## Degree theory

January 26, 2020

Degree theory is a part of mathematics which I have had little to do with up to now. The one result related to degree theory which has come up repeatedly in connection with my research interests in the past is the Brouwer fixed point theorem. I had an early contact with it through Hirsch’s book ‘Differential Topology’, which I mentioned in a previous post. One formulation of the theorem is that any continuous mapping from the closed unit ball centred at the origin in $n$-dimensional Euclidean space to itself has a fixed point. The statement is obviously topologically invariant and so we can replace the closed ball by any topological space homeomorphic to it. Since any closed bounded convex subset of Euclidean space is homeomorphic to a closed ball we get an apparently more general formulation concerning convex sets. For $n=1$ the theorem is easily proved using the intermediate value theorem. I find that it is already not intuitive for $n=2$. There are various different proofs.

A step towards one type of proof is as follows. Let $B$ be the open unit ball in $R^n$. If the mapping is called $f$ consider for any point $x$ of the closed unit ball $\bar B$ the straight line joining $x$ and $f(x)$. Extend it in the direction beyond $x$ until it meets the boundary $\partial B$ and call the intersection point $\phi(x)$. Then $\phi$ is continuous, it maps $\bar B$ onto $\partial B$ and its restriction to $\partial B$ is the identity. A map of this type is called a retraction. Thus to prove the Brouwer fixed point theorem it is enough the prove the ‘no retraction theorem’, i.e. that there is no retraction of the unit ball onto the unit sphere. My aim here is not to present a proof of the theorem which is as simple is possible but instead to use one proof of it (the one given in Smoller’s book) to try and throw some more light on degree theory.

The degree is an integer $d(f,B,y_0)$ which is associated to a continuous mapping $f:\bar B\to R^n$ and a point $y_0\in R^n\setminus f(\partial B)$. The proof of the no retraction theorem consists of putting together the following three statements. The first is that (i) since $y_0$ does not belong to the image of $\partial B$ we have $d(f,B,y_0)=d(I,B,y_0)$, where $I$ is the identity mapping. The second is that (ii) $d(I,B,y_0)=1$. The third is that (iii) the statement (ii) implies that $d(f,B,y_0)=1$ and $y_0\in f(B)$, a contradiction. It remains to consider how (i)-(iii) are proved. Central properties of the degree is that it varies continuously under certain types of deformations and that it is an integer. These two things together show that it is left unchanged by these deformations. The proof of (i) is as follows. The mappings $f$ and $I$ agree on $\partial D$ and $y_0$ does not belong to the image of $\partial D$ under either of them. It is possible to join these two mappings by a homotopy given by $tf+(1-t)I$. The degree at a given point is left unchanged by a homotopy whose image does not meet that point (iv). This property holds in the present case. Thus (i) follows from (iv). In the proof of (iv) in Smoller’s book it is assumed that the restriction of the homotopy to each fixed value of $t$ is $C^1$. Thus there is a gap in the argument at this point. In the book it is filled by Theorem 12.7 which says that various properties of the degree proved for $C^1$ functions also hold for continuous functions. The differentiability is used in the proof of (iv) via the fact that the degree can be expressed as the integral of the pull-back of a suitable differential form under the mapping. Property (ii) follows from the fact that at a regular point $y_0$ the degree is equal to an expression computed from the determinant of the linearization of the mapping at the inverse images of $y_0$. In the case of the identity mapping every point is regular and the computation is trivial. When $y_0\notin f(B)$ it follows that $y_0$ is regular and again the computation is trivial and gives the result zero. This completes the proof of (iii).

