Archive for August, 2022

Lyapunov-Schmidt reduction and stability

August 30, 2022

I discussed the process of Lyapunov-Schmidt reduction in a previous post. Here I give an extension of that to treat the question of stability. I again follow the book of Golubitsky and Schaeffer. My interest in this question has the following source. Suppose we have a system of ODE \dot y+F(y,\alpha)=0 depending on a parameter \alpha. The equation for steady state solutions is then F(y,\alpha)=0. Sometimes we can eliminate all but one of the variables to obtain an equation g(x,\alpha)=0 for a real variable x whose solutions are in one to one correspondence with those of the original equation for steady states. Clearly this situation is closely related to Lyapunov-Schmidt reduction in the case where the linearization has corank one. Often the reduced equation is much easier to treat that the original one and this can be used to obtain information on the number of steady states of the ODE system. This can be used to study multistationarity in systems of ODE arising as models in molecular biology. In that context we would like more refined information related to multistability. In other words, we would like to know something about the stability of the steady states produced by the reduction process. Stablity is a dynamical property and so it is not a priori clear that it can be investigated by looking at the equation for steady states on its own. Different ODE systems can have the same set of steady states. Note, however, that in the case of hyperbolic steady states the stability of a steady state is determined by the eigenvalues of the linearization of the function F at that point. Golubitsky and Schaeffer prove the following remarkable result. (It seems clear that results of this kind were previously known in some form but I did not yet find an earlier source with a clear statement of this result free from many auxiliary complications.) Suppose that we have a bifurcation point (y_0,\lambda_0) where the linearization of F has a zero eigenvalue of multiplicity one and all other eigenvalues have negative real part. Let x_0 be the corresponding zero of g. The result is that if g(x)=0 and g'(x)\ne 0 then for x close to x_0 the linearization of F about the steady state has a unique eigenvalue close to zero and its sign is the same as that of g'(x). Thus the stability of steady states arising at the bifurcation point is determined by the function g.

I found the proof of this theorem hard to follow. I can understand the individual steps but I feel that I am still missing a global intuition for the strategy used. In this post I describe the proof and present the partial intuition I have for it. Close to the bifurcation point the unique eigenvalue close to zero, call it \mu, is a smooth function of x and \alpha because it is of multiplicity one. The derivative g' is also a smooth function. The aim is to show that they have the same sign. This would be enough to prove the desired stability statement. Suppose that the gradient of g' at x_0 is non-zero. Then the zero set of g is a submanifold in a neighbourhood of x_0. It turns out that \mu vanishes on that manifold. If we could show that the gradient of \mu is non-zero there then it would follow that the sign of \mu off the manifold is determined by that of g'. With suitable sign conventions they are equal and this is the desired conclusion. The statement about the vanishing of \mu is relatively easy to prove. Differentiating the basic equations arising in the Lyapunov-Schmidt reduction shows that the derivative of F applied to the gradient of a function \Omega arising in the reduction process is zero. Thus the derivative of F has a zero eigenvalue and it can only be equal to \mu. For by the continuous dependence of eigenvalues no other eigenvalue can come close to zero in a neighbourhood of the bifurcation point.

After this the argument becomes more complicated since in general the gradients of g' and \mu could be zero. This is got around by introducing a deformation of the original problem depending on an additional parameter \beta and letting \beta tend to zero at the end of the day to recover the original problem. The deformed problem is defined by the function \tilde F(y,\alpha,\beta)=F(y,\alpha)+\beta y. Lyapunov-Schmidt reduction is applied to \tilde F to get a function \tilde g. Let \tilde\mu be the eigenvalue of D\tilde F which is analogous to the eigenvalue \mu of DF. From what was said above it follows that, in a notation which is hopefully clear, \tilde g_x=0 implies \tilde\mu=0. We now want to show that the gradients of these two functions are non-zero. Lyapunov-Schmidt theory includes a formula expressing \tilde g_{x\beta} in terms of F. This formula allows us to prove that \tilde g_{x\beta}(0,0,0)=\langle v_0^*,v_0\rangle>0. Next we turn to the gradient of \tilde \mu, more specifically to the derivative of \tilde \mu with respect to \beta. First it is proved that \tilde\Omega (0,0,\beta)=0 for all \beta. I omit the proof, which is not hard. Differentiating \tilde F and evaluating at the point (0,0,\beta) shows that v_0 is an eigenvalue of D\tilde F there with eigenvalue \beta. Hence \tilde\mu (0,0,\beta)=0 for all \beta. Putting these facts together shows that \tilde\mu (\tilde \Omega,0,\beta)=\beta and the derivative of \tilde\mu (\tilde \Omega,0,\beta) with respect to \beta at the bifurcation point is equal to one.

We now use the following general fact. If f_1 and f_2 are two smooth functions, f_2 vanishes whenever f_1 does and the gradients of both functions are non-zero then f_2/f_1 extends smoothly to the zero set of f_1 and the value of the extension there is given by the ratio of the gradients (which are necessarily proportional to each other). In our example we get \tilde\mu(\tilde\Omega,\alpha,\beta)=\tilde a(x,\alpha,\beta)\tilde g_x(x,\alpha,\beta) with \tilde a(0,0,0)=[\frac{\partial}{\partial\beta}(\tilde\Omega,0,0)]/\tilde g_{x\beta}(0,0,0)]>0. Setting \beta=0 in the first equation then gives the desired conclusion.


