In this paper,we give the definition of Maslov-type index of the discrete Hamiltonian system,and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of...In this paper,we give the definition of Maslov-type index of the discrete Hamiltonian system,and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of the caseω=1 toω∈U degenerate case via direct method which is different from that of the known literatures.Moreover the well-posedness of the splitting numbers Sh,ω±is proven,then the index iteration theories of Bott and Long are also valid for the discrete case,and those can be also applied to the study of the symplectic algorithm.展开更多
<正> 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 line...<正> 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
In this paper, we consider the relation of the Morse index of a closedgeodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closedgeodesic c on a Riemannian manifold M with its lin...In this paper, we consider the relation of the Morse index of a closedgeodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closedgeodesic c on a Riemannian manifold M with its linear Poincare map P (a symplectic matrix), weconstruct a symplectic path γ(t) starting from identity I and ending at P, such that the Morseindex of the closed geodesic c equals the Maslov-type index of γ. As an application of this result,we study the parity of the Morse index of any closed geodesic.展开更多
Based on the spectral now and the stratification structures of the symplectic group Sp(2n, c) , the Maslov-type index theory and its generalization, the w-index theory parameterized by all ω on the unit circle, for a...Based on the spectral now and the stratification structures of the symplectic group Sp(2n, c) , the Maslov-type index theory and its generalization, the w-index theory parameterized by all ω on the unit circle, for arbitrary paths in Sp(2n, C) are established. Then the Bott-type iteration formula of the Maslov-type indices for iterated paths in Sp(2n, C) is proved, and the mean index for any path in Sp(2n, C) is defined. Also, the relation among various Maslov-type index theories is studied.展开更多
In this paper, we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian s...In this paper, we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian systems.展开更多
The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space.Then a direct proof of the iteration formulae for the Masl...The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space.Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given.展开更多
In this article, we study the existence of nontrivial solutions for a class of asymptotically linear Hamiltonian systems with Lagrangian boundary conditions by the Galerkin approximation methods and the L-index theory...In this article, we study the existence of nontrivial solutions for a class of asymptotically linear Hamiltonian systems with Lagrangian boundary conditions by the Galerkin approximation methods and the L-index theory developed by the first author.展开更多
In this paper,we prove that for each positive k≡1 mod m there exists a P-symmetric kmτ-periodic solution xk for asymptotically linear mτ-periodic Hamiltonian systems,which are nonautonomous and endowed with a P-sym...In this paper,we prove that for each positive k≡1 mod m there exists a P-symmetric kmτ-periodic solution xk for asymptotically linear mτ-periodic Hamiltonian systems,which are nonautonomous and endowed with a P-symmetry.If the P-symmetric Hamiltonian function is semi-positive,one can prove,under a new iteration inequality of the Maslov-type P-index,that xk_(1) and xk_(2) are geometrically distinct for k_(1)/k_(2)≥(2n+1)m+1;and xk_(1),xk_(2) are geometrically distinct for k_(1)/k_(2)≥m+1 provided xk_(1) is non-degenerate.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
The authors establish the existence of nontrival periodic solntions of the asymptotically linear Hamiltonian systems in the general case that the asymptotic matrix may be degenerate and time-dependent.This is done by ...The authors establish the existence of nontrival periodic solntions of the asymptotically linear Hamiltonian systems in the general case that the asymptotic matrix may be degenerate and time-dependent.This is done by using the critical point theory,Galerkin approximation procedure and the Maslov-type index theory introduced and generalized by Conley,Zehnder and Long.展开更多
We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system ...We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.展开更多
文摘In this paper,we give the definition of Maslov-type index of the discrete Hamiltonian system,and obtain the relation of Morse index and Maslov-type index of the discrete Hamiltonian system which is a generalization of the caseω=1 toω∈U degenerate case via direct method which is different from that of the known literatures.Moreover the well-posedness of the splitting numbers Sh,ω±is proven,then the index iteration theories of Bott and Long are also valid for the discrete case,and those can be also applied to the study of the symplectic algorithm.