## Sard’s theorem

December 27, 2019

I have recently been reading Smoller’s book ‘Shock Waves and Reaction-Diffusion Equations’ as background for a course on reaction-diffusion equations I am giving. In this context I came across the subject of Sard’s theorem. This is a result which I had more or less forgotten about although I was very familiar with it while writing my PhD thesis more than thirty years ago. I read about it at that time in Hirsch’s book ‘Differential Topology’, which was an important reference for my thesis. Now I had the idea that this might be something which could be useful for my present research, without having an explicit application in mind. It is a technique which has a different flavour from those I usually apply. The theorem concerns a (sufficiently) smooth mapping between $n$-dimensional manifolds. It is a local result so that it is a enough to concentrate on the case where the domain of the mapping is a suitable subset of Euclidean space and the range is the same space. We define a regular value of $f$ to be a point $y$ such that the derivative of $f$ is invertible at each point $x$ with $f(x)=y$. A singular value is a point of the range which is not a regular value. The statement of Sard’s theorem is that the set $Z$ of singular values has measure zero. By covering the domain with a countable family of cubes we can reduce the proof to the case of a cube. Next we write the cube as the union of $N^n$ cubes, by dividing each side of the original cube into $N$ equal parts. We need to estimate the contribution to the measure of $Z$  from each of the small cubes. Suppose that $y_0$ is a singular value, so that there is a point $x_0$ where the derivative of $f$ is not invertible with $f(x_0)=y_0$. Consider now the contribution to the measure of the image from the cube in which $x_0$ lies. On that cube $f$ can be approximated by its first order Taylor polynomial at $x_0$. The image is contained in the product of a subset of a hyperplane whose volume is of the order $N^{-(n-1)}$ and an interval whose length is of the order $\epsilon N^{-1}$ for an $\epsilon$ which we can choose as small as desired. Adding over the at most $N^n$ cubes which contribute gives a bound for the measure of the set of singular values of order $\epsilon$. Since $\epsilon$ was arbitrary this completes the proof. In words we can describe this argument as follows. The volume of the image of a region which intersects the set of singular points under a suitable linear mapping is small compared to the volume of the region itself and the volume of the image under the nonlinear mapping can be bounded by the corresponding quantity for the linear mapping up to an error which is small compared to the volume defined by the linear mapping.

## Ernst Jünger

December 25, 2019

The book of Jünger which ignited my enthusiasm for his writing is ‘Afrikanische Spiele’. This is a work of fiction but it is based rather closely on Jünger’s own experiences. He was often bored in school and preferred to read adventure stories. For him Africa was the land of adventure and he wanted to go there. He ran away from school and travelled to Verdun, where he enlisted in the Foreign Legion. He was then stationed in Algeria. This was not his real aim and so he deserted and tried to travel further. He was caught and put into prison in solitary confinement. A doctor in the place he was stationed wrote to his father. Since in fact Jünger was not old enough to have joined the foreign legion and had only managed to do so by lying about his age his father was able to get him discharged and took him home. Shortly after he returned the First World War began and Jünger enlisted immediately and had his opportunity for adventure, as related in ‘In Stahlgewittern’. One thing which attracts me to Jüngers writing, in ‘Afrikanische Spiele’ and elsewhere, is the style. At the same time, the content is often remarkable. Here is a striking example. The hero of the book has taken some money with him when he left home. He feels the danger that he might give up and not dare to carry out his plan. To avoid this he takes all the money he has and puts in down a drain in Verdun. In this way he removes the chance of turning back. This reminds me a little of the story of how Nansen became the first to cross the Greenland icecap. He chose the direction of crossing in such a way that failure would have meant almost certain death.

I will mention a passage in ‘Gärten und Strassen’ which particularly struck me and which Banine mentions in her book. In this book Jünger described his experiences during the German invasion of France in the Second World War. This time, in contrast with what happened in the First World War, he was hardly involved in the fighting at all. At one time he was the commanding officer in the town of Laon. Laon has a magnificent gothic cathedral, which I have visited myself. He describes his experience of looking at this cathedral, for whose safety he was responsible at that time, and feeling that this huge building was like a small vulnerable creature. He was successful in preventing treasures from the cathedral being stolen or destroyed, helped by the fact that those who might have done so did not realize how valuable these things were.

Despite my admiration for Jünger’s writings there is one thing which I do not like and which I feel I have to mention. This is a tendency to esotericism which he shows from time to time and which I just try to ignore. Despite this I am sure that I will continue to read Jünger with pleasure in the future.

