  The dynamic mode decomposition (DMD) is a data-driven method used for
identifying the dynamics of complex nonlinear systems. It extracts important
characteristics of the underlying dynamics using measured time-domain data
produced either by means of experiments or by numerical simulations. In the
original methodology, the measurements are assumed to be approximately related
by a linear operator. Hence, a linear discrete-time system is fitted to the
given data. However, often, nonlinear systems modeling physical phenomena have
a particular known structure. In this contribution, we propose an
identification and reduction method based on the classical DMD approach
allowing to fit a structured nonlinear system to the measured data. We mainly
focus on two types of nonlinearities: bilinear and quadratic-bilinear. By
enforcing this additional structure, more insight into extracting the nonlinear
behavior of the original process is gained. Finally, we demonstrate the
proposed methodology for different examples, such as the Burgers' equation and
the coupled van der Pol oscillators.
