  The phase structure of magnetized cold quark matter is analyzed in the
framework of the two-flavor Nambu-Jona-Lasinio models paying special attention
to its dependence on the model parameters as different values within the
phenomenological allowed range are considered. We first discuss the simpler
chiral limit case, and then the more realistic situation of finite current
masses. We show that in spite of the difference in the nature of some
transitions, both cases are alike and exhibit a rather rich phase structure for
a significant range of acceptable parameters. A simplification of the phase
structure is obtained as parameters leading to larger values of the dressed
quark mass in the vacuum are considered. Finally, we consider the so-called
"inverse catalysis effect" showing that in some phases it implies an actual
decrease of the order parameter as the magnetic field increases.
