On the formation of singularities of solutions of nonlinear differential systems in antistokes directions

Abstract

We determine the position and the type of spontaneous singularities of solutions of generic analytic nonlinear differential systems in the complex plane, arising along antistokes directions towards irregular singular points of the system. Placing the singularity of the system at infinity we look at equations of the form $\mathbf{y}'=\mathbf{f}(x^{-1},\mathbf{y})$ with $\mathbf{f}$ analytic in a neighborhood of $(0,\mathbf{0})$, with genericity assumptions; $x=\infty$ is then a rank one singular point. We analyze the singularities of those solutions $\mathbf{y}(x)$ which tend to zero for $x\to \infty$ in some sectorial region, on the edges of the maximal region (also described) with this property. After standard normalization of the differential system, it is shown that singularities occuring in antistokes directions are grouped in nearly periodical arrays of similar singularities as $x\to\infty$, the location of the array depending on the solution while the (near-) period and type of singularity are determined by the form of the differential system.