In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, b...In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient- Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms （SPAs） for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge Kutta methods, this paper finds that there is an error term of order p＋l for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method （a second-order Runge-Kutta method） are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.展开更多
Based on the gradient-Hamiltonian decomposition （GHD） theory of vector fields, an algorithm （ called as GHD algorithm） is proposed in this paper. For the GHD algorithm, visual interpretations of the advantages in ...Based on the gradient-Hamiltonian decomposition （GHD） theory of vector fields, an algorithm （ called as GHD algorithm） is proposed in this paper. For the GHD algorithm, visual interpretations of the advantages in stability are given by using the eigenvalue curves. From the numerical results for linear decay systems, it reaches the conclusion that the GHD algorithm proposed in this paper has a better computational accuracy than other algorithms and presents a replication of long time qualitative properties of the underlying system.展开更多
An energy method is proposed to investigate the critical transformation condition from a Taylor cone to a cone-jet. Based on the kinetic theorem, the system power allocation and the electrohydrodynamics stability are ...An energy method is proposed to investigate the critical transformation condition from a Taylor cone to a cone-jet. Based on the kinetic theorem, the system power allocation and the electrohydrodynamics stability are discussed. The numerical results indicate that the energy of the liquid cone tip experiences a maximum value during the transformation. With the proposed jetting energy, we give the critical transformation condition under which the derivative of jetting energy with respect to the surface area is greater than or equal to the energy required to form a unit of new liquid surface.展开更多
Based on the weak Noether symmetry proposed by Mei F X, this paper discusses the weak Noether symmetry for nonholonomic system of non-Chetaev type, and presents expressions of three kinds of conserved quantities by we...Based on the weak Noether symmetry proposed by Mei F X, this paper discusses the weak Noether symmetry for nonholonomic system of non-Chetaev type, and presents expressions of three kinds of conserved quantities by weak Noether symmetry. Finally, the application of this new results is showed by a practical example.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No 10572021)the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No 20040007022)
文摘In this paper, the dissipative and the forced terms of the Duffing equation are considered as the perturbations of nonlinear Hamiltonian equations and the perturbational effect is indicated by parameter ε. Firstly, based on the gradient- Hamiltonian decomposition theory of vector fields, by using splitting methods, this paper constructs structure-preserving algorithms （SPAs） for the Duffing equation. Then, according to the Liouville formula, it proves that the Jacobian matrix determinants of the SPAs are equal to that of the exact flow of the Duffing equation. However, considering the explicit Runge Kutta methods, this paper finds that there is an error term of order p＋l for the Jacobian matrix determinants. The volume evolution law of a given region in phase space is discussed for different algorithms, respectively. As a result, the sum of Lyapunov exponents is exactly invariable for the SPAs proposed in this paper. Finally, through numerical experiments, relative norm errors and absolute energy errors of phase trajectories of the SPAs and the Heun method （a second-order Runge-Kutta method） are compared. Computational results illustrate that the SPAs are evidently better than the Heun method when e is small or equal to zero.
基金Sponsored by the National Natural Science Foundation of China (10572021)the Doctoral Programme Foundation of Institute of Higher Educationof China (20040007022)the Program for New Century Excellent Talents in University of Fujian Province
文摘Based on the gradient-Hamiltonian decomposition （GHD） theory of vector fields, an algorithm （ called as GHD algorithm） is proposed in this paper. For the GHD algorithm, visual interpretations of the advantages in stability are given by using the eigenvalue curves. From the numerical results for linear decay systems, it reaches the conclusion that the GHD algorithm proposed in this paper has a better computational accuracy than other algorithms and presents a replication of long time qualitative properties of the underlying system.
基金supported by the National Basic Research Program of China(Grant No.2013CB733004)
文摘An energy method is proposed to investigate the critical transformation condition from a Taylor cone to a cone-jet. Based on the kinetic theorem, the system power allocation and the electrohydrodynamics stability are discussed. The numerical results indicate that the energy of the liquid cone tip experiences a maximum value during the transformation. With the proposed jetting energy, we give the critical transformation condition under which the derivative of jetting energy with respect to the surface area is greater than or equal to the energy required to form a unit of new liquid surface.
基金supported by the National Natural Science Foundation of China (Grant Nos 10572021 and 10772025)the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No 20040007022)
文摘Based on the weak Noether symmetry proposed by Mei F X, this paper discusses the weak Noether symmetry for nonholonomic system of non-Chetaev type, and presents expressions of three kinds of conserved quantities by weak Noether symmetry. Finally, the application of this new results is showed by a practical example.
