for discussion about the variance decomposition in non-linear world, please read my communication with Dong-Eun on this matter in the attached file (bottom-up).

IRFs for a nonlin model are informally defined in dynare++ tutorial. It is a difference between paths with and without an initial shock. Note that this forms a distribution for each period, since the shocks are involved. In the linear case, the contributions of shocks in a path with and without the initial shock will cancel, so the distribution is mass-point (simply a line). In the non-linear case, the contributions of shocks depend also on the level of other variables which differ in paths with and without the initial shock, so the contributions will not cancel. Dynare++ reports mean and variance of these distributions for each period.