## Monotone systems revisited

December 4, 2019

There are some topics in mathematics and physics which are a lasting source of dissatisfaction for me since I feel that I have not properly understood them despite having made considerable efforts to do so. In the case of physics the reason is often that the physicists who understand the subject are not able to explain it in a way which provides what a mathematician sees as a comprehensible account. In mathematics the problem is a different one. Mathematicians frequently have a tendency (often justified) to discuss things on a level which is as general as possible. This leads to theorems which are loaded down with detail and where the many technical conditions make it difficult to see the wood for the trees. When confronted with such things I sometimes feel exhausted and give up. I prefer an account which builds up ideas step by step from simple beginnings. Here I return to a subject which I have written about more than once in this blog before but where the sense of dissatisfaction remains. I hope to reduce it here.

I start with a system of ordinary differential equations $\dot x_i=f_i(x)$. It should be defined on the $n$-dimensional Euclidean space or on one of its orthants. (An orthant is the subset of Euclidean space defined by making a choice of the signs of its components. It generalises a quadrant in the two-dimensional case.) The system is said to be cooperative if $\frac{\partial f_i}{\partial x_j}>0$ for all $i\ne j$. The name comes from the fact that the equations for the population dynamics of a set of species has this property if each species benefits the others. Suppose we now have two solutions $x$ and $\bar x$ of the system and that $x_i(t_0)\le\bar x_i(t_0)$ for all $i$ at some time time $t_0$. We may abbreviate this relation by $x(t_0)\le\bar x(t_0)$. Here we see a partial order on Euclidean space defined by the ordering of the components. A theorem of Müller and Kamke says that if the initial data for two solutions of a cooperative system at time $t_0$ satisfies this relation then $x(t)\le\bar x(t)$ for all $t\ge t_0$. Another way of saying this is that the time-$t$ flow of the system is preserves the partial order. A system of ODE with this property is called monotone. Thus the Müller-Kamke theorem says that a cooperative system is monotone.

The differential condition for monotonicity can be integrated. If $x$ and $\bar x$ are two points in Euclidean space with $x_i=\bar x_i$ for a certain $i$ and $x_j\le\bar x_j$ for $j\ne i$ then $f_i(x)\le f_i(\bar x)$. To see this we join $x$ to $\bar x$ by a piecewise linear curve where the coordinates other than the $i$th are increased successively from $x_j$ to $\bar x_j$. On each segment of this curve the value of $f_i$ does not decrease, as a consequence of the fundamental theorem of calculus. Hence its value at the end of the entire path is at least as big as its value at the beginning. We now want to prove that a certain inequality holds at all times $t\ge t_0$. In order to do this we would like to consider the first time $t_*>t_0$ where the inequality fails and get a contradiction. Unfortunately there might be no such time – in principle the condition might fail immediately. To get around this we deform the system for the solution $\bar x$ to $\frac{d\bar x_i}{dt}=f_i(\bar x)+\epsilon$. If we can prove the result for the deformed system the result for the initial system follows by continuous dependence of the solution on $\epsilon$. For the deformed system let $t_*$ be the supremum of the times where the desired inequality holds. If the inequality does not hold globally then the system is still defined at $t=t_*$. For $t=t_*$ we have $x_i=\bar x_i$ for some $i$ and we can assume w.l.o.g. that $x_j<\bar x_j$ for some $j$ since otherwise the two solutions would be equal and the result trivial. The integrated form of the cooperativity condition implies that at $t_*$ the right hand side of the evolution equation for $\bar x_i-x_i$ is positive. On the other hand the fact that it just reached zero coming from positive values implies that the right hand side of the evolution equation is non-positive and we get a contradiction.

