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
Abstract In the paper,a concept of the essential numerical range W_(e)(T)of a linear relation T in a Hilbert space is given,other various essential numerical ranges W_(ei)(T),i=1,2,3,4,are introduced,and relationships...Abstract In the paper,a concept of the essential numerical range W_(e)(T)of a linear relation T in a Hilbert space is given,other various essential numerical ranges W_(ei)(T),i=1,2,3,4,are introduced,and relationships among W_(e)(T)and W_(ei)(T)are established.These results generalize relevant results obtained by Bogli et al.in[Bogli,S.,Marletta,M.and Tretter,C.,The essential numerical range for unbounded linear operators,J.Funct.Anal.,279,2020,47-12].Moreover,several fundamental properties of closed relations related to its operator parts are presented.In addition,singular discrete linear Hamiltonian systems including non-symmetric cases are considered,several properties for the associated minimal relations H_(0)are derived,and the above results for abstract linear relations are applied to H_(0).展开更多
In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows ar...In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows are called symplectic intoprators. Several symplectic methods are known in the class of Runge-Kutta methods. However, no higll order symplectic integrators are known in the class of Linear Multistep Methods (LMMs). Here, by using LMMs as Boundary Value Methods (BVMs), we show that symplectic integrators of arbitrary high order are also available in this class. Moreover, these methods can be used to solve both initial and boundary value problems. In both cases, the properties of the flow of Hamiltonian systems are 'essentially' maintained by the discrete map, at least for linear problems.展开更多
In this paper,a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations,which mainly originate from...In this paper,a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations,which mainly originate from parameters dispersions and measurement errors.Taking the perturbations into account,the perturbed system is regarded as a modification of the nominal system.By combining the perturbation series expansion method with the deterministic linear Hamiltonian system,the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived.Eventually,the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes.The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system.The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm.Numerical examples illustrate the superiority of the proposed method in accuracy and stability,especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.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
A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic...A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic model and the continuously stratified baroclinic model. Since this system can simulate the baroclinic effect simply, it is widely used to study the large-scale dynamic process in atmosphere and ocean. The present paper is concerned with the linear stability of the multilayer quasi-geostrophic flow, and the associated linear stability criteria are established. Firstly, the nonlinear model is turned into the form of a Hamiltonian system, and a basic flow is defined. But it cannot be an extreme point of the Hamiltonian function since the system is an infinite-dimensional one. Therefore, it is necessary to reconstruct a new Hamiltonian function so that the basic flow becomes an extreme point of it. Secondly, the linearized equations of disturbances in the multilayer quasi-geostrophic flow are derived by introducing infinitesimal disturbances superposed on the basic flows. Finally, the properties of the linearized system are discussed, and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms.展开更多
The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessar...The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessary and sufficient condition for an autonomous system to have a representation in terms of linear autonomous Birkhoff equations is obtained. The methods of constructing Birkhoffian dynamical functions are given. Two examples are given to illustrate the application of the results.展开更多
This paper establishes several new Lyapunov-type inequalities for the system of nonlinear difference equations■,which extend/supplement and improve some related existing ones.
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supported by the National Natural Science Foundation of China(No.11971262)Shandong Province Natural Science Foundation(No.ZR2019MA038,ZR2024QA088)。
文摘Abstract In the paper,a concept of the essential numerical range W_(e)(T)of a linear relation T in a Hilbert space is given,other various essential numerical ranges W_(ei)(T),i=1,2,3,4,are introduced,and relationships among W_(e)(T)and W_(ei)(T)are established.These results generalize relevant results obtained by Bogli et al.in[Bogli,S.,Marletta,M.and Tretter,C.,The essential numerical range for unbounded linear operators,J.Funct.Anal.,279,2020,47-12].Moreover,several fundamental properties of closed relations related to its operator parts are presented.In addition,singular discrete linear Hamiltonian systems including non-symmetric cases are considered,several properties for the associated minimal relations H_(0)are derived,and the above results for abstract linear relations are applied to H_(0).
文摘In this paper we are concerned with finite difference schemes for the numerical approximation of linear Hamiltonian systems of ODEs. Numerical methods which preserves the qualitative properties of Hamiltonian flows are called symplectic intoprators. Several symplectic methods are known in the class of Runge-Kutta methods. However, no higll order symplectic integrators are known in the class of Linear Multistep Methods (LMMs). Here, by using LMMs as Boundary Value Methods (BVMs), we show that symplectic integrators of arbitrary high order are also available in this class. Moreover, these methods can be used to solve both initial and boundary value problems. In both cases, the properties of the flow of Hamiltonian systems are 'essentially' maintained by the discrete map, at least for linear problems.
基金the National Nature Science Foundation of China(No.11772026)the Defense Industrial Technology Development Program(Nos.JCKY2016204B101,JCKY2018601B001)+1 种基金the Beijing Municipal Science and Technology Commission via project(No.Z191100004619006)the Beijing Advanced Discipline Center for Unmanned Aircraft System for the financial supports.
文摘In this paper,a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations,which mainly originate from parameters dispersions and measurement errors.Taking the perturbations into account,the perturbed system is regarded as a modification of the nominal system.By combining the perturbation series expansion method with the deterministic linear Hamiltonian system,the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived.Eventually,the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes.The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system.The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm.Numerical examples illustrate the superiority of the proposed method in accuracy and stability,especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.
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 supported by the National Natural Science Foundation of China(Grant Nos.41575026,41275113,and 41475021)
文摘A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic model and the continuously stratified baroclinic model. Since this system can simulate the baroclinic effect simply, it is widely used to study the large-scale dynamic process in atmosphere and ocean. The present paper is concerned with the linear stability of the multilayer quasi-geostrophic flow, and the associated linear stability criteria are established. Firstly, the nonlinear model is turned into the form of a Hamiltonian system, and a basic flow is defined. But it cannot be an extreme point of the Hamiltonian function since the system is an infinite-dimensional one. Therefore, it is necessary to reconstruct a new Hamiltonian function so that the basic flow becomes an extreme point of it. Secondly, the linearized equations of disturbances in the multilayer quasi-geostrophic flow are derived by introducing infinitesimal disturbances superposed on the basic flows. Finally, the properties of the linearized system are discussed, and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10932002,11172120,and 11202090)
文摘The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessary and sufficient condition for an autonomous system to have a representation in terms of linear autonomous Birkhoff equations is obtained. The methods of constructing Birkhoffian dynamical functions are given. Two examples are given to illustrate the application of the results.
基金supported by the NNSF of China(Grant No.41405083)Hunan Provincial Natural Science Foundation of China(Grant No.2015JJ3098)
文摘This paper establishes several new Lyapunov-type inequalities for the system of nonlinear difference equations■,which extend/supplement and improve some related existing ones.
基金supported in part by the E-Institutes of Shanghai Municipal Education Commission(E03004)the NSF of China(11071170)+2 种基金the Ministry of Education of China(211058)the Specialized Research Fund for the Doctoral Program of Higher Education(20113127110003)Shanghai Municipal Education Commission(11ZZ118)
