Let me now come to the central theme of the book. Harald zur Hausen discovered that a type of viruses causing warts, the human papilloma virus (HPV), also cause the majority of cases of cervical cancer. He was also involved in the development of the vaccine against these viruses which can be seen as the second major cancer vaccine, following the vaccine against hepatitis B. For this work he got a Nobel prize in 2008. He pursued the idea that this class of viruses could cause cervical cancer single-mindedly for a long time while few people believed it could be true. The picture in the book is that while there were a number of people thinking about a viral cause for the disease they were fixated either on herpes viruses or retroviruses. Herpes viruses were popular in this context because the first human virus known to be associated with cancer was the Epstein-Barr virus (EBV) related to Burkitt’s lymphoma and EBV is a herpes virus. Early in his career zur Hausen worked in the laboratory of Werner and Gertrude Henle in Philadelphia. I studied (among other things) zoology in my first year at university and part of that, which appealed to me, was learning about anatomical structures and their names. From that time I remember the ‘loop of Henle’, a structure in the kidney. The Henle of the loop, Jakob Henle, was the grandfather of Werner. As I learned from a footnote in Blumberg’s book, the elder Henle was also the mentor of Robert Koch. Incidentally, Blumberg worked in Philadelphia starting in 1964 while zur Hausen went there in 1966. I did not notice any personal cross references between the two men in their books.

It seems that Gertrude Henle ruled with a strong hand. Once when a laboratory technician was ill for a few days she put on so much pressure that the young woman came into the lab one day just to show how ill she was. She did look convincingly ill and while she was there a blood sample was taken. This turned out to be a stroke of luck. Everyone in the lab had been tested for EBV as part of the research being done there and the technician was one of the few who had tested negative. After her illness she tested positive. In this way it was discovered that glandular fever, the illness she had, is caused by EBV. At that point it is natural to ask why EBV causes a relatively harmless disease in developed countries and cancer in parts of Africa. I have not gone into the background of this but I read that the areas where Burkitt’s lymphoma occurs tend to coincide with areas where malaria is endemic, suggesting a possible connection between the two.

One of the key insights which led to progress in the research on HPV was the recognition that this was not just one virus but a large family of related viruses. Those which turned out to be the biggest cause of cervical cancer are numbers 16 and 18. (After some initial arguments the viruses were named in the order of their discovery.) To obtain this insight it was necessary to have sufficiently good techniques for analysing DNA. The book gives a clear idea of how the progress in understanding in this field was intimately linked to the development of new techniques in molecular biology.

When zur Hausen won the Nobel prize it seemed that the German press and parts of the medical establishment had nothing better to do than to attack him, instead of celebrating his success. From the beginning it was suggested that he only got the prize because a member of the prize committee was on the board of one of the companies producing the vaccine and so would have a personal advantage from the publicity. It was also suggested that the vaccine was ineffective and/or dangerous. (The latter point actually led to a decrease in the number of people getting vaccinated and so, presumably, will mean that in the future many women will get a cancer that could have been prevented.) I do not believe that there was any justification for any of the criticism. So why did it happen? The explanation which occurs to me is the (latent or openly expressed) negative attitudes to science and technology which seem rather widespread in the German press and in German society. I find this surprising for a country which has contributed so much to science and technology and derives so much economic benefit from it.

After finishing the book I decided to try to get a small personal impression of Harald zur Hausen by watching the video of his Nobel lecture. It is untypical for such a lecture in that it contains relatively little about the work the prize was given for and instead concentrates on future research directions. According to the book zur Hausen’s co-laureate Luc Montagnier was suprised by that. The subject is zur Hausen’s lasting theme, the relation between infection and cancer. I found a lot of interesting ideas in it which were new to me. I mention just one. It is well known that there are statistics relating to a possible increase in the incidence of leukemia near nuclear power plants. Whether or not you find this data a convincing argument that there is an increased incidence it is fairly certain that you will link the increase in leukemia in this case (if any) to the effects of radiation. I was no exception to the tendency to make this connection. In his talk zur Hausen says that there are similar statistics showing an increase in leukemia near oil drilling platforms. So how does that fit together? If you cannot think of an answer and you would like to know then watch the video!

