The Dynamics Of Integrable Hamilton Systems And Of 3-dimensional Systems Having An Invariant Algebraic Surface

Integrable Hamiltonian system is an important branch of nonlinear science. They are applied widely in mechanics, acoustics, optics, biology, life sciences, society sciences, etc. Specially, in the fields of biology, astrodynamics, spaceflight engineering technology, lots of models are constructed in integrable Hamiltonian systems or their perturbations. In addition, large numbers of chaotic phenomena, nonlinear oscillations, biology mathematics models are described in 3-dimensional systems such as Lorenz systems, Rabinovich systems, Chen systems, Rikitake systems and Lotka-Volterra systems, etc. Although these systems look simple, but in fact their dynamics are extremely complex. Up to now, their dynamics cannot be understood clearly. To simplify the questions, ones usually study the system having invariants, for instance first integrals or invariant algebraic surfaces.For studying dynamical systems, ones are very interested in whether the systems have invariants. The study on the existence of invariants of a system can be go back to Poincare's and Hilbert's era. If a system has a first integral, then its study can be reduced in one dimension. If a system is integrable, it is possible to characterize its dynamics globally. If a system has an invariant algebraic surface, it is helpful to study the dynamics in whole space by studying the dynamics on the invariant algebraic surface. It is valuable to study the Liouvillian integrability of Hamiltonian systems and their topology, geometry and algebra and to study 3-dimensional systems having an invariant algebraic surface. But as Poincare had realized, it is difficult to find invariant algebraic surfaces and first integrals for a given system.In this paper, we study some integrable Hamiltonian systems and their topological entropy on special Riemannian manifolds, and the global topological structure of orbits of some famous 3-dimension systems having an invariant algebraic surface.In chapter 1, we introduce Hamiltonian systems associated with their integrability and topological entropy, and 3-dimensional systems having an invariant algebraic surface.In chapter 2, we characterize the Liouvillian integrable orthogonal separable Hamiltonian systems on T² for a given metric, and prove that the Hamiltonian flow restricted to any compact level hypersurface has zero topological entropy. Furthermore, by examples we show that the integrable Hamiltonian systems on T² can have complicated dynamical phenomena. For instance they can have several families of invariant tori, each family is bounded by the homoclinic-loop-like cylinders and heteroclinic-loop-like cylinders. As we know, it is the first concrete example to present families of invariant tori at the same time appearing in such a complicated way.In chapter 3, we will first characterize C^∞smoothly orthogonally separable Hamiltonian Systems with a potential energy on T²Ã—[0,1]. Then using these integrable Hamiltonian systems on T²Ã—[0,1] we obtain a class of integrable Hamiltonian systems on the Riemannian manifold M_A.Moreover, we prove that for the integrable natural Hamiltonian H with total energy no less than e_H there exists a subsetΩ(?)D:= {e∈R; e≥e_H} of Lebesgue measure zero such that the Hamiltonian flow restricted to each energy surface {H = e} with e∈D\Ωhas a positive topological entropy. As a result, we obtain the first example, as our knowledge, of C^∞Liouvillian integrable natural Hamiltonian flows on a Riemannian manifold which has a positive topological entropy.In chapter 4, we give the characterization of the integrable natural Hamiltonian systems, which orthogonal separable on Tⁿ×[0,1], with the configuration space a three dimensional quotient manifold induced by the Anosov map. Moreover, we prove that the Hamiltonian flow restricted to suitable regular energy surfaces has a positive topological entropy.In chapter 5, we characterize the global topological structure of orbits of Rabinovich system (?)= hy -v₁x + yz,(?) = hx - v₂y - xz,(?)= -v₃z + xy and the Chen system (?) = a(y - x), (?) = (c - a)x - xz + cy, (?) = xy - bz having an invariant algebraic surface. We complete the classification of dynamics of these two systems.