Characteristic Curves and the exponentiation in the Riordan Lie group: A connection through examples

We point out how to use the classical characteristic method, that is used to solve quasilinear PDE's, to obtain the matrix exponential of some lower triangle infinite matrices. We use the Lie Frechet structure of the Riordan group described in [4]. After that we describe some linear dynamical systems in $\mathbb{K}[[x]]$ with a concrete involution being a symmetry or a time-reversal symmetry for them. We take this opportunity to assign some dynamical properties to the Pascal Triangle.