Connecting Orbits And Topological Entropy Of Natural Hamiltonian Systems

In this thesis, we study the existence of heteroclinic orbits of natural Hamiltonian systems, Jacobian metrics of the configuration manifolds and existence of integrable natural Hamiltonian systems with positive topological entropy.In the first part, we will study the existence of heteroclinic orbits of natural Hamiltonian systems. In [Ann. Inst. H. Poincare Anal. Non Lineaire 6(1989),331-346] Rabinowitz investigated the existence of heteroclinic orbits for a second order autonomous Hamil-tonian system. His method strongly depends on the autonomous condition. In 2007 Izydorek and Janczewska [J. Diff. Eqns.238(2007), 381-393] provided a new tool to extend the Rabinowitz's result to non-autonomous case.We will study non-autonomous Hamiltonian systems on Rie-mannian manifold (M, g) where (gij)n×n is the inverse matrix of the Riemannian metric g= (gij)n×n, and (p,q) is the local coordinate on the cotangent bundle T*M. With the help of variational methods and critical point the-ory, by using properties of geodesics and the relations between weak convergence and strong convergence in Hilbert spaces, we proved the existence of heteroclinic orbit of the Hamiltonian systems un-der some suitable conditions. This result improves the results of Rabinowitz and Izydorek-Janczewska in two aspects. First our re-sult is presented on Riemannian manifolds, while theirs were given in Euclidean spaces. Second our conditions are more general than theirs, and so some Hamiltonian systems which cannot be applied by theirs, can be studied by ours.In the second part, we study the geometry and topology of the configuration manifold, on which the natural Hamiltonian system has positive topological entropy. Consider Hamiltonian system on where S1 is 1-dimensional torus and N is a closed compact Rie-mannian manifold. Based on the work of Bolotin and Rabinowitz given in [J. Diff. Eqns.148(1998),364-387], we prove that there exists a real numberδ> 0, such that for 0< h<δ, if either the Hamiltonian flow on the isoenergy level{H(p, q)= h} is ergodic, or the fundamental group of N is subexponential growth, then the manifold (M, (h - V)g) is not a nonpositively curvature one.In the third part, we study the existence of integrable nature Hamiltonian systems with positive topological entropy. Topologi-cal entropy is a notion to indicate the complexity of dynamics of a system. In the past ones conjectured that integral Hamiltonian sys-tems have vanishing topological entropy. Bolsinov and Taimanov [Invent. Math.140(2000),639-650] constructed the first example of three-dimensional Riemannian manifold on which there exists a smooth integrable geodesic flow having positive topological entropy.Using the properties of fundamental groups and hyperbolic toral automorphisms, we construct a class of Liouville integrable natural Hamiltonian systems which have positive topological en-tropy. As our knowledge, this is the first example of C∞integrable natural Hamiltonian system with potential energy having positive topological entropy.