What is the effective impact of the explosive orbital growth in discrete-time one-dimensional polynomial dynamical systems?

We study the distribution of periodic orbits in one-dimensional two-parameter maps. Specifically, we report an exact expression to quantify the growth of the number of periodic orbits for discrete-time dynamical systems governed by polynomial equations of motion of arbitrary degree. In addition, we compute high-resolution phase diagrams for quartic and for both normal forms of cubic dynamics and show that their stability phases emerge all distributed in a similar way, preserving a characteristic invariant ordering. Such coincidences are remarkable since our exact expression shows the total number of orbits of these systems to differ dramatically by more than several millions, even for quite low periods. All this seems to indicate that, surprisingly, the total number and the distribution of stable phases is not significantly affected by the specific nature of the nonlinearity present in the equations of motion.