Yet part of the data won't make it to the limit, and something is lost if one looks only at the monodromy of the local systems. In the example that would be the monodromy associated to any one of the loops encircling only one singularity $\pm a$.

So, where has the lost data gone? And, what is the link with Stokes lines? In the above example the Stokes data is trivial, but simply considering the modified ODE $$(x^2-a^2)y'+y=x$$ gives a non-trivial example. For $a=0$ the so-called Euler's equation has a unique power-series solution $$\hat y(x) = \sum_n (n!)x^{n+1} $$ which doesn't sum as an analytic object in the usual way. By Borel-Laplace summing this series you obtain two analytic solutions, each one defined on a sector containing a half-plane, from which you deduce two sectorial local systems. The Stokes data comes from the comparison between these two local systems where the sectors overlap.The overlapping location is determined by the bissecting lines of the sectors, i.e. the Stokes lines. In the example you can obtain a Liouvillian representation of the solutions by explicit integration, therefore providing an integral representation for the Stokes data. You end up with formulas with coefficients given by values of the Gamma function (more details are linked at the end).

The discussion above streeses the fact that monodromy data is not a good presentation since it doesn't pass to the limit when a regular system degenerates onto an irregular one. Moreover the distinction monodromy/Stokes data is rather artificial, since Stokes data has also a meaning as gluing of local systems. I prefer the view where everything is "Stokes data": one can always subsdivide $\mathbb P_1$ into "sectors" attached to the singular points, on which you have a trivial system, and the sectorial systems get compared in the pairwise intersections of said generalized sectors. In the case of a regular singularity, you can form a neighborhood around it by tiling contiguous sectors: the composition of the Stokes operators coming from crossing the corresponding overlaps attached at the singularity gives you the monondromy operator. All this data passes to the limit in cases of merging.

As the construction shows, the Stokes data is not attached to an element of the fundamental group of $X\setminus sing$, like the monodromy, but rather to the "dual" groupoid of paths linking singular point (with an explicit representation as a path-integral operator).

Yet part of the data won't make it to the limit, and something is lost if one looks only at the monodromy of the multivalued solutions. In the example that would be the monodromy associated to any one of the loops encircling only one singularity $\pm a$.

So, where has the lost data gone? And, what is the link with Stokes lines? In the above example the Stokes data is trivial, but simply considering the modified ODE $$(x^2-a^2)y'+y=x$$ gives a non-trivial example. For $a=0$ the so-called Euler's equation has a unique power-series solution $$\hat y(x) = \sum_n (n!)x^{n+1} $$ which doesn't sum as an analytic object in the usual way. By Borel-Laplace-summing this series you obtain two analytic solutions, each one defined on a sector containing a half-plane, from which you deduce two sectorial systems of solutions. The Stokes data comes from the comparison between these two systems where the sectors overlap.The overlapping location is determined by the bissecting lines of the sectors, i.e. the Stokes lines. In the example you can obtain a Liouvillian representation of the solutions by explicit integration, therefore providing an integral representation for the Stokes data. You end up with formulas with coefficients given by values of the Gamma function (more details are linked at the end).

The discussion above streeses the fact that monodromy data is not a good presentation since it doesn't pass to the limit when a regular system degenerates onto an irregular one. Moreover the distinction monodromy/Stokes data is rather artificial, since Stokes data has also a meaning as gluing of local solutions. I prefer the view where everything is "Stokes data": one can always subsdivide $\mathbb P_1$ into "sectors" attached to the singular points, on which you have a trivial system, and the sectorial systems get compared in the pairwise intersections of said generalized sectors. In the case of a singularity (regular or not) you can form a neighborhood around it by tiling contiguous sectors: the composition of the Stokes operators coming from crossing the corresponding overlaps attached at the singularity gives you the monondromy operator. All the Stokes data passes to the limit in cases of merging.

As the construction shows, the Stokes data is not attached to an element of the fundamental group of $X\setminus sing$, like the monodromy, but rather to the "dual" groupoid of paths linking singular points (with an explicit representation as a path-integral operator).

So, where has the lost data gone? And, what is the link with Stokes lines? In the above example the Stokes data is trivial, but simply considering the modified ODE $$(x^2-a^2)y'+y=x$$ gives a non-trivial example. For $a=0$ the so-called Euler's equation has a unique power-series solution $$\hat y(x) = \sum_n (n!)x^{n+1} $$ which doesn't sum as an analytic object in the usual way. By Borel-Laplace summing this series you obtain two analytic solutions, each one defined on a sector containing a half-plane, from which you deduce two sectorial local systems. The Stokes data comes from the comparison between these two local systems where the sectors overlap.The overlapping location is determined by the bissecting lines of the sectors, i.e. the Stokes lines.

The discussion above streeses the fact that monodromy data is not a good presentation since it doesn't pass to the limit when a regular system degenerates onto an irregular one. Moreover the distinction monodromy/Stokes data is rather artificial, since Stokes data has also a meaning as gluing of local systems. I prefer the view where everything is "Stokes data": one can always subsdivide $\mathbb P_1$ into "sectors" on which you have a trivial local system, that get compared in the pairwise intersections of said generalized sectors. In the case of a regular singularity, you can form a neighborhood around it by tiling contiguous sectors: the composition of the Stokes operators coming from crossing the corresponding overlaps attached at the singularity gives you the monondromy operator. All this data passes to the limit in cases of merging.

As the construction shows, the Stokes data is not attached to an element of the fundamental group of $X\setminus sing$, like the monodromy, but rather to the "dual" groupoid of paths linking singular points.

So, where has the lost data gone? And, what is the link with Stokes lines? In the above example the Stokes data is trivial, but simply considering the modified ODE $$(x^2-a^2)y'+y=x$$ gives a non-trivial example. For $a=0$ the so-called Euler's equation has a unique power-series solution $$\hat y(x) = \sum_n (n!)x^{n+1} $$ which doesn't sum as an analytic object in the usual way. By Borel-Laplace summing this series you obtain two analytic solutions, each one defined on a sector containing a half-plane, from which you deduce two sectorial local systems. The Stokes data comes from the comparison between these two local systems where the sectors overlap.The overlapping location is determined by the bissecting lines of the sectors, i.e. the Stokes lines. In the example you can obtain a Liouvillian representation of the solutions by explicit integration, therefore providing an integral representation for the Stokes data. You end up with formulas with coefficients given by values of the Gamma function (more details are linked at the end).

The discussion above streeses the fact that monodromy data is not a good presentation since it doesn't pass to the limit when a regular system degenerates onto an irregular one. Moreover the distinction monodromy/Stokes data is rather artificial, since Stokes data has also a meaning as gluing of local systems. I prefer the view where everything is "Stokes data": one can always subsdivide $\mathbb P_1$ into "sectors" attached to the singular points, on which you have a trivial system, and the sectorial systems get compared in the pairwise intersections of said generalized sectors. In the case of a regular singularity, you can form a neighborhood around it by tiling contiguous sectors: the composition of the Stokes operators coming from crossing the corresponding overlaps attached at the singularity gives you the monondromy operator. All this data passes to the limit in cases of merging.

As the construction shows, the Stokes data is not attached to an element of the fundamental group of $X\setminus sing$, like the monodromy, but rather to the "dual" groupoid of paths linking singular point (with an explicit representation as a path-integral operator).