A polycycle is a separatrix polygone of a vector field in the plane. Its monodromy transformation

is an analog of the Poincare map for a limit cycle. The difference is that the second map is

defined on an interval, and the first one on a half-interval. In the analytic case, the second map may be decomposed

in a Taylor series. The first one is much more complicated, and may have a correction (difference

with identity) equal to a tower of exponents. Namely, the first map may have a form

$ \Delta (x) = x + f_0^{\circ n} (x)$ for arbitrary $n$; here $ f_0 = \exp (-\frac {1}{x})$. Very specific

asymptotic series for these maps will be described in the talk.