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The one modern foreign language I studied at school was French and I liked that a lot. The system was that at the age of sixteen everyone chose between sciences and languages. I could not give up science and so I had to give up French after four years. At least officially. In my fifth and last year of secondary school I used to spend my lunch breaks with two girls, Ingrid and Joy, who were still studying French. Since I had been relatively far advanced I could help them with their homework and this naturally caused me to continue learning some more French. Apart from an intrinsic appreciation for the beauty of the French language which I already had then the association with spending time with two attractive girls certainly increased my interest further. After I went to university I started reading French literature and getting more and more into that. The culmination of this was ‘A la recherche du temps perdu’ and since then Proust has always been the author I appreciate most. Over the years I read the whole novel twice and parts of it more often. I would like to read it again but at some time (a long time ago now) I decided to put that off until my retirement.

At university I was a member of the Creative Writing Group. I wrote some poetry and short pieces of prose but nothing has remained of that. It was a chance to meet interesting people. For certain periods Bernard MacLaverty was writer in residence and part of the duties associated with that was to take part in the Creative Writing Group and give the students advice. I remember him arriving to meet us for the first time with a bottle of Scotch whisky as a present. Among the members of the group were Alison Smith and Alison Lumsden (commonly referred to by us as Ali Smith and Ali Lum). I recently saw that Alison Lumsden has gone back to Aberdeen University (where we studied) as professor of English. As for Ali Smith, she was clearly the most talented writer in the group and later she became a successful novelist. I last saw her quite a few years ago at a reading she gave in Berlin. Perhaps I will write something about my impressions of her novels in a later post. I recently remembered a story associated to another member of the group, Colin Donati. I was once visiting him in his flat in Aberdeen and I found a single loose page of a novel lying on the floor. Of course I was curious to read it and see if I could identify the author. It was not something I had read before but I thought I recognized the style as that of one of my favourite authors. Despite that I would not have been certain if it had not been for one specific subject mentioned on the page which appeared to me conclusive: rooks. These birds occur in several places in the writings of Virginia Woolf (the errant page was from her novel ‘Jacob’s room’), notably in ‘To the Lighthouse’. At the moment I am living in a small furnished flat until our house is built and the final move to Mainz can take place. Near that flat there is a roost of Jackdaws and Rooks and I enjoy hearing them through the open window in the evenings. It occurres to me that I will probably miss those pleasant companions when I move to the house.

These days I do not find much time for reading novels. The last one I can remember reading which I really liked is ‘Ungeduld des Herzens’ by Stefan Zweig. That was about a year ago. Perhaps I should take some time again for reading beyond the confines of science.

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The theorem is proved as follows. The problem is transformed to polar coordinates and then is written as a function of . In this way a non-autonomous scalar equation with -periodic coefficients is obtained and the aim is to find a -periodic solution. The first step is to reformulate the task as a fixed-point problem with the property that if a fixed point is periodic it will be a solution of the original problem. Then it is shown using the Banach fixed point theorem(in a minor variant of the local existence theorem for ODE using Picard iteration) that there always exists a fixed point depending on a certain new parameter. This fixed point is only periodic if the result of substituting it into the right hand side of the original equation has mean value zero. This condition can be written as . Applying the implicit function theorem to shows the existence of a solution of for small. This completes the proof.

Summing up, there are two types of theorem about Hopf bifurcation, a ‘coarse’ theorem of the type just sketched with weak hypotheses and a weak but still very interesting conclusion and a ‘fine’ theorem which gives stronger conclusions but needs a stronger hypothesis (non-vanishing of the Lyapunov coefficient and its sign). In his original paper Hopf proved both types. Are there also ‘rough’ versions of theorems about other bifurcations?

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There was one very interesting subject which was not the subject of a talk at the conference (at least not of one I heard – it was briefly referred to in the talk of Collins mentioned above) but was a subject of conversation. It is a paper called ‘Paradoxical Results in Perturbation-Based Signaling Network Reconstruction’ by Sudhakaran Prabakaran, Jeremy Gunawardena and Eduardo Sontag which appeared in Biophys. J. 106, 2720. It suggests that the ways in which biologists deduce the influence of substances on each other on the basis of experiments are quite problematic. The mathematical content of the paper is rather elementary but its consequences for the way in which theoretical ideas are applied in biology may be considerable. The system studied in the paper is an in vitro reconstruction of part of the MAP kinase cascade and so not so far from some of my research.

