Positive nonlinear systems, response maps and realizations

Positivity of a nonlinear control system with output for an arbitrary time scale is studied. This includes in particular continuous-time and discrete-time systems. Positivity of the system is characterized with the aid of natural operators associated to the system: a shift and a skew derivation. For different time scales the criteria take different forms. The results on positivity are then used to characterize globally positive realizability of nonlinear response maps.