Explicit solutions and periodicity in a class of rational difference-equation systems

We investigate four second-order rational difference-equation systems obtained from the four possible sign configurations inExplicit solution formulas are derived for the two same-sign systems.For the two mixed-sign systems, the orbit closes after four iterations.We identify explicit admissibility conditions on the initial data that guarantee global well-definedness and distinguish the statement "period dividing four" from minimal period four.The two same-sign systems have, respectively, divergence of both components to +∞ and to -∞ along every globally defined orbit.Numerical examples are included to illustrate these theoretical conclusions.