  Taylor-Couette flow is often used as a simplified model for complex rotating
flows in the interior of stars and accretion disks. The flow dynamics in these
objects is influenced by magnetic fields. For example, quasi-Keplerian flows in
Taylor-Couette geometry become unstable to a travelling or standing wave in an
external magnetic field if the fluid is conducting; there is an instability
even when the flow is hydrodynamically stable. This magnetorotational
instability leads to the development of chaotic states and, eventually,
turbulence, when the cylinder rotation is sufficiently fast. The transition to
turbulence in this flow can be complex, with the coexistence of parameter
regions with spatio-temporal chaos and regions with quasi-periodic behaviour,
involving one or two additional modulating frequencies. Although the unstable
modes of a periodic flow can be identified with Floquet analysis, here we adopt
a more flexible equation-free data-driven approach. We analyse the data from
the transition to chaos in the magnetized Taylor-Couette flow and identify the
flow structures related to the modulating frequencies with Dynamic Mode
Decomposition; this method is based on approximating nonlinear dynamics with a
linear infinite-dimensional Koopman operator. With the use of these structures,
one can construct a nonlinear reduced model for the transition.
