Study On The Lyness Difference Equation

A class of difference equation named Lyness equation is studied in this paper, so are the same class of two examples. Through studying on the two examples, several properties, such as oscillation, periodicity and global attractivity of this class of difference equation are discussed.Numerical computations and simulations are used to study the periodicity of a new revision of a class of Lyness equation. An IngegerPart function was set to compare to. Through this method, different iteration values of the new revision of a class of Lyness equation with different conditions are given. Several different graphs are given which based on the different values. From the graphs, we can find out the periodicity of the new revision of a class of Lyness equation directly. Numerical computations and simulations show that the new revision of a class of Lyness equation has some rule along with the increasing of the order. That is, when it's the k th order Lyness equation, the period of the k th order Lyness equation is 3k + 2. Also, Through the numerical computations and simulations, we find that when the computer run many steps, the graphs show some periodicity. From this, we extrapolate that after some step, the data of the k th order Lyness equation has periodicity, which are right for all k∈N. And then, theoretical analysis on the new revision of a class of Lyness equation are given, when the k th order is small. Every solution of the equation is solved when k is zero.The following remark gives an interesting discovery. That is, the same periodicity is found between the WU-XING systems of our country and the Lyness equation which we studied in the paper. We find that the theory of the WU-XING systems suggested a scientific judgment about periodicity, which about the same 5-period.Finally, another nonlinear Lyness equation is studied. We firstly give the linearization of the equation, and then find out a necessary and sufficient condition for the oscillation of the nonlinear Lyness equation. And then we show that the equation has no positive solution with prime period two. Finally, we show that the equation is permanent, and the unique positive equilibrium of the equation is a global attractor in the invariant interval.