Schwartzian derivative for multidimensional maps and flows

A generalization of Schwartzian derivative to maps and flows in the space $\mathbb R^n$ and in infinite-dimensional spaces is introduced. It is used to study the type of stability loss (soft or hard) for fixed points and periodic trajectories of diffeo-morphisms and flows. In particular, an example of a partial differential equation of reaction-diffusion type is presented for which the conditions of soft loss of stability of a spatially homogeneous solution are verified.