Chin. Phys. Lett.  2008, Vol. 25 Issue (11): 3886-3889    DOI:
Original Articles |
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
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GANG Tie-Qiang, CHEN Li-Jie, MEI Feng-Xiang 2008 Chin. Phys. Lett. 25 3886-3889
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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.
Keywords: 05.45.Pq      02.60.Lj     
Received: 23 March 2008      Published: 25 October 2008
PACS:  05.45.Pq (Numerical simulations of chaotic systems)  
  02.60.Lj (Ordinary and partial differential equations; boundary value problems)  
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https://cpl.iphy.ac.cn/       OR      https://cpl.iphy.ac.cn/Y2008/V25/I11/03886
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GANG Tie-Qiang
CHEN Li-Jie
MEI Feng-Xiang
[1] Trotter H F 1959 Proc. Am. Math. Soc. 10 545
[2] Hairer E, Lubich C and Wanner G 2002 Geometric
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[5] Qin M Z and Zhu W J 1992 Computing 47 309
[6] Gang T Q, Mei F X and Chen L J 2007 Chin. Phys.
Lett. 25 866
[7] Strang G 1968 SIAM J. Numer. Anal. 5 506
[8] Robinson C R 2007 An Introduction to Dynamical
Systems: Continuous and Discrete (Beijing: China Machine Press) (in
Chinese)
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