Among the parallel sessions those which were most relevant for me were one entitled ‘Algebra in the Life Sciences’ and organized by Elisenda Feliu, Nicolette Meshkat and Carsten Wiuf and one called ‘Developments in the Mathematics of Biochemical Reaction Networks’ organized by Casian Pantea and Maya Mincheva. My talk was in the second of these. These sessions were very valuable for me since they allowed me to meet a considerable number of people working in areas close to my own research interests, including several whose papers were well known to me but whom I had never met. I think that this will bring me to a new level in my work in mathematical biology due to the various interactions which took place. I will not discuss the contents of individual talks here. It is rather the case that what I learned form them will flow into my research effort and hence indirectly influence future posts in this blog. I feel that this conference has gained me entrance into a (for me) new research community which could be the natural habitat for my future research. I am very happy about that. The whole conference was an enjoyable and stimulating experience for me. I noticed no jet lag at all but I must be suffering from a lack of sleep due to the fact that the many things going on here just did not leave me the eight hours of sleep per night I am used to.

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Blumberg did a lot of work with blood coming from people living in unusual or extreme conditions. For this he travelled to exotic places such as Suriname, northern Alaska and remote parts of Nigeria. He seems to have had a great appetite for exciting travel and a corresponding dose of courage. He has plenty of adventures to relate. The second protein he found he names the ‘Australia antigen’ since it was common among aborigines. A good source of antibodies was the blood of people who had had many blood transfusions since their immune systems had been confronted with many antigens. In particular they often carried the Australia antigen.

Pursuing the nature of the Australia antigen led to the realization that it was part of the hepatitis B virus, a virus which causes liver disease and can be spread by blood contact, in particular blood transfusions. The transfusion recipients had become infected with hepatitis B and had produced antibodies to it. Hepatitis B was the first hepatitis virus to be discovered and so why is it labelled ‘B’? In fact people had noticed cases of hepatitis after tranfusions and suspected two viruses, ‘A’ transmitted by contaminated food or water and ‘B’ transmitted by blood contact. There were researchers who had been ‘hunting’ intensively for these viruses and many of them were understandibly not happy when an outsider beat them to it.

For many years Blumberg worked at the Fox Chase Cancer Center in Philadelphia. It was generously funded and the fact that his research had little obvious relation to cancer was not a problem. Once the director of the institute warned that a serious funding cut might be coming. This led Blumberg and colleagues to the idea of developing a vaccine against hepatitis B as a way of making money. Just as Blumberg had not been a virologist when he discovered the virus he was not an expert on vaccines when he developed the vaccine. At that time the need for a vaccine did not seem so urgent since hepatitis B was known as an acute disease which was rarely life-threatening. Later the vaccine acquired a very different significance. There are very many chronic carriers (hundreds of millions worldwide) and a significant proportion of these develop liver cancer after many years. Thus, surprisingly, the hepatitis B vaccine has attained the status of an ‘anti-cancer vaccine’ and has had a huge medical impact.

This book has a very different flavour from the book of Francois Jacob I wrote about in a previous post. Blumberg gives the impression of being a highly cultured person but more than that of an adventurer and man of action. (Along the way he was Master of Balliol College Oxford and director of the NASA Astrobiology Institute.) Jacob also had enough adventures but appears to belong to a more intellectual type, concentrating more on his inner life. In his book Blumberg does not reveal too much which is really personal and always maintains a certain distance to the reader.

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The first plenary talk, by Charles Fefferman, was on a subject related to a topic I was interested in many years ago. I learned that a lot has happened since I last thought about this. The attempt to model a body of fluid with a free surface leads to considerable mathematical difficulties. When I started working on dynamical models for this kind of situation few people seemed to be interested in proving theorems on the subject. The source of my interest in the subject was the influence of Jürgen Ehlers, who always had a clear vision of what were the important problems. In this way I found myself in the position of a pioneer in a certain research area. Being in that situation has the advantage of not being troubled by strong competition. On the other hand it can also mean that whatever you achieve can be largely ignored and it is not the best way to get wide recognition. Often finishing mathematical research directions gets more credit than starting them. This could no doubt be compensated by suitable advertizing but that was never my strong point. This is a configuration which I have often found myself in and in fact, comparing advantages and disadvantages, I do not feel I need to change it. Coming back to the fluids with free surface, this is now a hot topic and played a prominent role at the conference. When I was working on this the issue of local existence in the case of inviscid fluids was still open. A key step was the work of Sijue Wu on water waves. I learned from the talk of Fefferman that this has been extended in the meantime to global existence for small data. The question which is now the focus of interest is formation of singularities (i.e. breakdown of classical solutions) for large data. Instead of considering the breaking of one wave the idea is to consider two waves which are approaching each other while turning over until they meet. There are already analytical results on parts on this process by Fefferman and collaborators and they plan to extend this to a more global picture by using a computer-assisted proof. Another plenary was by Ingrid Daubechies, who talked about applications of image processing to art history. I must admit that beforehand the theme did not appear very attractive to me but in fact the talk was very entertaining and I am glad I went to hear it.

