## Hopf bifurcations and Lyapunov-Schmidt theory

In a previous post I wrote something about the existence proof for Hopf bifurcations. Here I want to explain another proof which uses Lyapunov-Schmidt reduction. This is based on the book ‘Singularities and Groups in Bifurcation Theory’ by Golubitsky and Schaeffer. I like it because of the way it puts the theorem of Hopf into a wider context. The starting point is a system of the form $\dot x+f(x,\alpha)=0$. We would like to reformulate the problem in terms of periodic functions on an interval. By a suitable normalization of the problem it can be supposed that the linearized problem at the bifurcation point has periodic solutions with period $2\pi$. Then the periodic solutions whose existence we wish to prove will have period close to $2\pi$. To be able to treat these in a space of functions of period $2\pi$ we do a rescaling using a parameter $\tau$. The rescaled equation is $(1+\tau)\dot x+f(x,\alpha)=0$ where $\tau$ should be thought of as small. A periodic solution of the rescaled equation of period $2\pi$ corresponds to a periodic solution of the original system of period $2\pi/(1+\tau)$. Let $X_1$ be the Banach space of periodic $C^1$ functions on $[0,2\pi]$ with the usual $C^1$ norm and $X_0$ the analogous space of continuous functions. Define a mapping $\Phi:X_1\times{\bf R}\times{\bf R}\to X_0$ by $\Phi(x,\alpha,\tau)=(1+\tau)\dot x+f(x,\alpha)$. Note that it is equivariant under translations of the argument. The periodic solutions we are looking for correspond to zeroes of $\Phi$. Let $A$ be the linearization of $f$ at the origin. The linearization of $\Phi$ at $(0,0,0)$ is $Ly=\frac{dy}{dt}+Ay$. The operator $L$ is a (bounded) Fredholm operator of index zero.

Golubitsky and Schaeffer prove the following facts about this operator. The dimension of its kernel is two. It has a basis such that the equivariant action already mentioned acts on it by rotation. $X_0$ has an invariant splitting as the direct sum of the kernel and range of $L$. Moreover $X_1$ has splitting as the sum of the kernel of $L$ and the intersection $M$ of the range of $L$ with $X_1$. These observations provide us with the type of splitting used in Lyapunov-Schmidt reduction. We obtain a reduced mapping $\phi:{\rm ker}L\times{\bf R}\times{\bf R}\to {\rm ker}L$ and it is equivariant with respect to the action by translations. The special basis already mentioned allows this mapping to be written in a concrete form as a mapping ${\bf R}^2\times{\bf R}\times{\bf R}\to {\bf R}^2$. It can be written as a linear combination of the vectors with components $[x,y]$ and $[-y,x]$ where the coefficients are of the form $p(x^2+y^2,\alpha,\tau)$ and $q(x^2+y^2,\alpha,\tau)$. These functions satisfy the conditions $p(0,0,0)=0$, $q(0,0,0)=0$, $p_\tau(0,0,0)=0$, $q_\tau(0,0,0)=-1$. It follows that $\phi$ is only zero if either $x=y=0$ or $p=q=0$. The first case corresponds to the steady state solution at the bifurcation point. The second case corresponds to $2\pi$-periodic solutions which are non-constant if $z=x^2+y^2>0$. By a rotation we can reduce to the case $y=0, x\ge 0$. Then the two cases correspond to $x=0$ and $p(x^2,\alpha,\tau)=q(x^2,\alpha,\tau)=0$. The equation $q(x^2,\alpha,\tau)=0$ can be solved in the form $\tau=\tau (x^2,\alpha)$, which follows from the implicit function theorem. Let $r(z,\alpha)=p(z,\tau(z,\alpha))$ and $g(x,\alpha)=r(x^2,\alpha)x$. Then $\phi(x,y,\tau,\alpha)=0$ has solutions with $x^2+y^2>0$ only if $\tau=\tau(x^2+y^2,\alpha)$. All zeroes of $\phi$ can be obtained from zeroes of $g$. This means that we have reduced the search for periodic solutions to the search for zeroes of the function $g$ or for those of the function $r$.

If the derivative $r_\alpha(0,0)$ is non-zero then it follows from the implicit function theorem that we can write $\alpha=\mu(x^2)$ and there is a one-parameter family of solutions. There are two eigenvalues of $D_xf$ of the form $\sigma(\alpha)-i\omega(\alpha)$ where $\sigma$ and $\omega$ are smooth, $\sigma (0)=0$ and $\omega (0)=1$. It turns out that $r_\alpha (0,0)=\sigma_\alpha(0,0)$ which provides the link between the central hypothesis of the theorem of Hopf and the hypothesis needed to apply the implicit function theorem in this situation. The equation $r=0$ is formally identical to that for a pitchfork bifurcation, i.e. a cusp bifurcation with reflection symmetry. The second non-degeneracy condition is $r_z(0,0)=0$. It is related to the non-vanishing of the first Lyapunov number.