In a post I wrote a long time ago I described matched asymptotic expansions as being like black magic. Now I have understood some more about how to get from there to rigorous mathematics. My main guide in doing so has been Chapter 7 of the book ‘Classical Methods in Ordinary Differential Equations’ by Hastings and McLeod. There they give an extensive treatment of a model problem by Kaplun and Lagerstrom. The ultimate source of this is work of Stokes on hydrodynamics around 1850. In his calculations he found some paradoxical phenomena. Roughly speaking, attempting to obtain an asymptotic expansion for a solution led to inconsistencies. These things remained a mystery for many years. A big step forward came in the work of Kaplun and Lagerstrom in 1957. There they introduced an ODE model which, while having no direct physical interpretation, provides a relatively simple mathematical context in which to understand these phenomena. It is this model problem which is treated in detail by Hastings and McLeod. The model is a boundary value problem for the equation 
Consider the case 
. Suppose that for some value 
the solution tends to 
for 
and for another value 
the solution tends to 
. Then we might expect that in between there is a value 
for which the solution is bounded. We could try to home in on the value of 
. Let 
be the subset where 
where 
and 
are non-empty and open and 
then 
has a periodic solution. In fact this is related to the Brouwer fixed point theorem specialised to dimension one (which of course is elementary to prove). The next example is to show that 
has a periodic solution. After that this is generalised to the case where 
is replaced by an arbitrary bounded continuous function on 
and we look for a bounded solution. The next example is a kind of forced pendulum equation 
and the aim is to find a solution which is at the origin at two given times. In the second chapter a wide variety of examples is presented, including those just mentioned, and used to illustrate a number of general points. The key point in a given application is to find a good choice for the subsets. There is also a discussion of two-parameter shooting and its relation to the topology of the plane. This has a very different flavour from the arguments I am familiar with. It is related to Wazewski’s theorem (which I never looked at before) and the Conley index. The latter is a subject which has crossed my path a few times in various guises but where I never really developed a warm relationship. I did spend some time looking at Conley’s book. I found it nicely written but so intense as to require more commitment than I was prepared to make at that time. Perhaps the book of Hastings and McLeod can provide me with an easier way to move in that direction.
be a dynamical system on 
and 
a point with 
. Let 
. Then 
, such that the real parts of the eigenvalues of the restrictions of 
is the centre subspace. The centre manifold theorem states that there exists an invariant manifold of the system passing through 
is trivial although he states vaguely that this restriction is not necessary. The other concerns the issue of regularity. Carr assumes that the system is 
and states that the centre manifold obtained is also 
for any 
. No proof is given there. Perko cites a book of Guckenheimer and Holmes and one of Ruelle for this but as far as I can see neither of them contains a proof of this statement. Looking through the literature the situation of what order of differentiability is required to get a result and whether the regularity which comes out is equal to that which goes in or whether it is a bit less seems quite chaotic. Having been frustrated by this situation a trip to the library finally allowed me to find what I now see as the best source. This is a book called ‘Normal forms and bifurcations of planar vector fields’ by Chow, Li and Wang. Despite the title it offers an extensive treatment of the existence theory in any (finite) dimension and proves, among other things, the result stated by Perko. I feel grateful to those authors for their effort.
. The cut-off is done only in the 
variables. If a suitable function 
is chosen then setting 
. Substituting the solution into the nonlinearity in the evolution equation for 
. This leads to an integral equation of the general form 
. If 
, i.e. 
for them to work. It is only necessary to replace 
by 
in some places.
depending on a parameter 
. The equation for steady state solutions is then 
. Sometimes we can eliminate all but one of the variables to obtain an equation 
for a real variable 
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 
where the linearization of 
. The result is that if 
and 
then for 
. Thus the stability of steady states arising at the bifurcation point is determined by the function 
, is a smooth function of 
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 
arising in the reduction process is zero. Thus the derivative of 
and letting 
. Lyapunov-Schmidt reduction is applied to 
to get a function 
. Let 
be the eigenvalue of 
which is analogous to the eigenvalue 
. From what was said above it follows that, in a notation which is hopefully clear, 
implies 
. We now want to show that the gradients of these two functions are non-zero. Lyapunov-Schmidt theory includes a formula expressing 
in terms of 
. Next we turn to the gradient of 
, more specifically to the derivative of 
for all 
shows that 
is an eigenvalue of 
for all 
and the derivative of 
with respect to 
and 
are two smooth functions, 
extends smoothly to the zero set of 
with ![\tilde a(0,0,0)=[\frac{\partial}{\partial\beta}(\tilde\Omega,0,0)]/\tilde g_{x\beta}(0,0,0)]>0](https://s0.wp.com/latex.php?latex=%5Ctilde+a%280%2C0%2C0%29%3D%5B%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%5Cbeta%7D%28%5Ctilde%5COmega%2C0%2C0%29%5D%2F%5Ctilde+g_%7Bx%5Cbeta%7D%280%2C0%2C0%29%5D%3E0&bg=ffffff&fg=333333&s=0&c=20201002)
. Setting 
in the first equation then gives the desired conclusion.