I gave a talk on my recent work with Juliette Hell on the MAPK cascade in a session organized by Bernold Fiedler and Atsushi Mochizuki. I found the session very interesting and the highlight for me was Mochizuki’s talk on his work with Fiedler. The subject is how much information can be obtained about a network of chemical reactions by observing a few nodes, i.e. by observing a few concentrations. What I find particularly interesting are the direct connections to biological problems. Applied to the gene regulatory network of an ascidian (sea squirt) this theoretical approach suggests that the network known from experimental observations is incomplete and motivates searching for the missing links experimentally. Among the many other talks I heard at the conference, one which I found particularly impressive concerned the analysis of successive MRT pictures of patients with metastases in the lung. The speaker was using numerical simulations with these pictures as input to provide the surgeon with indications which of the many lesions present was likely to develop in a dangerous way and should therefore be removed. One point raised in the talk is that it is not really clear what information about the tissue is really contained in an MRT picture and that this could be an interesting mathematical problem in itself. In fact there was an encouragingly (from my point of view) large number of sessions and other individual talks at the conference on subjects related to mathematical biology.

The conference took place on the campus of the Universidad Autonoma somewhat outside the city. A bonus for me was hearing and seeing my first bee-eater for many years. It was quite far away (flying high) but it gave me real pleasure. I was grateful that the temperatures during the week were very moderate, so that I could enjoy walking through the streets of Madrid in the evening without feeling disturbed by heat or excessive sun.

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The signalling network involved in the activation of T cells is very complex but over time I have become increasingly familiar with it. I want to review now some of the typical features to be found in this and related networks. Phosphorylation and dephosphorylation play a very important role. Phosphate groups can be added to or removed from many proteins, replacing (in animals) the hydroxyl groups in the side chains of the amino acids serine, threonine and tyrosine. The enzymes which add and remove these groups are the kinases and phosphatases, respectively. Often the effect of (de-)phosphorylation is to switch the kinase or phosphatase activity of the protein on or off. This kind of process has been studied from a mathematical point of view relatively frequently, with the MAPK cascade being a popular example. Another phenomenon which is controlled by phosphorylation is the binding of one protein to another, for instance via SH2 domains. An example involved in T cell activation is the binding of ZAP-70 to the -chain associated to the T cell receptor. This binding means that certain proteins are brought into proximity with each other and are more likely to interact. Another type of players are linker or adaptor proteins which seem to have the main (or exclusive?) function of organising proteins spatially. One of these I was aware of is LAT (linker of activated T cells). While reading the Itk paper I came across Slp76, which did not strike me as familiar. Another element of signalling pathways is when one protein cleaves another. This is for instance a widespread mechanism in the complement system.

Now back to Itk (IL2-inducible T cell kinase). It is a kinase and belongs to a family called the Tec kinases. Another member of the family which is more prominent medically is Btk, which is important for the function of B cells. Mutations in Btk cause the immunodeficiency disease X-linked agammaglobulinemia. This is the subject of the first chapter of the fascinating book ‘Case studies in Immunology’ by Geha and Notarangelo. As the name suggests this gene is on the X chromosome and correspondingly the disease mainly affects males. In some work I did I looked at the pathway leading to the transcription factor NFAT. However I only looked at the more downstream part of the pathway. This is related to the fact that in experimental work the more upstream part is often bypassed by the use of ionomycin. This substance causes a calcium influx into the cytosol which triggers the lower part of the pathway. In the natural situation the calcium influx is caused by binding to receptors on the endoplasmic reticulum. The comes from the cleavage of by . This I knew before, but what comes before that? In fact is activated through phosphorylation by Itk and Itk is activated through phosphorylation by Lck, a protein I was very familar with due to some of its other effects in T cell activation.

