Dynamical Behaviors Of Some Difference Equations And The Heat Equation

As numerical solutions of differential equations or simulations of discrete systems, difference equations are widely used in various subjects of natural science and social science, including biology, economics, ecology and medical science etc. People study the properties, such as the oscillation, boundedness, asymptotical stability and periodicity etc., for the solutions of mathematical models defined by various kinds of difference equations.However, because of the complexity of the dynamics defined by the difference equations and the extremely difficult for the corresponding research work, quite a lot of references confine themselves to investigate those difference equations with relatively simpler solution structures such as possessing a unique positive equilibrium, which is globally asymptotically stable. More references are restricted to the study of the oscillation for the solution of the difference equations. Few references are concerned with the existence of the invariant manifolds of the difference equations, particularly, the existence and the attractiveness of the center manifold of the positive equilibrium. While, it is exactly the later, which is very important in the theory of the dynamical systems.In this thesis, we first investigate that the solutions of the rational equationare globally attracted to the center manifold consisting of all the 2-periodic orbits, by defining a new kind of distance. And we conclude that any orbit starting from the first quadrant monotonously approaches to the central manifold under the meaning of the distance mentioned above, and the center manifold is globally asymptotically stable.Secondly, we conclude that the rational equationhas a unique positive equilibrium which possesses an 1-dimensional center manifold filled with the 2-periodic orbits, and the center manifold is asymptotically stable. The complex dynamical behavior, such as the global attractiveness, is discussed. The numerical computation shows that the center manifold is also globally asymptotically stable.At last, we investigate some properties of the sub-solution for the heat equation. Some conclusions and their proofs are given.