Document Type


Date of Award



College of Science, Engineering, and Technology (COSET)

Degree Name

MS in Mathematics

First Advisor

Robert Nehs


The objective of our research is to analyze the periodic nature of solutions of the second order maxi-type difference equation: xn+1 = max{f(xn),[(xn-1)}, n = 0,1,2,3, ... (1) where f is a function of order p = 2, 3 or 4 and max is the maximum function; max{x, y} = the larger of x and y. Our conjecture that every order 2, 3, and 4 functions produced solutions that were eventually periodic was later confirmed. We also observed that the length of the eventual cycle depended on the order of the function. The value of the order p of f determined the bounds for the lengths of the cycles within each solution. As a consequence, we offer a method that will efficiently determine the solutions to the system (1) thereby, allowing us to determine the lengths of the cycle for higher order functions. We also introduced order-vector notation that will help to simplify this process. In summary, Theorem 4.1 states that every order 2 function produces an eventual3-cycle solution. Moreover, Theorem 4.2 asserts that every order 3 function yields an eventual4-cycle and 5-cycle solutions. Finally, Theorem 4.3 affirms that every order 4 function generates an eventual 5-cycle, 6-cycle, and 7-cycle solutions