Archive for December, 2020

The Bogdanov-Takens bifurcation, part 2

December 30, 2020

This post continues the discussion in the previous post on this subject. It turns out that the (exact) normal form is a system whose dynamics are not easy to analyse. The bifurcation diagram can be found as Figure 8.8 in the book of Kuznetsov. The case presented there is the supercritical one. The diagram can easily be converted into that describing the subcritical case, the most important thing being that the time direction is reversed so that the types of stability are interchanged. The bifurcation point is the origin in the plane of the parameters $(\beta_1,\beta_2)$. The plane can be written as the union of four open regions $R_i$ separated by bifurcation curves. These regions are distinguished by the types of dynamical behaviour which occur for the given values of the parameters. $R_1$ is simplest, having a trivial dynamics, without steady states or periodic solutions. The boundary of $R_1$ is a smooth curve $T$ passing through the origin. If the origin is deleted the remainder consists of two connected components $T_-$ and $T_+$. The curve $T_-$ is the boundary between $R_1$ and $R_2$. There a fold bifurcation takes place, producing a saddle point and a sink. The region $R_2$ is that between $T_-$ and the negative part of $\beta_2$ axis, which we call $H$. On that axis a Hopf bifurcation takes place. The sink turns into a source and a stable periodic solution is born. This is a supercritical Hopf bifurcation. For the other sign of the parameter $s$ there is an unstable periodic solution and the Hopf bifurcation is subcritical. This is the reason that we have also adopted the terms sub- and supercritical in the case of a BT bifurcation. (Note that the sign of $s$ can be computed from the parameters $a_{ij}$ and $b_{ij}$.) On the other side of $H$ we come to $R_3$. In that region there is a saddle point and the periodic solution already mentioned. It is the unique periodic solution for given values of the parameters. The other boundary of $R_3$ is a curve $C$ where a non-local bifurcation takes place. On the curve there is an orbit which is homoclinic to the saddle. On the other side of $C$ is $R_4$ where the homoclinic loop has broken and the qualitative behaviour is similar to that in $R_2$ except that there is a source instead of a sink. From $R_4$ we can pass through $T_+$ and return to $R_1$ by means of a fold bifurcation. It is perhaps worth mentioning what happens on the upper half of the $\beta_2$ axis. There is no bifurcation there but the two eigenvalues of the linearization are equal and opposite (a neutral saddle). This type of situation may be dangerous when looking for Hopf bifurcations because it shares certain algebraic properties with a Hopf point without being one. For this reason it could make sense to call it a ‘false Hopf point’, which is my internal name for it.

The proofs of the statements about the homoclinic orbit and the periodic solution are hard and I will just make a few comments on that. There is a subtle rescaling which, in particular, ensures that the two steady states are a fixed distance apart, independent of the parameters. Doing this transformation brings the system into the form of a perturbation of a Hamiltonian system. In the latter case the homoclinic orbit is obtained as a level surface of the Hamiltonian. This perturbation can be analysed by a method due to Pontryagin. Use is made of elliptic functions. These things are sketched in some detail by Kuznetsov in an appendix. He also mentions an alternative method for proving the uniqueness of the periodic orbit due to Dumortier and Rousseau. Concerning the fact that the approximate normal form can be transformed to the exact normal form Kuznetsov only gives a very brief sketch of the proof. Combining what is known about the qualitative behaviour of the solutions in normal form and the fact that any system satisfying the Bogdanov-Takens conditions can be reduced to this normal form provides a method for proving the existence and stability of periodic solutions and the existence of a homoclinic loop in given dynamical systems. The genericity conditions can be easier to check than for a Hopf bifurcation since they are conditions on the second derivatives rather than on the third derivatives. How can it be that it is easier to analyse a more complicated bifurcation than a simpler one? The point is that the hardest work is hidden in the proofs of the theorems about the normal form and does not have to be repeated when analysing a concrete system. The use of a BT bifurcation to help prove the existence of a Hopf bifurcation is connected to the idea of an organizing centre. The idea is to obtain insight into a dynamical system by looking for points with extremely special properties. In a given system a point like this of a given type may not exist but when it exists it may be easier to find than a point with a lower degree of speciality, even if the latter occurs more commonly. For instance we can look for a BT point instead of looking for a Hopf point. This type of strategy may be especially useful in systems where there are many parameters over which there is a lot of control. It gives a way of focussing the search in the parameter space. It is the opposite of the situation where you feel that you are looking for something you expect to be plentiful but do not know where to start.

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.