A key source of information about monotone dynamical systems is the book of Hal Smith with this title. I have repeatedly looked at this book but always got bogged down quite quickly. Now I realise that for my purposes it would have been much better if I had started with chapter 3. The Müller-Kamke theorem is discussed in section 3.1. The range of application of this theorem can be extended considerably by the following trick, discussed in section 3.5. Suppose that we define $y_i=(-1)^{m_i}x_i$ where each of the $m_i$ are zero or one. This transforms the signs of $Df$ in a certain way and so cooperativity of the system for $y$ corresponds to a certain sign pattern for the entries of $Df$. A first important condition is that each off-diagonal element of $Df(x)$ should be either non-negative or non-positive. Next, the sign of $\frac{\partial f_i}{\partial x_j}\frac{\partial f_j}{\partial x_i}$ is not changed be the transformation and must thus be non-negative. In the context of population models this can be interpreted as saying that there is no pair of species which are in a predator-prey relationship. Given that these two conditions are satisfied we consider a labelled graph where the nodes are the numbers from $1$ to $n$ and there is an edge between two nodes if at least one of the corresponding partial derivatives is non-zero at some point. The edge is then labelled with the sign of this non-zero value. A loop in the graph can be assigned the sign which is the product of those of its edges. It turns out that a system can be transformed to a cooperative system in the way indicated if and only if the graph contains no negative loops. I will call a system of this type ‘cooperative up to sign reversal’. The system can be transformed by a permutation of the variables into one where $Df$ has diagonal blocks with non-negative entries and off-diagonal elements with non-positive entries.

If all elements of $Df$ are required to be non-positive we get the class of competitive systems. It should be noted that being competitive leads to less restrictions on the dynamics of a system (towards the future) than being cooperative. We can define a class of systems which are competitive up to sign reversal. An example of such a system is the basic model of virus dynamics. In that system the unknowns are the populations of uninfected cells $x$, infected cells $y$ and virus particles $v$. The transformation $y\mapsto -y$ makes it into a competitive system. In various models of virus dynamics including the immune response the target cells of the virus and the immune cells are in a predator-prey relationship and so these systems can be neither cooperative up to sign or competitive up to sign.

## Cedric Villani’s autobiography

November 1, 2019

I have just read Cédric Villani’s autobiographical book ‘Théorème Vivant’. I gave the German translation of the book to Eva as a present. I thought it might give her some more insight into what it is like to be a mathematician and give her some fortitude in putting up with a mathematician as a husband. Since I had not read the book before I decided to read it in parallel. I preferred to read the original and so got myself that. With hindsight I do not think it made so much difference that I read it in French instead of German. I think that the book is useful for giving non-experts a picture of the life of a mathematician (and not just that of a mathematician who is as famous as Villani has become). For this I believe that it is useful that the book contains some pieces of mathematical text which are incomprehensible for the lay person and some raw TeX source-code. I think that they convey information even in the absence of an understanding of the content. On the other hand, this does require a high level of tolerance on the part of the reader. Fortunately Eva was able to show this tolerance and I think she did enjoy the book and learn something more about mathematics and mathematicians.

For me the experience was of course different. The central theme of the book is a proof of Villani and Clement Mouhot of the existence of Landau damping, a phenomenon in plasma physics. I have not tried to enter into the details of that proof but it is a subject which is relatively close to things I used to work on in the past and I was familiar with the concept of Landau damping a long time ago. I even invested quite a lot of time into the related phenomenon of the Jeans instability in astrophysics, unfortunately without significant results. Thus I had some relation to the mathematics. It is also the case that I know many of the people mentioned in the book personally. Sometimes when Villani mentions a person without revealing their name I know who is meant. As far as I remember the first time I met Villani was at a conference in the village of Anogia in Crete in the summer of 2001. At that time he struck me as the number one climber of peaks of technical difficulty in the study of the Boltzmann equation. I do not know if at that time he already dressed in the eccentric way he does today. I do not remember anything like that.

For me the book was pleasant to read and entertaining and I can recommend it to mathematicians and non-mathematicians. If I ask myself what I really learned from the book in the end then I am not sure. One thing it has made me think of is how far I have got away from mainstream mathematics. A key element of the book is that the work described there got Villani a Fields medal, the most prestigious of mathematical prizes. These days the work of most Fields medallists is on things to which I do not have the slightest relation. Villani was the last exception to that rule. Of course this is a result of the general fact that communication between different mathematical specialities is so hard. The Fields medal is awarded at the International Congress of Mathematicians which takes place every four years. That conference used to be very attractive for me but now I have not been to one since that in 2006 in Madrid and I imagine that I will not go to another. That one was marked by the special excitement surrounding Perelman’s refusal of the Fields medal which he was offered for his work on the Poincaré conjecture. Another sign of the change in my orientation is that I am no longer even a member of the American Mathematical Society, probably the most important such society in the world. I will continue to follow my dreams, whatever they may be. Villani is also following his dreams. I knew that he had gone into politics, becoming a member of parliament. I was surprised to learn that he has recently become a candidate for the next election to become mayor of Paris.

