## The fold-Hopf and Hopf-Hopf bifurcations

Bifurcations of a dynamical system $\dot x=f(x,\alpha)$ can be classified according to their codimension. The intuitive idea is that the set of vector fields exhibiting a given type of bifurcation form a submanifold of the space of all vector fields of the given codimension. Well-known examples of bifurcations of codimension one are the cusp bifurcation, which can already occur when the variables $x$ and $\alpha$ are one-dimensional, and the Hopf bifurcation where the variable $x$ must be two-dimensional but the variable $\alpha$ can be one-dimensional. In fact the minimal dimension of the variable $\alpha$ required corresponds to the codimension of the bifurcation. In this post I want to discuss two bifurcations of codimension 2, the fold-Hopf bifurcation, where $x$ must be at least three-dimensional and the Hopf-Hopf bifurcation where $x$ must be at least four-dimensional

The first typical feature of these bifurcations is the configuration of eigenvalues of $D_x f$ at the bifurcation point. For a fold there is an eigenvalue zero. For a Hopf bifurcation there is a complex conjugate pair of non-zero purely imaginary eigenvalues. For a fold-Hopf bifurcation there is a zero eigenvalue and a complex conjugate pair of non-zero purely imaginary eigenvalues. For a Hopf-Hopf bifurcation there are two complex conjugate pairs of non-zero purely imaginary eigenvalues. The basic hope now is that if some genericity conditions are satisfied the system can be locally reduced to a normal form by a transformation of variables. This is true for the fold and Hopf cases but for fold-Hopf and Hopf-Hopf it is no longer true. A weaker goal which can be attained is to reduce the system to an approximate normal form so that the right hand side is the sum of a simple explicit expression and a higher order error term. The genericity assumptions are as follows. For the cusp the steady state should move with non-zero velocity when the parameter is changed and the steady state of the system for the bifurcation value of the parameter should be as non-degenerate as possible. This means that although $f_x=0$ there (which is the bifurcation condition) $f_{xx}\ne 0$. For the Hopf case the eigenvalues which are on the imaginary axis at the bifurcation value should move off the axis with non-zero velocity when the parameter is changed. At the same time the steady state at the bifurcation value should be as non-degenerate as possible. Solutions close to this steady state circle it and a corresponding Poincare mapping can be defined which describes how the distance from the steady state changes when the solution circles it once. Call this $p(x)$. The fact that there is a bifurcation means that $p'(0)=0$ and $p''(0)$ is automatically zero. The non-degeneracy condition is that $p'''(0)\ne 0$.

Now we come to the fold-Hopf bifurcation. One non-degeneracy condition combines conditions from the two simpler bifurcations in a simple way. It says that the position of the steady state and the real part of the eigenvalue move independently as the two parameters are changed. This is condition ZH0.3 in Theorem 8.7 in the book of Kuznetsov. I was confused by the fact that this condition involves a quantity $\gamma (\alpha)$ which is apparently nowhere defined in the book. It does occur on one other page. On that page there is also a $\Gamma (\alpha)$ which is defined and I think that the solution to the problem is that these two are equal. Assuming that that is correct then $\gamma (\alpha)$ is the projection of the position of the steady state onto the kernel of the linearization at the bifurcation point. The remaining non-degeneracy conditions are conditions on the system at the bifurcation value of the parameter. At the moment I do not have an intuition for the meaning of those conditions.

In the case of the Hopf-Hopf bifurcation a genericity assumption which is qualitatively different from those we have seen up to now is a non-resonance condition, condition HH.0 of Kuznetsov. It says that the two imaginary parts of the eigenvalues at the bifurcation point should not exhibit linear relations with integer coefficients. The next condition is that the real parts of the eigenvalues move independently as the two parameters are changed (HH.5 of Kuznetsov). As in the fold-Hopf case the remaining non-degeneracy conditions are conditions on the system at the bifurcation value of the parameter which I do not understand intuitively.

When analysing the Hopf bifurcation it turns out that after doing a suitable transformation of variables and discarding some terms which can be considered small the resulting system is rotationally invariant. In polar coordinates the angular component is constant and the radial component is cubic in the radius. For the fold-Hopf bifurcation it is natural to proceed as follows. We do a linear transformation so that the new axes belong to the eigenspaces of the linearization. Moreover this transformation is chosen so that the restriction of the linearization to the plane correponding to the complex eigenvalues is in standard form. Then the normal form is rotationally invariant and can be expressed in cylindrical polar coordinates. The component in the angular direction depends only on the coordinate $\xi$ along the axis while the other two components depend only on $\xi$ and the radial coordinate $\rho$. Thus the analysis of the phase portrait of the system near the bifurcation point can be reduced to the analysis of a two-dimensional dynamical system on the half-plane $\rho\ge 0$ called the amplitude system. A steady state of the amplitude system with $\rho=0$ corresponds to a steady state of the full system while a steady state of the amplitude system with $\rho>0$ corresponds to a periodic solution of the full system.

The amplitude system contains a parameter $s=\pm 1$ and a parameter $\theta$. The signs of these two quantities are of crucial importance. If $s=1$ and $\theta>0$ then the system can be reduced to normal form. If $s=-1$ and $\theta<0$ then the situation is still relatively simple but adding a small perturbation typically causes a heteroclinic orbit to break. The most difficult case is that where $s\theta<0$ and it is in that case that chaos may occur. A steady state of the amplitude system away from the axis can undergo a Hopf bifurcation and this corresponds to the occurrence of an invariant torus in the full system and is a Neimark-Sacker bifurcation of the limit cycle. This torus can break up for some value of the parameters and this is what leads to chaos.

In the Hopf-Hopf case the normal form involves two angular coordinates where the dynamics are trivial and two radial coordinates $r_1$ and $r_2$. Thus we again obtain a two-dimensional amplitude system, this time defined on a quadrant. Kuznetsov distinguishes between a ‘simple’ and a ‘difficult’ case according to the parameters but at my present level of understanding they both look very difficult. For Hopf-Hopf the truncated normal form is generically never topologically equivalent to the full system. A Neimark-Sacker bifurcation is always present.

As mentioned in a previous post, both bifurcations discussed here have been observed numerically in an ecological model and Hopf-Hopf bifurcations (but not fold-Hopf bifurcations, this was stated incorrectly in the previous post) in a model for the MAPK cascade.

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