Firstly, by using the Liouville formula, we prove that the Jacobian matrix determinants of splitting methods are equal to that of the exact flow. However, for the explicit Runge-Kutta methods, there is an error term o...Firstly, by using the Liouville formula, we prove that the Jacobian matrix determinants of splitting methods are equal to that of the exact flow. However, for the explicit Runge-Kutta methods, there is an error term of order p + I for the Jacobian matrix determinants. Then, the volume evolution law of a given region in phase space is discussed for different algorithms. It is proved that splitting methods can exactly preserve the sum of Lyapunov exponents invariable. Finally, a Poincaré map and its energy distribution of the Duffing equation are computed by using the second-order splitting method and the Heun method (a second-order Runge-Kutta method). Computation illustrates that the results by splitting methods can properly represent systems' chaotic phenomena.展开更多
Based on a splitting method and a composition method, we construct some structure-preserving algorithms with first-order, second-order and fourth-order accuracy for a Lorenz system. By using the Liouville's formula, ...Based on a splitting method and a composition method, we construct some structure-preserving algorithms with first-order, second-order and fourth-order accuracy for a Lorenz system. By using the Liouville's formula, it is proven that the structure-preserving algorithms exactly preserve the volume of infinitesimal cube for the Lorenz system. Numerical experimental results illustrate that for the conservative Lorenz system, the qualitative behaviour of the trajectories described by the classical explicit fourth-order Runge-Kutta (RK4) method and the fifth-order Runge-Kutta-Fehlberg (RKF45) method is wrong, while the qualitative behaviour derived from the structure-preserving algorithms with different orders of accuracy is correct. Moreover, for the small dissipative Lorenz system, the norm errors of the structure-preserving algorithms in phase space axe less than those of the Runge-Kutta methods.展开更多
A symmetry and a conserved quantity of the Birkhoff system are studied. The symmetry is called the Birkhoff symmetry. Its definition and criterion are given in this paper. A conserved quantity can be deduced by using ...A symmetry and a conserved quantity of the Birkhoff system are studied. The symmetry is called the Birkhoff symmetry. Its definition and criterion are given in this paper. A conserved quantity can be deduced by using the symmetry. An example is given to illustrate the application of the result.展开更多
The purpose of this paper is to study the solution of the celebrated Whittaker equations by using analytical mechanics methods, including the Lagrange-Noether method, Hamilton-Poisson method and potential integral met...The purpose of this paper is to study the solution of the celebrated Whittaker equations by using analytical mechanics methods, including the Lagrange-Noether method, Hamilton-Poisson method and potential integral method.展开更多
This paper studies the Hojman conserved quantity, a non-Noether conserved quantity, deduced by special weak Noether symmetry for Lagrange systems. Under special infinitesimal transformations in which the time is not v...This paper studies the Hojman conserved quantity, a non-Noether conserved quantity, deduced by special weak Noether symmetry for Lagrange systems. Under special infinitesimal transformations in which the time is not variable, its criterion is given and a method of how to seek the Hojman conserved quantity is presented. A Hojman conserved quantity can be found by using the special weak Noether symmetry.展开更多
Perturbation differential equations of motion of a general nonholonomic system subjected to the ideal nonholonomic constraints of Chetaev's type are established, and the equation of variation of energy is deduced by ...Perturbation differential equations of motion of a general nonholonomic system subjected to the ideal nonholonomic constraints of Chetaev's type are established, and the equation of variation of energy is deduced by using the perturbation equations of the system. A criterion of the stability is obtained and an example is given to illustrate the application of the result.展开更多
基金Supported by the National Natural Science Foundation of China under Grant No 10572021, and the Specialized Research Fund for the Doctoral Programme of Higher Education of China under Grant No 20040007022.
文摘Firstly, by using the Liouville formula, we prove that the Jacobian matrix determinants of splitting methods are equal to that of the exact flow. However, for the explicit Runge-Kutta methods, there is an error term of order p + I for the Jacobian matrix determinants. Then, the volume evolution law of a given region in phase space is discussed for different algorithms. It is proved that splitting methods can exactly preserve the sum of Lyapunov exponents invariable. Finally, a Poincaré map and its energy distribution of the Duffing equation are computed by using the second-order splitting method and the Heun method (a second-order Runge-Kutta method). Computation illustrates that the results by splitting methods can properly represent systems' chaotic phenomena.
基金Supported by the National Natural Science Foundation of China under Grant No 10572021, and the Doctoral Programme Foundation of Institute of Higher Education of China under Grant 20040007022.
文摘Based on a splitting method and a composition method, we construct some structure-preserving algorithms with first-order, second-order and fourth-order accuracy for a Lorenz system. By using the Liouville's formula, it is proven that the structure-preserving algorithms exactly preserve the volume of infinitesimal cube for the Lorenz system. Numerical experimental results illustrate that for the conservative Lorenz system, the qualitative behaviour of the trajectories described by the classical explicit fourth-order Runge-Kutta (RK4) method and the fifth-order Runge-Kutta-Fehlberg (RKF45) method is wrong, while the qualitative behaviour derived from the structure-preserving algorithms with different orders of accuracy is correct. Moreover, for the small dissipative Lorenz system, the norm errors of the structure-preserving algorithms in phase space axe less than those of the Runge-Kutta methods.
基金Project supported by the National Natural Science Foundation of China (Grant No 10572021) and the Special Research Fund for the Doctoral Program of Higher Education of China (Grant No 20040007022).
文摘A symmetry and a conserved quantity of the Birkhoff system are studied. The symmetry is called the Birkhoff symmetry. Its definition and criterion are given in this paper. A conserved quantity can be deduced by using the symmetry. An example is given to illustrate the application of the result.
基金Project supported by the National Natural Science Foundation (Grant No 10572021) and the Doctoral Program Foundation of Institution of Higher Education of China (Grant No 20040007022).
文摘The purpose of this paper is to study the solution of the celebrated Whittaker equations by using analytical mechanics methods, including the Lagrange-Noether method, Hamilton-Poisson method and potential integral method.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10572021 and 10772025)the Doctoral Program Foundation of Institution of Higher Education of China (Grant No 20040007022)
文摘This paper studies the Hojman conserved quantity, a non-Noether conserved quantity, deduced by special weak Noether symmetry for Lagrange systems. Under special infinitesimal transformations in which the time is not variable, its criterion is given and a method of how to seek the Hojman conserved quantity is presented. A Hojman conserved quantity can be found by using the special weak Noether symmetry.
基金Supported by the National Natural Science Foundation of China under Grant No 10572021, and the Doctoral Programme Foundation of Institute of Higher Education of China under Grant No 20040007022.
文摘Perturbation differential equations of motion of a general nonholonomic system subjected to the ideal nonholonomic constraints of Chetaev's type are established, and the equation of variation of energy is deduced by using the perturbation equations of the system. A criterion of the stability is obtained and an example is given to illustrate the application of the result.