An investigation of the parameter space for a family of dissipative mappings.  The parameter plane investigation for a family of two-dimensional, nonlinear, and area contracting map is made. Several dynamical features in the system such as tangent, period-doubling, pitchfork, and cusp bifurcations were found and discussed together with cascades of period-adding, period-doubling, and the Feigeinbaum scenario. The presence of spring and saddle-area structures allow us to conclude that cubic homoclinic tangencies are present in the system. A set of complex sets such as streets with the same periodicity and the period-adding of spring-areas are observed in the parameter space of the mapping.