Integrability analysis of the Shimizu(-M)orioka system
Integrability analysis of the Shimizu(-M)orioka system
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Shimizu(-M)orioka系统的可积性分析
DOI：
10.1016/j.cnsns.2019.105101
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发表时间：
2020
影响因子：
3.9
通讯作者：
Li Wenlei
中科院分区：
文献类型：
--
作者：
Huang Kaiyin;Shi Shaoyun;Li Wenlei
本文的目的是从可积性的角度对Shimizu-Morioka系统x˙= y, y˙= x−λ y−x z, z˙= - α z+ x2给出一些新的认识。首先，我们提出了在时间和坐标上的线性标度，将Shimizu-Morioka系统转化为α≠0时Rucklidge系统的特例，并讨论了Shimizu-Morioka系统与Rucklidge系统之间的关系。在此基础上，从Rucklidge系统上的相应结果推导出了α≠0的Shimizu-Morioka系统的Darboux可积性。当α= 0时，利用代数几何中的Gröbner基研究了Shimizu-Morioka系统的Darboux可积性。其次，利用奇异点和周期轨道的稳定性研究了Shimizu-Morioka系统全局c1第一积分的不存在性。最后，在α≠0的情况下，利用扩展的Morales-Ramis理论证明了它对几乎所有参数值都是不可积的；在α= 0的情况下，利用拟齐次分解和Kowalevski指数证明了它是不可积的。我们的结果与该系统在大范围参数范围内允许混沌行为的事实相一致。
The aim of this paper is to give some new insights into the Shimizu–Morioka system x˙= y, y˙= x− λ y− x z, z˙=− α z+ x 2, from the integrability point of view. Firstly, we propose a linear scaling in time and coordinates which converts the Shimizu–Morioka system into a special case of the Rucklidge system when α≠ 0 and discuss the relationship between Shimizu–Morioka system and Rucklidge system. Based on this observation, Darboux integrability of the Shimizu–Morioka system with α≠ 0 is trivially derived from the corresponding results on the Rucklidge system. When α= 0, we investigate Darboux integrability of the Shimizu–Morioka system by the Gröbner basis in algebraic geometry. Secondly, we use the stability of the singular points and periodic orbits to study the nonexistence of global C 1 first integrals of the Shimizu–Morioka system. Finally, in the case α≠ 0, we prove it is not rationally integrable for almost all parameter values by an extended Morales-Ramis theory, and in the case α= 0, we show that it is not algebraically integrable by quasi-homogeneous decompositions and Kowalevski exponents. Our results are in accord with the fact that this system admits chaotic behaviors for a large range of its parameters.