Epilogue: Stokes Phenomena. Dynamics, Classification Problems and Avatars

We describe the Stokes phenomenon, its main significance, and its emergence in various landscapes. We explain how to use Stokes phenomena to enrich the classical dynamics of complex dynamical systems, defining some wild dynamics. In the same line, we describe how, adding Stokes multipliers to the classical monodromy, it is possible to classify a lot of complex dynamical systems (linear or not), up to “gauge transformations” (using some cohomological invariants), and to get generalizations of the Riemann-Hilbert correspondence, with a lot of applications. In the linear case, we describe the relations between Stokes phenomena and differential Galois theory and some important consequences. We end with a short description of many other incarnations of Stokes phenomenon (singular perturbations, resurgence, difference and q-difference equations, theoretical physics …), with some insights in the several variables cases. All along our article, we insist on the historical roots and on some simple geometric ideas. The interpretation of the divergence of some power series as expressing a form of branching for an analytic function is a leitmotiv and a red thread all along the text.