## Article on mathematical economics

September 28, 2019

I now find myself in the for me a priori surprising situation of being one of the authors of an article in the Journal of Mathematical Economics (85, 17-37). The title is ‘The invariant distribution of wealth and employment status in a small open economy with precautionary savings’. Without having a global overview of the subject I want to explain roughly what is being described by the mathematical model in this article. The idea is that we have a population of individuals who may lose their jobs and find new ones in a way which involves a lot of randomness. It is supposed that when they have employment these individuals save some money in order to be able to support themselves better during periods where they are unemployed. They question is whether these assumptions lead to a long-term stationary distribution of wealth and employment in the population. The starting model is a stochastic one but the consideration of stationary distributions lead to an ODE problem.

So how did I get involved in this project? Klaus Wälde, a professor of economics at the University of Mainz, had a project on this subject with Christian Bayer, a mathematician from the Weierstrass Institute in Berlin. The two of them submitted a paper which contained some partial results on stationary distributions. However they did not have a complete existence proof and the referees insisted that they try to improve on that. Wälde looked for some assistance on this point in the mathematics department in Mainz and ended up talking to me about the matter. The concrete mathematical problem to be solved concerned a certain boundary value problem for a system of ODE with rather singular boundary conditions. At this point the universality of mathematics comes in. It turned out that this problem coming from economics could be solved with the help of a theorem which I proved with Bernd Schmidt in 1991 to treat a problem coming from astrophysics. Along the way I was also able to show that an analyticity assumption which had been made in the formulation of the economics model was unnecessary. I have no intention of going further into the area of mathematical economics – I want to remain focussed on mathematical biology. It is nevertheless a nice feeling to think that with the mathematical training I have I could in principle contribute to almost any field of science.

## Science as a literary pursuit

August 24, 2019

I found something in a footnote in the book of Oliver Sacks I mentioned in the previous post which attracted my attention. There is a citation from a letter of Jonathan Miller to Sacks with the idea of a love of science which is purely literary. Sacks suggests that his own love of science was of this type and that that is the reason that he had no success as a laboratory scientist. I feel that my own love for science has a strong literary component, or at least a strong component which is under the control of language. In molecular biology there are many things which have to be named and people have demonstrated a lot of originality in inventing those names. I find the language of molecular biology very attractive in a way which has a considerable independence from the actual meaning of the words. I expect that there are other people for whom this jungle of terminology acts as a barrier to entering a certain subject. In my case it draws me in. In my basic field, mathematics, the terminology and language is also a source of pleasure for me. I find it stimulating that everyday words are often used with a quite different meaning in mathematics. This bane of many starting students is a charm of the subject for me. Personal taste plays a strong role in these things. String theory is another area where there is a considerable need for inventing names. There too a lot of originality has been invested but in that case the result is not at at all to my taste. I emphasize that when I say that I am not talking about the content, but about the form.

The idea of using the same words with different meanings has a systematic development in mathematics in context of topos theory. I learned about this through a lecture of Ioan James which I heard many years ago with the title ‘topology over a base’. What is the idea? For topological spaces $X$ there are many definitions and many statements which can be formulated using them, true or false. Suppose now we have two topological spaces $X$ and $B$ and a suitable continuous mapping from $X$ to $B$. Given a definition for a topological space $X$ (a topological space is called (A) if it has the property (1)) we may think of a corresponding property for topological spaces over a base. A topological space $X$ over a base $B$ is called (A) if it has property (2). Suppose now that I formulate a true sentence for topological spaces and suppose that each property which is used in the sentence has an analogue for topological spaces over a base. If I now interpret the sentence as relating to topological spaces over a base under what circumstances is it still true? If we have a large supply of statements where the truth of the statement is preserved then this provides a powerful machine for proving new theorems with no extra effort. A similar example which is better known and where it is easier (at least for me) to guess good definitions is where each property is replaced by one including equivariance under the action of a certain group.

