Connection problem of transseries solution of Hamiltonian system

This paper studies a connection problem for the transseries solution of the nonintegrable Hamiltonian system which is the analytic counterpart of the one studied by Taimanov related with a geodesic flow. Our purpose is to show the connection formula and the representation of the Stokes operator in the category of a transseries. As for the connection problem for the Painlevé equation, a general theory like a Riemann–Hilbert approach or an exact asymptotic analysis is well-known. Since our Hamiltonian system has a general degree of freedom and is nonintegrable, our equation does not fall into the framework of the general theory. The principal results are the connection formula in a transseries for a first integral in Sect. 4 and those for the transseries solution in Sect. 5. We also show the representation formula of the Stokes operator via the elliptic function, from which we prove the appearance of infinite singular points for a solution after the connection across a Stokes line. Our major conclusion is that, by a similar method like an isomonodromy method together with a moment summability method we have connection formulas for a transseries solution.