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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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| Cite this article: |
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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.
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| Keywords:
05.45.Pq
02.60.Lj
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Received: 23 March 2008
Published: 25 October 2008
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| PACS: |
05.45.Pq
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(Numerical simulations of chaotic systems)
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02.60.Lj
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(Ordinary and partial differential equations; boundary value problems)
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Chinese) |
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