Slow–fast dynamics in a perturbation model of double pendulum system with singularity of triple zero eigenvalues
Slow–fast dynamics in a perturbation model of double pendulum system with singularity of triple zero eigenvalues
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DOI：
10.1007/s11071-022-08020-2
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发表时间：
2022-10
影响因子：
5.6
通讯作者：
Weipeng Lyu;Lu Zhang;Haibo Jiang;Qinsheng Bi
中科院分区：
文献类型：
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作者：
Weipeng Lyu;Lu Zhang;Haibo Jiang;Qinsheng Bi
在许多物理和工程系统中，快慢动力学（如突发行为）是常见的。在以往的研究中，一些人关注余维1分岔引起的突发行为，另一些人则关注特殊结构引起的突发行为。然而，系统在临界条件下可能会表现出许多复杂的动力学，由于高余维分岔。我们的研究目的是研究三重零特征值奇点附近的爆发振荡。以具有外激励的双摆系统的扰动模型为例，研究了爆发行为的动力学机制。由于激振频率与固有频率之间存在阶数差，因此，外激励作用下的摄动模型可以看作是一个广义自治系统。通过将变换后的相图与平衡分支重叠，确定了四种类型的爆发振荡：折叠/折叠型、零-Hopf/零-Hopf型、对称零-Hopf/sup-Hopf/折叠周期型和对称零-Hopf/sup-Hopf/折叠周期/sub-Hopf型。首先，我们发现由于复合褶皱条件的奇异性，极限环的许多分支发生，导致许多复杂的动力学，如二维环面破裂，对称结构的破缺，混沌破裂。这些结果在理解具有高余维分岔条件的系统的稳定性方面起着至关重要的作用。为制定控制策略提供了理论依据。
Slow–fast dynamics such as bursting behaviors are common in many physical and engineering systems. In the previous study, some focused on the bursting behavior caused by the codimension-1 bifurcations, and others focus on the bursting behavior due to the particular structures. However, systems under critical conditions may exhibit many complicated dynamics due to the high co-dimensional bifurcations. Our research aims to investigate the bursting oscillations near a triple zero eigenvalues singularity. A perturbation model of the double pendulum system with external excitation is taken as an example and investigated the dynamical mechanism of bursting behaviors. Because of the order gap between the exciting frequency and natural frequency, the perturbation model with the external excitation can be regarded as a generalized autonomous system. By overlapping the transformed phase portrait and the equilibrium branch, four types of bursting oscillations are determined: fold/fold type, zero-Hopf/zero-Hopf type, symmetrical zero-Hopf/sup-Hopf/fold-cycle type, and symmetrical zero-Hopf/sup-Hopf/fold-cycle/sub-Hopf type. Primarily, we find that due to the singularity of compound fold conditions, many bifurcations of the limit cycle occur, which cause many complex dynamics such as 2-D tori bursting, breaking of symmetric structure, and chaotic bursting. These results play an essential role in understanding the stability of a system with high co-dimensional bifurcation conditions. They can be expected to provide a theoretical basis for formulating a control strategy.