January | 2022 | Hydrobates

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.

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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.