\eta_1=\eta_2$ and $\dot\eta_2=\beta_1+\beta_2\eta_1+\eta_1^2+s\eta_1\eta_2+O(|\eta|^3)$, which is the (approximate) normal form of Bogdanov. Takens introduced a somewhat different normal form. The parameter $s$ is plus or minus one. Because of relations to the Hopf bifurcation I call the case $s=-1$ supercritical and the case $s=1$ subcritical. Let us denote the coefficients in the quadratic contribution to $R$ by $a_{ij}(\alpha)$ and $b_{ij}(\alpha)$. The condition BT.1 is that $a_{20}(0)+b_{11}(0)\ne 0$ and it is required to allow an application of the implicit function theorem. The condition BT.2 is that $b_{20}\ne 0$ and it is required to allow a change of time coordinate. The final transformation involves a rescaling of both the unknowns and the parameters. The existence of the new parameters $\beta$ as a function of the old parameters $\alpha$ is guaranteed by the implicit function theorem and it turns out that the non-degeneracy condition is equivalent to the condition that the derivative of a certain mapping is invertible at the bifurcation point. If the equation is $\dot y=g(y,\alpha)$ then the mapping is $(y,\alpha)\mapsto (g,{\rm tr}D_y g,\det D_y g)$. This condition is BT.3. This equivalence is the subject of a lemma in the book which is not proved there. As far as I can see proving this requires some heavy calculations and I do not have a conceptual explanation as to why this equivalence holds. Carrying out all these steps leads to the approximate normal form. At this point there is still a lot more to be understood about the BT bifurcation. It remains to understand how to convert the approximate normal form to an exact one and how to analyse the qualitative behaviour of the solutions of the system defined by the normal form. I will leave discussion of these things to a later post.