  Swept-frequency pulses have found applications in a wide range of areas
including spectroscopic techniques where efficient control of spins is
required. For many of these applications, a good understanding of the evolution
of spin systems during these pulses plays a vital role, not only in describing
the mechanism of techniques, but also in enabling new methodologies. In
magnetic resonance spectroscopy, broadband inversion, refocusing, and
excitation using these pulses are among the most used applications in NMR, ESR,
MRI, and $in$ $vivo$ MRS. In the present survey, a general expression for
chirped pulses will be introduced, and some numerical approaches to calculate
the spin dynamics during chirped pulses via solutions of the well-known
Liouville-von Neumann equation and the lesser-explored Wei-Norman Lie algebra
along with comprehensive examples are presented. In both cases, spin state
trajectories are calculated using the solution of differential equations.
Additionally, applications of the proposed methods to study the spin dynamics
during the PSYCHE pulse element for broadband homonuclear decoupling and the
CHORUS sequence for broadband excitation will be presented.