It seems that in knockout mice which lack Itk T cell development is still possible but the immune system is seriously compromised. Effects can be seen in the differentiation of T-helper cells into the types Th1, Th2 and Th17. The problems are less in the case of Th1 responses because Itk can be replaced by another Tec kinase called Rlk. In the case of Th2 responses this does not work and the secretion of the typical Th2 cytokine IL4 is seriuously affected. The Th17 cells are in an intermediate position, with IL17A being affected but IL17F not. Itk also has important effects during the maturation of T cells. Despite the many roles of Itk there are few cases known where mutations in the corresponding genes leads to medical problems in humans. This kind of mutation is a unique opportunity to learn about the role of various substances in humans, where direct experiments are not possible.

In a 2009 paper of Huck et. al. (J. Exp. Med. 119, 1350) the case of two sisters who suffered from serious problems with immunity is described. In particular they had strong infections with Epstein-Barr virus which could not be overcome despite intensive treatment. They also has an excess of B cells. The older sister died at the age of ten. The younger sister was even more severely affected and stem cell transplantation was attempted when she was six years old. Unfortunately she did not survive that. After extensive investigations it was discovered that both sisters were homozygous for the same mutation in the gene for Itk and that was the source of their problems. Their medical history offers clues to what Itk does in humans. The gene is on chromosome 5 and thus it is natural that its mutations are much more rarely discovered than those of Btk. The mutation must occur in both copies of the gene in order to have a serious effect and this can happen just as easily in females as in males.

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More recently I decided to try to finally find out what happens with the Higgins-Selkov oscillator itself. Reading Selkov’s paper I originally had the impression that he had proved the essential properties of the solutions. This turned out to be mistaken. One obstacle for me was that Selkov cited a theorem from a famous Russian textbook of Andronov et. al. and I did not know what the theorem was. An English translation of the book exists in the university library here but since Selkov only cites a page number I did not know how to find the theorem. I was able to get further when Jan Fuhrmann got hold of a copy of the page in question from the Russian original. This page has an easily identifiable picture on it and this allowed me to identify the corresponding page of the English translation and hence the theorem. I found that, as far as it is applicable to the oscillator problem this was something I could prove myself by a simple centre manifold argument. Thus I realized that the results quoted by Selkov only resolve some of the simpler issues in this problem.

At this stage I decided to follow the direction pointed out by Selkov on my own. The first stage, which can be used to obtain information about solutions which tend to infinity, is to do a Poincare compactification. This leads to a dynamical system on a compact subset of Euclidean space. In general it leads to the creation of new stationary points on the boundary which are not always hyperbolic. In this particular example two new stationary points are created. One of these has a one-dimensional centre manifold and it is relatively easy to determine its qualitative nature. This is what Selkov was quoting the result of Andronov et. al. for. The other new stationary solution is more problematic since the linearization of the system at that point is identically zero. More information can be obtained by transforming to polar coordinates about that point. This creates two new stationary points. One is hyperbolic and hence unproblematic. The linearization about the other is identically zero. Passing to polar coordinates about that point creates three new stationary points. One of them is hyperbolic while the other two have one-dimensional centre manifolds. The process comes to an end. When trying this kind of thing in the past I was haunted by the nightmare that the process would never stop. Is there a theorem which forbids that? In any case, in this example it is possible to proceed in this way and determine the qualitative behaviour near all points of the boundary. The problem is that this does not seem to help with the original issue. I see no way in which, even using all this information, it is possible to rule out that every solution except the stationary solution tends to infinity as tends to infinity.

Given that this appeared to be a dead end I decided to try an alternative strategy in order to at least prove that there are some parameter values for which there exists a stable periodic solution. It is possible to do this by showing that a generic supercritical Hopf bifurcation occurs and I went to the trouble of computing the Lyapunov coefficient needed to prove this. I am not sure how much future there is for the Higgins-Selkov oscillator since there are more modern and more complicated models for glycolysis on the market which have been studied more intensively from a mathematical point of view. More information about this can be found in a paper of Kosiuk and Szmolyan, SIAM J. Appl. Dyn. Sys. 10, 1307.