Different mathematicians have different channels by which they make contact with their subject. There is an algebraic channel which means starting to calculate, to manipulate symbols, as a route to understanding. There is a geometric channel which means using schematic pictures to aid understanding. There is a combinatoric channel which means arranging the mathematical objects to be studied in a certain way. There is a linguistic channel, where the names of the objects play an important role. There is a logical channel, where formal implications are the centre of the process. There may be many more possibilities. For me the linguistic channel is very important. The intriguing name of a mathematical object can be enough to provide me with a strong motivation to understand what it means. The geometric channel is also very important. In my work schematic pictures which may be purely mental are of key importance for formulating conjectures or carrying out proofs. By contrast the other channels are less accessible to me. The algebraic channel is problematic because I tend to make many mistakes when calculating. I find it difficult enough just to transfer a formula correctly from one piece of paper to another. As a child I was good in mental arithmetic but somehow that and related abilities got lost quite early. The combinatoric channel is one where I have a psychological problem. Sometimes I see myself surrounded by a large number of mathematical objects which should be arranged in a clever way and this leads to a feeling of helplessness. Of course I use the logical channel but that is usually on a relatively concrete level and not the level of building abstract constructs.

Does all this lead to any conclusion? It would make sense for me to think more about my motivations in doing (and teaching) mathematics in one way or another. This might allow me to do better mathematics on the one hand and to have more pleasure in doing so on the other hand.

## Encounter with an aardvark

August 21, 2019

When I was a schoolboy we did not have many books at home. As a result I spent a lot of time reading those which were available to me. One of them was a middle-sized dictionary. It is perhaps not surprising that I attached a special significance to the first word which was defined in that dictionary. At that time it was usual, and I see it as reasonable, that articles did not belong to the list of words which the dictionary was responsible for defining. For this reason ‘a’ was not the first word on the list and instead it was ‘aardvark’. From the dictionary I learned that an aardvark is an animal and roughly what kind of animal it is. I also learned something about its etymology (it was an etymological dictionary) and that it originates from Dutch words meaning ‘earth’ and ‘pig’. Later in life I saw pictures of aardvarks in books and saw them in TV programmes, but without paying special attention to them. The aardvark remained more of an intriguing abstraction for me than an animal.

Yesterday, in Saarbrücken zoo, I walked into a room and saw an aardvark in front of me. Suddenly the abstraction turned into a very concrete animal pacing methodically around its enclosure. I had a certain feeling of unreality. I do not know if aardvarks always walk like that or whether it was just a habit which this individual had acquired by being confined to a limited space. Each time it returned (reappearing after having disappeared into a region not visible to me) the impression of unreality was heightened. I was reminded of the films of dinosaurs which sometimes come on TV, where the computer-reconstructed movements of the animals look very unrealistic to me. Seeing the aardvark I asked myself, ‘if mankind only knew this animal from fossil remains would it ever have been possible to reconstruct the gait I now see before me?’

Another animal I encountered in the Saarbrücken zoo is a species whose existence I did not know of before. This is Pallas’s cat. This is a wild cat with a very unusual and engaging look. The name Pallas has a special meaning for me for the following reason. When I was young and a keen birdwatcher some of the birds which were most exciting for me were rare vagrants from Siberia which had been brought to Europe by unusual weather conditions. A number of these are named after Pallas. I knew almost nothing about the man Pallas. Now I have filled in some background. In particular I learned that he was a German born in Berlin who was sent on expeditions to Siberia by Catherine the Great.