, where 
is a basic reproductive ratio. It was left open, whether the number of steady states is ever greater than the minimum compatible with this parity condition. If there existed backward bifurcations (see here for the definition) it might be expected that there are cases with 
and two positive solutions. We proved that in fact this model does not admit backward bifurcations. It is known that a related model for HIV with therapy (Nonlin. Anal. RWA 17, 147) does admit backward bifurcations and it would be interesting to have an intuitive explanation for this difference.
of a set 
which is either the positive orthant in Euclidean space (the subset where all coordinates are non-negative) or the intersection of that set with an affine subspace. The second case is included to allow the imposition of fixed values of some conserved quantities. Let 
be the boundary of 
there is a unique solution 
with 
. If 
let 
, where 
is the Euclidean distance. Let 
. A first attempt to define persistence (PD1) says that it means that 
for all 
such that 
for all 
for all 
-limit point in 
such that all components of all solutions are bounded by 
, where 
such that for any positive solution 
for 
. A criterion for permanence which is sometimes useful will now be stated without proof. If a system whose solutions are uniformly bounded at infinity has the property that 
then it is permanent. Here 
is the 
there is a solution through that point on an interval 
which is non-negative. For some systems points like this can be ruled out directly and this gives a way of proving that they are permanent.
be the flow of the dynamical system, i.e. 
is the value at time 
of the solution with 
. Its powers define a discrete semi-dynamical system with flow 
, where 
be the closure of the set of points 
. This is a compact set and because of persistence it lies in ![K'=\phi (K\times [0,1])](https://s0.wp.com/latex.php?latex=K%27%3D%5Cphi+%28K%5Ctimes+%5B0%2C1%5D%29&bg=ffffff&fg=333333&s=0&c=20201002)
. This is also a compact subset of ![\phi([t],x)\in K](https://s0.wp.com/latex.php?latex=%5Cphi%28%5Bt%5D%2Cx%29%5Cin+K&bg=ffffff&fg=333333&s=0&c=20201002)
where ![[t]](https://s0.wp.com/latex.php?latex=%5Bt%5D&bg=ffffff&fg=333333&s=0&c=20201002)
is the largest integer smaller that 
and proves that the system of ODE is persistent. A similar argument shows that if the time-one map is permanent the system of ODE is permanent. We only have to start with the compact set 
. 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 
. Then the periodic solutions whose existence we wish to prove will have period close to 
. The rescaled equation is 
where 
. Let 
be the Banach space of periodic ![[0,2\pi]](https://s0.wp.com/latex.php?latex=%5B0%2C2%5Cpi%5D&bg=ffffff&fg=333333&s=0&c=20201002)
with the usual 
the analogous space of continuous functions. Define a mapping 
by 
. Note that it is equivariant under translations of the argument. The periodic solutions we are looking for correspond to zeroes of 
. Let 
at the origin. The linearization of 
is 
. The operator 
is a (bounded) Fredholm operator of index zero.
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 
. It can be written as a linear combination of the vectors with components ![[x,y]](https://s0.wp.com/latex.php?latex=%5Bx%2Cy%5D&bg=ffffff&fg=333333&s=0&c=20201002)
and ![[-y,x]](https://s0.wp.com/latex.php?latex=%5B-y%2Cx%5D&bg=ffffff&fg=333333&s=0&c=20201002)
where the coefficients are of the form 
and 
. These functions satisfy the conditions 
, 
, 
, 
. It follows that 
is only zero if either 
or 
. The first case corresponds to the steady state solution at the bifurcation point. The second case corresponds to 
. By a rotation we can reduce to the case 
. Then the two cases correspond to 
and 
. The equation 
can be solved in the form 
, which follows from the implicit function theorem. Let 
and 
. Then 
has solutions with 
only if 
. All zeroes of 
is non-zero then it follows from the implicit function theorem that we can write 
and there is a one-parameter family of solutions. There are two eigenvalues of 
of the form 
where 
and 
and 
. It turns out that 
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 
is formally identical to that for a pitchfork bifurcation, i.e. a cusp bifurcation with reflection symmetry. The second non-degeneracy condition is 
. It is related to the non-vanishing of the first Lyapunov number.
, 
, where 
, 
shows that there can be infinitely many periodic solutions and that the precise question has to be a little different. A periodic solution is called a limit cycle if there is another solution which converges to the image of the first as 
of limit cycles is always finite. A second is whether an inequality of the form 
holds, where 
depends only on the degree of the polynomials. 
. Until recently it was not known whether 
was finite. A third question is to obtain an explicit bound for 
and that 
for a particular cubic polynomial 
. Both claims were false. As shown by Ilyashenko in 1982 there was a gap in Dulac’s proof. After 60 years there was almost no progress on this problem. Écalle worked on it intensively for a long time. For this reason he produced few publications and almost lost his job. At this point I pause to give a personal story. Some years ago I was invited to give a talk in Bayrischzell in the context of the elite programme in mathematical physics of the LMU in Munich. The programme of such an event includes two invited talks, one from outside mathematical physics (which in this case was mine) and one in the area of mathematical physics (which in this case was on a topic from string theory). In the second talk the concept of resurgence, which was invented by Écalle in his work on Hilbert’s sixteenth problem, played a central role. I see this as a further proof of the universality of mathematics.
. There is an example with three limit cycles which contain a common steady state and another which does not. It is problematic to find (or to portray) these with a computer since three are very small and one very large. More about the history can be found in an excellent article of Iyashenko (Centennial history of Hilbert’s 16th problem. Bull. Amer. Math. Soc. 39, 301) which was the main source for what I have written. In March 2021 a preprint by Pablo Pedregal appeared (
Hydrobates 