1Department of Mechanics, Beijing Institute of Technology, Beijing 1000812Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005
Poincaré Map Based on Splitting Methods
GANG Tie-Qiang1, CHEN Li-Jie2, MEI Feng-Xiang1
1Department of Mechanics, Beijing Institute of Technology, Beijing 1000812Department of Mechanical and Electrical Engineering, Xiamen University, Xiamen 361005
摘要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+1 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.
Abstract：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+1 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.
(Ordinary and partial differential equations; boundary value problems)
引用本文:
GANG Tie-Qiang;CHEN Li-Jie;MEI Feng-Xiang. Poincaré Map Based on Splitting Methods[J]. 中国物理快报, 2008, 25(11): 3886-3889.
GANG Tie-Qiang, CHEN Li-Jie, MEI Feng-Xiang. Poincaré Map Based on Splitting Methods. Chin. Phys. Lett., 2008, 25(11): 3886-3889.
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2008, Vol. 25 
