Archive for January, 2022

Looking for a good read

January 9, 2022

In the recent past my reading has mostly been limited to mathematics and other scientific subjects. I have harldly found time to read literature. I have also tended to read only authors I already knew I liked. A few weeks ago I was in a second hand book shop and tried to do something against these tendencies. One of the things I did was to buy a book called ’50 Great Short Stories’ edited by Milton Crane. My motivation was less a desire to read short stories than to get to know authors I might like to read more of. Of course it was clear to me that reading a short story by an author may not give a very useful impression of what a novel by that author might be like. I was disappointed to find that there were few of the short stories in the book I liked very much. Those I did not appreciate much included ones by a number of authors who have written books I like very much, for instance E. M. Forster, Henry James, Guy de Maupassant, James Joyce, and Virginia Woolf. I long time ago I was keen on Aldous Huxley and read most of his books. This collection contains a short story by him ‘The Gioconda Smile’, which I am sure I had read before. I was not very enthusiastic about it this time but at least I did find it good. In the end there were only two stories in the collection which were by authors I had not previously read and which I liked enough so as to want to read more. The first is ‘The Other Two’ by Edith Wharton and I found the psychological subtlety of the writing attractive. I read a little about the author, who I found out has often been compared with Henry James. My superficial reading on this subject indicates to me that Wharton might have some of those qualities of James which I like while lacking some of those I dislike. Thus I am now motivated to read more of Wharton. The other story which made a very positive impression on me was ‘A Good Man is Hard to Find’ by Flannery O’Connor. I find it difficult to pin down what it was that I liked so much about it. In fact I think both aspects of the form and content were involved. I think a part of it was the impression of reading ‘something very different’. Here again I want to read more.

The other book I bought on that day was a collection of writings of Jean Paul, ‘Die wunderbare Gesellschaft in der Neujahrsnacht’. Unfortunately I found it completely inaccessible for me and I only read a small part of it.


Hopf bifurcations and Lyapunov-Schmidt theory

January 1, 2022

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.