This paper introduces a Maslov-type index theory for paths in the symplectic groups, especially for the degenerate paths via rotational perturbation method, therefore gives a full classification of the linear Hamilton...This paper introduces a Maslov-type index theory for paths in the symplectic groups, especially for the degenerate paths via rotational perturbation method, therefore gives a full classification of the linear Hamiltonian systems with continuous, periodic, and symmetric coefficients. Associating this index with each periodic solution, we establish the existence of muhiple periodic solutions of asymptotically linear Hamihonian systems.展开更多
A symplectic matrix M is singular, if det(M-I)=0. In this paper we study the struc-ture of the singular set of symplectic mtrices. We discuss the changes of the dimension ofthe null space and the determinant of the di...A symplectic matrix M is singular, if det(M-I)=0. In this paper we study the struc-ture of the singular set of symplectic mtrices. We discuss the changes of the dimension ofthe null space and the determinant of the difference between a singular symplectic matrixand the identity matrix under rotational perturbations. The results obtained will be used todefine a Maslov-type index theory for (degenerate) paths in symplectic groups, and thereforeto establish the existence of periodic solutions of asymptotically linear Hamiltonian systems.展开更多
In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula of C. Viterbo for no...In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula of C. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R2n with n ＞ 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics when n=2.展开更多
基金Partially supported by NSF (10801079)Partially supported by RFDP (20080551002)+1 种基金Partially supported by LPMC of MOE of ChinaPartially supported by the 973 Program of MOST, NNSF, MCME, RFDP, LPMC of MOE of China, S. S. Chern Foundation, and Nankai University
文摘In this paper, the concavity of closed geodesics proposed by M. Morse in 1930s is studied.
文摘This paper introduces a Maslov-type index theory for paths in the symplectic groups, especially for the degenerate paths via rotational perturbation method, therefore gives a full classification of the linear Hamiltonian systems with continuous, periodic, and symmetric coefficients. Associating this index with each periodic solution, we establish the existence of muhiple periodic solutions of asymptotically linear Hamihonian systems.
基金Project supported by the National Natural Science Foundation of China.
文摘A symplectic matrix M is singular, if det(M-I)=0. In this paper we study the struc-ture of the singular set of symplectic mtrices. We discuss the changes of the dimension ofthe null space and the determinant of the difference between a singular symplectic matrixand the identity matrix under rotational perturbations. The results obtained will be used todefine a Maslov-type index theory for (degenerate) paths in symplectic groups, and thereforeto establish the existence of periodic solutions of asymptotically linear Hamiltonian systems.
基金This work was partially supported by the 973 Program of the Ministryof Science and Technology, the Mathematical Center of the Ministry of Education, the Research Fund for the Doctorial Program of High Education, the Research Fund for the Doctorial Prog
文摘In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula of C. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R2n with n ＞ 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics when n=2.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10071040)Ph.D. Fund of ME of China, PMC Key Lab of ME of China, the 973 Program of STM, RFDP, MCME of China, CEC of Tianjin, S. S. Chern Foundation.
