  The present work studies the non-linear dynamics of a shear layer, driven by
a body force and confined between parallel walls, a simplified setting to study
transitional and turbulent shear layers. It was introduced by Nogueira \&
Cavalieri (J. Fluid Mech. 907, A32, 2021), and is here studied using a
reduced-order model based on a Galerkin projection of the Navier-Stokes system.
By considering a confined shear layer with free-slip boundary conditions on the
walls, periodic boundary conditions in streamwise and spanwise directions may
be used, simplifying the system and enabling the use of methods of dynamical
systems theory. A basis of eight modes is used in the Galerkin projection,
representing the mean flow, Kelvin-Helmholtz vortices, rolls, streaks and
oblique waves, structures observed in the cited work, and also present in shear
layers and jets. A dynamical system is obtained, and its transition to chaos is
studied. Increasing Reynolds number $Re$ leads to pitchfork and Hopf
bifurcations, and the latter leads to a limit cycle with amplitude modulation
of vortices, as in the DNS by Nogueira \& Cavalieri. Further increase of $Re$
leads to the appearance of a chaotic saddle, followed by the emergence of
quasi-periodic and chaotic attractors. The chaotic attractors suffer a merging
crisis for higher $Re$, leading to chaotic dynamics with amplitude modulation
and phase jumps of vortices. This is reminiscent of observations of coherent
structures in turbulent jets, suggesting that the model represents dynamics
consistent with features of shear layers and jets.