Finally I want to say something about the concept of feedback, something I find very confusing. Often it is said in the literature that oscillations are related to negative feedback. On the other hand the oscillations in glycolysis are often said to result from positive feedback. How can this be consistent? The most transparent definition of feedback I have seen is the one from a paper of Sontag which I discussed in the context of monotone systems. In that sense the feedback in the Higgins-Selkov oscillator is definitely negative. An increase in the concentration of the substrate leads to an increase in the rate of production of the product. An increase in the concentration of the product leads to an increase of the rate of consumption of the substrate. The combination of a positive and a negative sign gives a negative loop. The other way of talking about this seems to be related to the fact that an increase in the concentration of the product leads to an increase in the reaction rate between substrate and product. This is consistent with what was said before. The difference is what aspects of the system are being regarded as cause and effect, which can lead to a different assignment of the labels positive and negative. The problem as I see it is that feedback is frequently invoked but rarely defined, with the implicit suggestion that the definition should be obvious to anyone with an ounce of understanding. I seem to be lacking that ounce.

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The system of equations which is fundamental for this work is that of Huang and Ferrell discussed in my previous post on the subject. I call it the MM-MA system (for Michaelis-Menten via mass action). When this system can be reduced to a smaller system by means of a quasistationary approximation the result will be called the MM system (for Michaelis-Menten) (cf. this post). With a suitable formulation the MM system is a singular limit of the MM-MA system. The MAPK cascade consists of three coupled layers. The first main result of our paper concerns the dual futile cycle, which can be thought of as the second layer of the cascade in isolation (cf. this post). We proved that the MM system for the dual futile cycle exhibits a generic cusp bifurcation and hence that for suitable values of the parameters there exist two different stable stationary solutions (bistability). Using the fact that this system is a singular limit of the system arising from the MM-MA description of the same biological system we then used geometric singular perturbation theory (cf. this post) to conclude that the MM-MA system also shows bistability.

The second main result concerns the system obtained by truncating that of Huang-Ferrell by keeping only the first two layers. It is subtle to find a useful quasistationary approximation for this system and we were put on the right track by a paper of Ventura et. al. (PLoS Comp. Biol. 4(3):e1000041). This allowed us to obtained an MM system which is a limit of the MM-MA system in a way which allows geometric singular perturbation theory to be applied. This leads to the following relative statement: if the MM system for the truncated MAPK cascade has a hyperbolic periodic solution then the same is true for the MM-MA system. To get an absolute statement it remains to prove the existence of periodic solutions of the MM system, which in this case is of dimension three. That there are solutions of this kind is indicated by numerical work of Ventura et. al.

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Consider a dynamical system defined by a smooth vector field on a manifold . Let be a compact subset of the manifold which is invariant under the flow generated by the vector field. The aim here is to define the maximum Lyapunov exponent of a point . The derivative of the flow, is a linear mapping from to . In the Euclidean space picture is treated as a matrix and this matrix is multiplied by its transpose. What is this transpose in an invariant setting? It could be taken to be the mapping from to naturally associated to by duality. The product of the matrices could be associated with the composition of the linear mappings but unfortunately the domains and ranges do not match. To overcome this I introduce a Riemannian metric on a neighbourhood of . It is then necessary to show at the end of the day that the result does not depend on the metric. The key input for this is that since is compact the restrictions of any two metrics and to are uniformly equivalent. In other words, there exists a positive constant such that for all tangent vectors at points of . Once the metric has been chosen it can be used to identify the tangent and cotangent spaces with each other at the points and and thus to compose and its ‘transpose’ to get a linear mapping on the vector space . This vector space does not depend on . The eigenvalues of the mapping are easily shown to be positive. The maximum Lyapunov exponent is the maximum over of the limes superior for of times the logarithm of . Note that the ambiguity of a multiplicative constant in the definition of becomes an ambiguity of an additive constant in the definition of the logarithms and because of the factor this has no effect on the end result.

In general if the maximum Lyapunov exponent at a point is positive this is regarded as a sign of instability of the solution starting at that point (sensitive dependence on initial conditions) and if the exponent is negative this is regarded as a sign of stability. Unfortunately in general these criteria are not reliable, a fact which is known as the Perron effect. This is connected with the question of reducing the study of the asymptotic behaviour of a non-autonomous linear system of ODE to that of the autonomous systems obtained by freezing the coefficients at fixed times.

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