Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems
Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems
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DOI：
10.1088/1361-6544/ac24e4
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发表时间：
2021-08
期刊：
影响因子：
1.7
通讯作者：
Shoya Motonaga;K. Yagasaki
中科院分区：
文献类型：
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作者：
Shoya Motonaga;K. Yagasaki
研究了依赖于小参数ɛ>；0的动力系统的周期轨道、同宿轨道、第一积分和交换向量场的持久性，给出了它们持久性的几个必要条件。在这里，我们不仅处理同宿轨道的平衡点，而且处理周期轨道。我们还讨论了这些结果与单自由度哈密顿系统时间周期摄动的标准次调和同宿Melnikov方法以及多自由度哈密顿系统自治摄动的另一种形式的同宿Melnikov方法之间的一些关系。特别地，如果次调和函数或同宿Melnikov函数在连通开集上不恒为零，则在扰动系统的未受扰动的周期轨道或同宿轨道附近不存在当扰动趋于零时收敛到哈密顿量或另一个第一积分的第一积分。我们用四个例子来说明我们的理论：周期受迫Duffing振子、两个相同的钟摆与谐振子耦合、周期性受迫刚体和屈曲梁的三模截断。
We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter ɛ > 0 and give several necessary conditions for their persistence. Here we treat homoclinic orbits not only to equilibria but also to periodic orbits. We also discuss some relationships of these results with the standard subharmonic and homoclinic Melnikov methods for time-periodic perturbations of single-degree-of-freedom Hamiltonian systems, and with another version of the homoclinic Melnikov method for autonomous perturbations of multi-degree-of-freedom Hamiltonian systems. In particular, we show that a first integral which converges to the Hamiltonian or another first integral as the perturbation tends to zero does not exist near the unperturbed periodic or homoclinic orbits in the perturbed systems if the subharmonic or homoclinic Melnikov functions are not identically zero on connected open sets. We illustrate our theory for four examples: the periodically forced Duffing oscillator, two identical pendula coupled with a harmonic oscillator, a periodically forced rigid body and a three-mode truncation of a buckled beam.