By comparing with symplectic different methods, the quadratic element is an approximately symplectic method which can keep high accuracy approximate of symplectic structure for Hamiltonian chaos, and it is also energy conservative when there have chaos phenomenon. We use the quadratic finite element method to solve the H[Formula: see text]non–Heiles system, and this method was never used before. Combining with Poincar[Formula: see text] section, when we increase the energy of the systems, KAM tori are broken and the motion from regular to chaotic. Without chaos, three kinds of methods to calculate the Poincar[Formula: see text] section point numbers are the same, and the numbers are different with chaos. In long-term calculation, the finite element method can better keep dynamic characteristics of conservative system with chaotic motion.

中文翻译：

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混沌哈密顿系统的有限元方法研究

通过与辛不同方法的比较，二次元是一种近似辛方法，它可以对哈密顿混沌保持辛结构的高精度近似，并且在出现混沌现象时也是能量守恒的。我们使用二次有限元法求解H[公式：见正文]non-Heiles系统，这种方法以前从未使用过。结合Poincar[公式：见正文]部分，当我们增加系统的能量时，KAM tori被打破，运动由规则变为混沌。没有混沌，三种计算Poincar的方法[公式：见正文]截面点数相同，混沌时数不同。在长期计算中，有限元法可以更好地保持具有混沌运动的保守系统的动态特性。