Dynamical Behaviors Of Some Differential, Difference Equations

This thesis is devoted to the study of dynamical behaviors of some differential, difference equations. The research work focuses mainly on two parts. The first part discusses dynamical behaviors near the heteroclinic or homoclinic cycle under reversible conditions. The second part investigates the qualitative properties of some higher-order rational difference equations. The fundamental contributions of these works are summarized in the following aspects:Homoclinic and heteroclinic orbits are important in dynamical systems. From their existence one may, under certain conditions, infer the existence of chaos nearby or the bifurcation behaviors of periodic orbits. There are more homoclinic or heteroclinic orbits in reversible system. This thesis considers some 4-dimensional reversible systems. First we study a reversible system which has a heteroclinic loop connecting a saddle-focus and a saddle. Under some generic conditions, we prove the existence of a countable set of reversible 1-homoclinic orbits to the saddle, 2- and 4-homoclinic orbits to the saddle-focus, and the nonexistence of 2-homoclinic orbits to the saddle. Moreover, each 2-homoclinic orbit is the limit of a family of reversible 2-periodic orbits. The intersections of those 2-periodic orbits with Fix(R) are spiral segments, while one or two ends of them are the intersections of 2-homoclinic orbits with Fix(R).Then we study the dynamical behaviors near a homoclinic cycle to a first-order fine saddle-focus. We clarify the existence of denumerable (2n+1)- and 2ⁿ-homoclinic orbits and the existence of countable families of reversible (2n + 1)- and 2ⁿ-periodic orbits near the homoclinic loop. Because the fine saddle-focus is not structurally stable, we also study the bifurcation of homoclinic orbits accompanying the Hopf bifurcation of the singularity. We prove that this system yields three periodic orbits Γ_s, Γ_u and Γ₀ under R-invariant perturbation, and those orbits lie on stable manifold, unstable manifold and Fix(R) respectively. At the same time, the system has continuum of heteroclinic orbits connecting Γ_s and Γ_u and a homoclinic orbit to Γ₀. While under non-R-invariant perturbation, this system yields one periodic orbit Γ_u¹ near the singularity, continuum of heteroclinic orbits connecting Γ_u¹ and O and one homoclinic orbit to O.Still then we study the globally asymptotic stability of two rational difference equa-...