文摘<正> 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 10071040 supported by NNSF,200014 supported by Excellent.Ph.D.Funds of ME of ChinaPMC Key Lab.of ME of China
文摘In this paper, we consider the relation of the Morse index of a closedgeodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closedgeodesic c on a Riemannian manifold M with its linear Poincare map P (a symplectic matrix), weconstruct a symplectic path γ(t) starting from identity I and ending at P, such that the Morseindex of the closed geodesic c equals the Maslov-type index of γ. As an application of this result,we study the parity of the Morse index of any closed geodesic.
基金National Natural Science Foundation of China MCSEC of China Qiu Shi Science and Technology Foundation.
文摘Based on the spectral now and the stratification structures of the symplectic group Sp(2n, c) , the Maslov-type index theory and its generalization, the w-index theory parameterized by all ω on the unit circle, for arbitrary paths in Sp(2n, C) are established. Then the Bott-type iteration formula of the Maslov-type indices for iterated paths in Sp(2n, C) is proved, and the mean index for any path in Sp(2n, C) is defined. Also, the relation among various Maslov-type index theories is studied.
基金This work was partially supported by the National Natural Science Foundation of China (Grant No.20060390014)
文摘In this paper, we study the Maslov-type index theory for linear Hamiltonian systems with brake orbits boundary value conditions and its applications to the existence of multiple brake orbits of nonlinear Hamiltonian systems.
基金supported by the National Natural Science Foundation of China(Nos.11221091,11471169)the Key Laboratory of Pure Mathematics and Combinatorics,the Ministry of Education of China
文摘The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space.Then a direct proof of the iteration formulae for the Maslov-type indices of symplectic paths is given.
基金Partially supported by NFS of China (11071127, 10621101)973 Program of STM (2011CB808002)
文摘In this article, we study the existence of nontrivial solutions for a class of asymptotically linear Hamiltonian systems with Lagrangian boundary conditions by the Galerkin approximation methods and the L-index theory developed by the first author.
基金Partially supported by the National Natural Science Foundation of China(11226156)the “New Start” Academic Research Projects of Beijing Union University(ZK201218,ZK10201304)
基金Partially supported by the National Natural Science Foundation of China(11226156)“New Start”Academic Research Projects of Beijing Union University(ZK201218)
基金partially supported by National Key R&D Program of China(Grant No.2020YFA0713300)NSFC Grants(Grant Nos.17190271 and 11171341)+2 种基金LPMC of Nankai Universitypartially supported by the NSFC Grants(Grant Nos.12171253 and 17190271)LPMC of Nankai University。
文摘In this paper,we prove that for each positive k≡1 mod m there exists a P-symmetric kmτ-periodic solution xk for asymptotically linear mτ-periodic Hamiltonian systems,which are nonautonomous and endowed with a P-symmetry.If the P-symmetric Hamiltonian function is semi-positive,one can prove,under a new iteration inequality of the Maslov-type P-index,that xk_(1) and xk_(2) are geometrically distinct for k_(1)/k_(2)≥(2n+1)m+1;and xk_(1),xk_(2) are geometrically distinct for k_(1)/k_(2)≥m+1 provided xk_(1) is non-degenerate.
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.
文摘The authors establish the existence of nontrival periodic solntions of the asymptotically linear Hamiltonian systems in the general case that the asymptotic matrix may be degenerate and time-dependent.This is done by using the critical point theory,Galerkin approximation procedure and the Maslov-type index theory introduced and generalized by Conley,Zehnder and Long.
基金Acknowledgements The authors would like to thank the referees for their critical reading and very helpful comments and suggestions. This work was partially supported by the National Natural Science Foundation of China （Grant Nos. 11221091, 11471169） and LPMC of MOE of China.
文摘We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.
