Very Discrete Dynamical Systems

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Robert Hesse, Mathematics


The chaotic behavior of discrete dynamical systems, such as that associated with the logistic difference equation, is fairly well understood. However, what happens when this equation is modified to become very discrete? In particular, we consider the floor logistic map: Floor[4x(1-x/M)] where M = 3, 4, 5, ... and discuss the dynamics, particularly fixed points and cycles, of this map of the set of integers from 0 to M onto itself. Results show that the function exhibits modular behavior: a non-trivial fixed point occurs only when M mod 4 = 0 or 3. A theorem and proof are included. Cycles can be found by organizing iterates into matrix form and computing eigenvalues, which indicate the length of the cycles. A second theorem stating that the eigenvalues will be either 0 (which indicate eventually periodic points) or roots of unity (which indicate cycles) is written and proved. However, through 1000 cases of M, we are unable to predict the order cycles will occur in and which cycles are associated with any particular value of M. We can, however, based on results, develop a function to predict the number of cycles associated with a given M. It is unsatisfying that there is no discernible pattern to when cycles occur, but more research may uncover the solution.

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