We prove that every positive solution of the difference equation xn=maxi∈ 1,k Aixn-i is eventually periodic, and that the prime period is bounded for all positive initial points. A lower bound, growing faster than polynomially, on the maximum prime period for a system of size k is given, based on a model designed to generate long periods. Conditions for systems to have unbounded preperiods are given. All cases of nonpositive systems, with either the A values and/or initial x values allowed to be negative, are analyzed. For all cases conditions are given for solutions to exist, for the system to be bounded, and for it to be eventually periodic. Finally, we examine several other difference systems, to see if the methods developed in this paper can be applied to them.
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