scholarly journals Exponential Discrete Gradient Schemes for Simulating Stochastic Hamiltonian Systems Preserving Hamiltonian Functions

2018 ◽  
Author(s):  
Jia-lin RUAN ◽  
Li-jin WANG
Keyword(s):  
Hamiltonian Systems ◽  
Discrete Gradient ◽  
Stochastic Hamiltonian Systems ◽  
Hamiltonian Functions

High-Order Symplectic Schemes for Stochastic Hamiltonian Systems

2014 ◽  
Vol 16 (1) ◽  
pp. 169-200 ◽  
Author(s):  
Jian Deng ◽  
Cristina Anton ◽  
Yau Shu Wong
Keyword(s):  
Hamiltonian Systems ◽  
High Order ◽  
Symplectic Integrators ◽  
Symplectic Methods ◽  
Computational Tools ◽  
Fully Implicit ◽  
Stochastic Hamiltonian Systems ◽  
Hamiltonian Functions ◽  
Symplectic Schemes

AbstractThe construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochastic Hamiltonian systems preserving Hamiltonian functions, the high-order symplectic methods have simpler forms than the explicit Taylor expansion schemes. A theoretical analysis of the convergence and numerical simulations are reported for several symplectic integrators. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

scholarly journals Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

2021 ◽  
Author(s):  
Wojciech Szumiński ◽  
Andrzej J. Maciejewski
Keyword(s):  
Hamiltonian Systems ◽  
Gaussian Curvature ◽  
Necessary Conditions ◽  
Constant Gaussian Curvature ◽  
Local Coordinates ◽  
Hamiltonian Functions

AbstractIn the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

scholarly journals Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems with Some Twisted Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Qi Wang ◽  
Qingye Zhang
Keyword(s):  
Hamiltonian Systems ◽  
Index Theory ◽  
Maslov Index ◽  
Homoclinic Orbits ◽  
Asymptotically Linear ◽  
Existence And Multiplicity ◽  
Hamiltonian Functions

By the Maslov index theory, we will study the existence and multiplicity of homoclinic orbits for a class of asymptotically linear nonperiodic Hamiltonian systems with some twisted conditions on the Hamiltonian functions.

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