  We consider propagation of transverse magnetic electromagnetic waves through
a nonlinear metamaterial slab, of subwavelength thickness and very small
average dielectric permittivity, made up of alternating metal and dielectric
nonlinear layers. Exploiting the effective-medium approach, we prove that the
output field intensity is a multi-valued function of both the input field
intensity and incidence angle so that the transmissivity exhibits a novel kind
of multi-stability and a pronounced two-parameter hysteresis behavior. The
predicted two-parameter hysteresis physically stems from the fact that, since
the linear and nonlinear contributions to the overall dielectric response can
be comparable, the slab can host more than one electromagnetic configuration
compatible with the incident field and the boundary conditions, regardless of
the slab thickness. We investigate slab transmission also through full wave
simulations proving the robustness of the two-parameter hysteresis beyond the
effective-medium approximation.
