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.

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