Dynamic mode decomposition for self-similar dynamics

Dynamic mode decomposition (DMD) has had enormous excess as a post-processing tool for both linear and nonlinear dynamical systems, where the output can in some circumstances be connected to a Koopman decomposition. Here, we explore the utility of Koopman and DMD in systems which collapse onto a self-similar solution. These dynamics are usually algebraic in time and it has been unclear how one might use DMD to identify the long-time, self-similar solution given early-time observations, or to estimate a collapse time onto such a solution. We demonstrate that both of these objectives can be accomplished by performing DMD on snapshots spaced equally in logarithmic time, and we will introduce several `rules of thumb’ for robust results. Our numerical experiments are supported by full analytical Koopman decompositions of the self-similar dynamical system for (1) a linear diffusion and (2) Burgers equation on the real line. Time permitting, we will also present some applications to two-dimensional vortex dynamics.