Trip to Scotland with some obstacles

August 21, 2022

Recently Eva and I travelled to Scotland. It was an organized tourist trip although I was also able to meet my sister and some other family for dinner on one evening. We flew from Frankfurt to Edinburgh via Brussels. We had heard a lot about chaos at airports in the recent past and were pleasantly surprised when everything went smoothly in Frankfurt. This did not last long. Our flight to Brussels was late and we had to run as fast as we could and jump the queue for immigration to just get to the gate in time for boarding. Our luggage was not so successful as we were. When we arrived in Edinburgh we waited a long time for our luggage to come and it did not. There was no information on the display and the information office was closed. There were no airport employees visible. Eventually we had no choice but to leave the baggage area and file a lost baggage claim. During the trip we were moving from one hotel to another and it was not possible to leave more than one forwarding address. We had to make several calls about this matter in the days that followed and often got the information that the person we were calling was not responsible and that we should call another number. The people I talked to often had strong Indian accents and I suppose that they were all sitting in call centres in India. We arrived in Edinburgh on a Sunday and in the end we got a message the following Saturday that our luggage would be delivered to our hotel in Aberdeen that day at a certain time. This was later revised to say that it would arrive on the Sunday between 00.02 and 02.22. It did arrive and was received by the hotel. Of course we had to buy various things during the week to replace those sitting in our suitcases. This whole business cost us a lot of time and nerves. On the way back we had to cross the EU border in Brussels.The only problem was that there was nobody there, only a written message that it was not possible to call anyone. In the end, fearing that we would miss our flight I spoke to a security guard who was buying a sandwich. He was very helpful. He made several phone calls. Then he took us through a security gate and passed us on to an immigration official who checked our passports. Again we just managed to reach the gate in time. In fact the flight was late so that we would have had a bit more time. The flight was almost empty. In Frankfurt there was a message that the luggage would arrive in thirty minutes. Then thirty suddenly changed to eight and the luggage came even faster than that. Thus positive surprises are also possible.

What conclusions do I draw from this? Firstly, I do not believe that we had specially bad luck but rather that this is the usual state of affairs at the moment. (The luggage of several other members of our group, arriving from different airports with different airlines, also took many days to arrive, in one case even a day longer than ours.) We also experienced a number of other things while in Scotland, such as lifts or coffee machines in hotels which were not working and had been waiting for months to be repaired. Many hotels in Scotland, especially in rural areas, have closed, at least for the season and maybe for ever. For these reasons we had to stay in some cases at hotels much further away from the points we wanted to visit than planned and there were long drives. The situation with logistics is dire. We were not organizing the trip alone. The organization was being done by a company which has many years of experience organizing trips of this kind in Scotland and all over the world. We know from previous experience that this company is very good. Thus things are very difficult even for the experts. If things do not change quickly this type of tourism is threatened. In future I will think very carefully about flying anywhere. This has nothing to do with the frequently discussed environmental issues but simply with the doubt that I will arrive successfully with my luggage and without an excessive amount of stress. If I do fly anywhere then I will be prepared to pay a higher price to get a direct flight. This is then the analogue of my present practise with train trips where I try to minimize the number of connections which have to be reached since the trains cannot be expected to be on time. It seems that these days the most reasonable thing is to expect that everything that can go wrong will go wrong. Travelling has become an adventure again. Will this change soon? I do not expect it will.

We arrived in Edinburgh in the midst of the Festival. The streets were full of people and the atmosphere good. On the evening of the second day we went to the Edinburgh Military Tattoo. The spectacle was impressive as was the way the arrival and departure of the mass of spectators was coordinated. The weather was dry and not too cold and so we were in luck. We heard that at the corresponding performance one week later it rained the whole time. Since almost all the spectators are sitting out in the open the weather makes a big difference. After leaving Edinburgh we crossed into Fife over the old road bridge which was only open for buses due to repairs. We briefly visited St. Andrews where I had not been before and then continued to Pitlochry where we spent a couple of nights. While there we had an excursion to visit the house of Walter Scott. I am not an admirer of Scott. One time years ago I felt the duty to read at least something by him and I read ‘Heart of Midlothian’. It did not leave a lasting impression on me. The main thing I remember about Scott is how the father in ‘To the Lighthouse’ often talks about his novels. From Pitlochry we drove to Braemar and then down Deeside to Aberdeen. We also made an extra little excursion to Dunnottar Castle. I had never been there before although it is so close to Aberdeen where I lived for seven years. The excursions I made from there were generally to the north or to the west. In Aberdeen we had a guided tour from a local which was quite entertaining.We then went into the Machar Bar (a place where I spent many hours as a student), ate stovies (which I had forgotten about for many years) and drank whisky. The guide recited some Burns and we did some singing. Together with him I sang ‘The Northern Lights of Old Aberdeen’. Up to that time we had almost only warm weather and sun (apart from a little coastal fog). After that we crossed over to the west coast. A rear view mirror of our bus was destroyed by another passing bus and this lead to some delays. In the end we took the most direct route from Inverness to Gairloch where our next hotel was. Perhaps we actually profited from the accident since the landscape on that route was spectacular. After the stress with our luggage I felt a great relaxation in Gairloch. The impression it made on me was of somewhere which is really far away and isolated from my usual everyday world. From our hotel room we could see Gannets fishing in the bay and in the night I heard Ringed Plovers calling on the beach. This is nature in the form I appreciate it most. The next day we crossed to Skye. It was rather foggy but what else can be expected from Skye? Our next hotel was in Tyndrum. From there we made a day trip to Iona via Oban and Mull. The general impression of the participants (and it was also my impression) was that the day was too hectic in order to enjoy it properly. For reasons already indicated the hotel was too far away and as a consequence the time was too short. I did not really get a feeling about what it might have been like for St. Columba to arrive on Iona and do what he did there. Another religious figure I would like to know more about after this trip is John Knox. Knowing very little about him I had the feeling that he was a bigot and an extremist. Now I wonder if he might not be responsible for some of most positive aspects of Scottish culture, aspects which I have profited from in my life. We spent our last night in Stirling, where we visited the castle.