Multi-resurgence and connection-to-Stokes formulæ for some linear meromorphic differential systems (Q2410309)

The author considers a linear system of meromorphic equations \(x^{r+1}dY/dx=A(x)Y\), where \(A\in M_n(\mathbb{C}\{ x\} )\), \(A(0)\neq 0\) (\(r\geq 1\) and \(n\geq 2\) being integers). A gauge transformation \(Y\mapsto T(x^{\nu})Y\) depending meromorphically on some fractional power of \(x\) brings the system to a formal normal form. The latter defines the levels \(r_1<\cdots <r_p\) of the system. The author shows that for any \(k\) the Borel transforms of the system's \(r_k\)-reduced formal solutions are resurgent. He gives the general form of all their singularities and, under some convenient hypotheses on the geometric configuration of the singular points, he presents exact formulae expressing some of the Stokes multipliers of level \(r_k\) of the initial system in terms of connection constants in the Borel plane. This generalizes formulae previously obtained by M. Loday-Richaud and him for systems with a single level. The results are illustrated by a numerical example.