Mathematical Sciences: A Research Initiation for Undergraduates Students: Difference Equations and Chaos

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An REU site project supporting six students will receive funding through this award. Research directed toward problems arising in the field of difference equations with a focus on understanding chaotic behavior of solutions will be carried out. Difference equations occur naturally in mathematical analysis from many sources. One of the most important occurs in discrete approximations to differential equations in which derivatives are replaced by differences. The oscillation theory of these equations requires very little mathematical background and the study of dynamical properties of first order difference equations can be reduced to the study of iterations of continuous functions over finite intervals. This project will emphasize two points of view. The first concerns qualitative properties of higher order difference equations, especially those of third and fourth order. The second thrust of the project will be the study of first order difference equations as examples of simple dynamical systems, where the orbit structure and bifurcation properties of specific families of equations will be taken up. The importance of difference equations in the study of mechanical vibrations, mathematical biology and economics will be emphasized.