## SMB meeting in Montreal

July 27, 2019

This week I have been attending the SMB meeting in Montreal. There was a minisymposium on reaction networks and I gave a talk there on my work with Elisenda Feliu and Carsten Wiuf on multistability in the multiple futile cycle. There were also other talks related to that system. A direction which was new to me and was discussed in a talk by Elizabeth Gross was using a sophisticated technique from algebraic geometry (the mixed volume) to obtain an upper bound on the number of complex solutions of the equations for steady states for a reaction network (which is then of course also an upper bound for the number of positive real solutions). There were two talks about the dynamics of complex balanced reaction networks with diffusion. I have the impression that there remains a lot to be understood in that area.

At this conference the lecture rooms were usually big enough. An exception was the first session ‘mathematical oncology from bench to bedside’ which was completely overfilled and had to move to a different room. In that session there was a tremendous amount of enthusiasm. There is now a subgroup of the SMB for cancer modelling which seems to be very active with its own web page and blog. I should join that subgroup. Some of the speakers were so full of energy and so extrovert that it was a bit much for me. Nevertheless, it is clear that this is an exciting area and I would like to be part of it. There was also a session of cancer immunotherapy led by Vincent Lemaire from Genentech. He and two others described the mathematical modelling being done in cancer immunotherapy in three major pharmaceutical companies (Genentech, Pfizer and Glaxo-Smith-Kline). These are very big models. Lemaire said that at the moment that there are 2500 clinical trials going on for therapies related to PD-1. A recurring theme in these talks was the difference between mice and men.

This morning there was a talk by Hassan Jamaleddine concerning nanoparticles used to present antigen. These apparently primarily stimulate Tregs more than effector T cells and can thus be used as a therapy for autoimmune diseases. He showed some impressive pictures illustrating clearance of EAE using this technique. A central theme was interference between attempts to use the technique in animals with two autoimmune diseases in different organs, e.g. brain and liver. I was interested by the fact that for what he was doing steady state analysis was insufficient for understanding the biology.

This afternoon, the conference being over, I took to opportunity to visit Paul Francois at McGill, a visit which was well worthwhile.

## The fold-Hopf and Hopf-Hopf bifurcations

June 30, 2019

Bifurcations of a dynamical system $\dot x=f(x,\alpha)$ can be classified according to their codimension. The intuitive idea is that the set of vector fields exhibiting a given type of bifurcation form a submanifold of the space of all vector fields of the given codimension. Well-known examples of bifurcations of codimension one are the cusp bifurcation, which can already occur when the variables $x$ and $\alpha$ are one-dimensional, and the Hopf bifurcation where the variable $x$ must be two-dimensional but the variable $\alpha$ can be one-dimensional. In fact the minimal dimension of the variable $\alpha$ required corresponds to the codimension of the bifurcation. In this post I want to discuss two bifurcations of codimension 2, the fold-Hopf bifurcation, where $x$ must be at least three-dimensional and the Hopf-Hopf bifurcation where $x$ must be at least four-dimensional

The first typical feature of these bifurcations is the configuration of eigenvalues of $D_x f$ at the bifurcation point. For a fold there is an eigenvalue zero. For a Hopf bifurcation there is a complex conjugate pair of non-zero purely imaginary eigenvalues. For a fold-Hopf bifurcation there is a zero eigenvalue and a complex conjugate pair of non-zero purely imaginary eigenvalues. For a Hopf-Hopf bifurcation there are two complex conjugate pairs of non-zero purely imaginary eigenvalues. The basic hope now is that if some genericity conditions are satisfied the system can be locally reduced to a normal form by a transformation of variables. This is true for the fold and Hopf cases but for fold-Hopf and Hopf-Hopf it is no longer true. A weaker goal which can be attained is to reduce the system to an approximate normal form so that the right hand side is the sum of a simple explicit expression and a higher order error term. The genericity assumptions are as follows. For the cusp the steady state should move with non-zero velocity when the parameter is changed and the steady state of the system for the bifurcation value of the parameter should be as non-degenerate as possible. This means that although $f_x=0$ there (which is the bifurcation condition) $f_{xx}\ne 0$. For the Hopf case the eigenvalues which are on the imaginary axis at the bifurcation value should move off the axis with non-zero velocity when the parameter is changed. At the same time the steady state at the bifurcation value should be as non-degenerate as possible. Solutions close to this steady state circle it and a corresponding Poincare mapping can be defined which describes how the distance from the steady state changes when the solution circles it once. Call this $p(x)$. The fact that there is a bifurcation means that $p'(0)=0$ and $p''(0)$ is automatically zero. The non-degeneracy condition is that $p'''(0)\ne 0$.

