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陈娅昵, 孟文静, 钱有华. 一类 Duffing 型系统的不动点混沌和Fold/Fold 簇发现象及机理分析[J]. 力学学报, 2020, 52(5): 1475-1484. DOI: 10.6052/0459-1879-20-098
引用本文: 陈娅昵, 孟文静, 钱有华. 一类 Duffing 型系统的不动点混沌和Fold/Fold 簇发现象及机理分析[J]. 力学学报, 2020, 52(5): 1475-1484. DOI: 10.6052/0459-1879-20-098
Chen Yani, Meng Wenjing, Qian Youhua. FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1475-1484. DOI: 10.6052/0459-1879-20-098
Citation: Chen Yani, Meng Wenjing, Qian Youhua. FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1475-1484. DOI: 10.6052/0459-1879-20-098

一类 Duffing 型系统的不动点混沌和Fold/Fold 簇发现象及机理分析

FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS

  • 摘要: 本文主要探究了一类含有两个慢变量的双稳态 Duffing 型系统,通过时间历程图、相图、分岔图等对系统进行数值模拟,然后从理论上分析不同参数下系统的动力学机理. 首先,研究发现当振幅参数取值大于 1 时,系统会表现出不动点混沌现象,并进一步解释了产生不动点混沌的机理. 其次, 介绍了参数空间中的簇发振荡现象,即系统穿过鞍结曲面的一侧到达另一侧所发生的行为,这里也称为鞍结簇发振荡. 事实上,当系统穿过鞍结曲面的时候,它的平衡点个数发生了变化. 然后,使用纵向抛物线路径说明了 Fold/Fold 簇发振荡产生的机理,发现无论常系数项和振幅的取值为多少,只要满足一定的关系,总会产生 Fold/Fold 簇发振荡,之后使用线性路径阐明了新增常系数项会使得系统发生簇发振荡的原因. 并且发现路径与鞍结曲面交点的位置会影响簇发振荡的对称性;路径的跨度会影响簇发振荡的大小. 最后,使用多拐折曲线路径讨论当两个激励项存在 n 倍关系时系统产生的现象. 结果表明当 n=3 时,常系数项的变化会使得系统表现出不同重数的 Fold/Fold 簇发振荡,最高可达到三重簇发振荡. 并且发现在理想状况下如果可以找到一条路径可以分割为 n 段,并且每一段都会与鞍结曲面有交点,那么会产生 n 重 Fold/Fold 簇发振荡.

     

    Abstract: In this paper, a class of bistable Duffing type system with two slow variables under new materials is explored. The system is simulated by time history diagram, phase diagram and bifurcation diagram, then the dynamic mechanism of the system under different parameters is analyzed theoretically. Firstly, this manuscript describes that when the amplitude parameter value is greater than 1, the system may exhibit fixed point chaos and explains the reason of fixed point chaos. Secondly, this manuscript introduces the phenomenon of Fold/Fold bursting in parameter space which is caused by the movement of the system from one side of the saddle-node surface to the other side. We also call it saddle-node bursting. In fact, when the system passes through the saddle-node surface, the number of equilibrium points changes. Then this manuscript uses the path of longitudinal parabolic to explain the mechanism of Fold/Fold bursting. And it is found that regardless of the value of constant coefficient term and amplitude, as long as a certain relationship is satisfied, there will always be Fold/Fold bursting. Next this manuscript uses the linear path to discuss the influence of newly added constant coefficient term. It is found that the position where the path intersects the saddle-node surface will affect the symmetry of the bursting, and the span of the path will affect the magnitude of the bursting oscillation. Finally, this manuscript uses the multiple inflection curve path to discuss the phenomenon when two incentive terms have specific relation. When n=3, the change of the constant coefficient term will make the system show Fold/Fold bursting with different times, and the maximum can reach triple bursting. Moreover, it is found that if you can find a path that can be divided into n segments, and each segment will have an intersection with the saddle surface, then n times Fold/Fold bursting will occur.

     

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