  In classical mechanics, solutions can be classified according to their
stability. Each of them is part of the possible trajectories of the system.
However, the signatures of unstable solutions are hard to observe in an
experiment, and most of the times if the experimental realization is adiabatic,
they are considered just a nuisance. Here we use a small number of XY magnetic
dipoles subject to an external magnetic field for studying the origin of their
collective magnetic response. Using bifurcation theory we have found all the
possible solutions being stable or unstable, and explored how those solutions
are naturally connected by points where the symmetries of the system are lost
or restored. Unstable solutions that reveal the symmetries of the system are
found to be the culprit that shape hysteresis loops in this system. The
complexity of the solutions for the nonlinear dynamics is analyzed using the
concept of boundary basin entropy, finding that the damping time scale is
critical for the emergence of fractal structures in the basins of attraction.
Furthermore, we numerically found domain wall solutions that are the smallest
possible realizations of transverse walls and vortex walls in magnetism. We
experimentally confirmed their existence and stability showing that our system
is a suitable platform to study domain wall dynamics at the macroscale.