Now we come to the fold-Hopf bifurcation. One non-degeneracy condition combines conditions from the two simpler bifurcations in a simple way. It says that the position of the steady state and the real part of the eigenvalue move independently as the two parameters are changed. This is condition ZH0.3 in Theorem 8.7 in the book of Kuznetsov. I was confused by the fact that this condition involves a quantity $\gamma (\alpha)$ which is apparently nowhere defined in the book. It does occur on one other page. On that page there is also a $\Gamma (\alpha)$ which is defined and I think that the solution to the problem is that these two are equal. Assuming that that is correct then $\gamma (\alpha)$ is the projection of the position of the steady state onto the kernel of the linearization at the bifurcation point. The remaining non-degeneracy conditions are conditions on the system at the bifurcation value of the parameter. At the moment I do not have an intuition for the meaning of those conditions.

In the case of the Hopf-Hopf bifurcation a genericity assumption which is qualitatively different from those we have seen up to now is a non-resonance condition, condition HH.0 of Kuznetsov. It says that the two imaginary parts of the eigenvalues at the bifurcation point should not exhibit linear relations with integer coefficients. The next condition is that the real parts of the eigenvalues move independently as the two parameters are changed (HH.5 of Kuznetsov). As in the fold-Hopf case the remaining non-degeneracy conditions are conditions on the system at the bifurcation value of the parameter which I do not understand intuitively.

When analysing the Hopf bifurcation it turns out that after doing a suitable transformation of variables and discarding some terms which can be considered small the resulting system is rotationally invariant. In polar coordinates the angular component is constant and the radial component is cubic in the radius. For the fold-Hopf bifurcation it is natural to proceed as follows. We do a linear transformation so that the new axes belong to the eigenspaces of the linearization. Moreover this transformation is chosen so that the restriction of the linearization to the plane correponding to the complex eigenvalues is in standard form. Then the normal form is rotationally invariant and can be expressed in cylindrical polar coordinates. The component in the angular direction depends only on the coordinate $\xi$ along the axis while the other two components depend only on $\xi$ and the radial coordinate $\rho$. Thus the analysis of the phase portrait of the system near the bifurcation point can be reduced to the analysis of a two-dimensional dynamical system on the half-plane $\rho\ge 0$ called the amplitude system. A steady state of the amplitude system with $\rho=0$ corresponds to a steady state of the full system while a steady state of the amplitude system with $\rho>0$ corresponds to a periodic solution of the full system.

The amplitude system contains a parameter $s=\pm 1$ and a parameter $\theta$. The signs of these two quantities are of crucial importance. If $s=1$ and $\theta>0$ then the system can be reduced to normal form. If $s=-1$ and $\theta<0$ then the situation is still relatively simple but adding a small perturbation typically causes a heteroclinic orbit to break. The most difficult case is that where $s\theta<0$ and it is in that case that chaos may occur. A steady state of the amplitude system away from the axis can undergo a Hopf bifurcation and this corresponds to the occurrence of an invariant torus in the full system and is a Neimark-Sacker bifurcation of the limit cycle. This torus can break up for some value of the parameters and this is what leads to chaos.

In the Hopf-Hopf case the normal form involves two angular coordinates where the dynamics are trivial and two radial coordinates $r_1$ and $r_2$. Thus we again obtain a two-dimensional amplitude system, this time defined on a quadrant. Kuznetsov distinguishes between a ‘simple’ and a ‘difficult’ case according to the parameters but at my present level of understanding they both look very difficult. For Hopf-Hopf the truncated normal form is generically never topologically equivalent to the full system. A Neimark-Sacker bifurcation is always present.

As mentioned in a previous post, both bifurcations discussed here have been observed numerically in an ecological model and Hopf-Hopf bifurcations (but not fold-Hopf bifurcations, this was stated incorrectly in the previous post) in a model for the MAPK cascade.