The Alekseev-Meinrenken diffeomorphism arising from the Stokes phenomenon

The Alekseev-Meinrenken diffeomorphism is a distinguished diffeomorphism from the space of $n\times n$ Hermitian matrices to the space of $n\times n$ positive definite Hermitian matrices. This paper derives the explicit expression of the diffeomorphism, via the Stokes phenomenon of meromorphic linear systems of ordinary differential equations with Poncar\'{e} rank $1$.