  We demonstrate that the integration of the recently developed dynamic mode
decomposition (DMD) with a multi-resolution analysis allows for a decomposition
method capable of robustly separating complex systems into a hierarchy of
multi-resolution time-scale components. A one-level separation allows for
background (low-rank) and foreground (sparse) separation of dynamical data, or
robust principal component analysis. The multi-resolution dynamic mode
decomposition is capable of characterizing nonlinear dynamical systems in an
equation-free manner by recursively decomposing the state of the system into
low-rank terms whose temporal coefficients in time are known. DMD modes with
temporal frequencies near the origin (zero-modes) are interpreted as background
(low-rank) portions of the given dynamics, and the terms with temporal
frequencies bounded away from the origin are their sparse counterparts. The
multi-resolution dynamic mode decomposition (mrDMD) method is demonstrated on
several examples involving multi-scale dynamical data, showing excellent
decomposition results, including sifting the El Ni\~no mode from ocean
temperature data. It is further applied to decompose a video data set into
separate objects moving at different rates against a slowly varying background.
These examples show that the decomposition is an effective dynamical systems
tool for data-driven discovery.
