不同尺度Duffing系统的分岔分析

发布时间：2018-04-18 17:59

本文选题：非光滑 + 多时间尺度　；参考：《江苏大学》2017年硕士论文

【摘要】：多时间尺度问题涉及到科学和工程技术等多个领域,具有广泛的应用背景,多尺度系统存在的复杂簇发振荡及其产生的分岔机制是非线性科学的前沿研究热点之一。迄今为止,对光滑动力系统的研究已形成了一套较为完整的描述其分岔行为的方法理论。然而,工程实际中有许多的动力系统含有非光滑的因素,如力学系统中的碰撞运动,考虑摩擦因素时的粘滑振动,以及电路中的开关等等,故而对非光滑动力系统的进一步探究有着深远的意义。基于以上背景,本文考虑一类多时间尺度下的非光滑动力系统,并构建了周期外激励下的非自治Duffing系统,当周期激励系统中的激励频率与系统的固有频率之间存在量级差距时,则可观察到一种表现为大幅振荡和微幅振荡组合的簇发振荡现象。选取适当的参数值,分别研究了三组不同参数条件下系统具体的簇发现象,即双涡卷情形、三涡卷情形和四涡卷情形。基于转换相图,并考虑到非光滑因素的影响,讨论了相应快子系统的分岔模式,揭示了该类非光滑系统中不同簇发振荡的产生及其沉寂态与激发态之间相互转迁的分岔机理。另外,针对于该类非光滑系统中的多平衡态共存现象,还考虑到了吸引子自身的演化行为,从而进一步揭示了在非光滑因素影响下系统产生特殊振荡现象的原因。同样以Duffing振子为原型,并引入一个参数激励项和一个周期外激励项,建立了一个参、外联合激励下的非光滑动力系统模型。利用deMoivre公式将系统中存在的两个慢变量等价转换为一个慢变量,从而可以直接应用传统的快慢分析法来讨论系统中存在的簇发振荡,并揭示不同类型簇发振荡的产生机理。文中选取了具有代表性的六组激励频率进行对比分析,在参数激励频率取定为Ω_1 = 0.01时,结果表明:随着Ω_2/Ω_1成倍增加,系统振荡结构越来越复杂,并且系统轨迹经历沉寂态与激发态之间相互转迁的次数也是成倍增加的;而当参数激励频率固定为Ω_1=0.02时,得到一般性的结论:在引入慢变量的过程中,对于所构造的两个函数f_1(x)= f_m~*(x)和f_2(x)= f_n~*(x),若m和n都为偶数,则与系统对应的转换相图一定是轴对称的,且与原系统的对称性无关。
[Abstract]:Multi-time scale problems involve many fields, such as science and engineering technology, and have a wide application background. The complex cluster oscillation and its bifurcation mechanism of multi-scale systems are one of the frontier research hotspots in nonlinear science.Up to now, the study of smooth dynamical system has formed a complete set of method theory to describe its bifurcation behavior.However, in engineering practice, many dynamic systems contain non-smooth factors, such as collision motion in mechanical systems, stick-slip vibration when friction factors are considered, switches in circuits, etc.Therefore, it is of great significance to further explore the non-smooth power system.Based on the above background, this paper considers a class of non-smooth dynamical systems with multiple time scales, and constructs a non-autonomous Duffing system under extraneous periodic excitation. When there is an order of magnitude difference between the excitation frequency of the periodic excitation system and the natural frequency of the system,A cluster oscillation with large amplitude oscillation and micro amplitude oscillation can be observed.In this paper, we select appropriate parameter values and study the specific cluster discovery images of the system under three groups of different parameter conditions, that is, the two-scroll case, the three-scroll case and the four-scroll case.Based on the transformation phase diagram and taking into account the influence of non-smooth factors, the bifurcation modes of the corresponding fast subsystems are discussed. The generation of different cluster oscillations and the bifurcation mechanism between the silent state and the excited state in this kind of non-smooth system are revealed.In addition, in view of the coexistence of multi-equilibrium states in this class of non-smooth systems, the evolutionary behavior of the attractor itself is also taken into account, which further reveals the causes of the special oscillation in the system under the influence of non-smooth factors.Taking the Duffing oscillator as the prototype and introducing a parametric excitation term and a periodic extrinsic excitation term, a non-smooth dynamic system model under the combined parametric and external excitation is established.By using deMoivre formula, the two slow variables in the system can be converted into one slow variable, thus the traditional fast and slow analysis method can be directly used to discuss the cluster oscillation in the system, and to reveal the mechanism of different types of cluster oscillation.In this paper, six groups of typical excitation frequencies are selected for comparative analysis. When the parameter excitation frequency is taken as 惟 _ s _ 1 = 0.01, the results show that the oscillation structure of the system becomes more and more complex with the multiplicity of 惟 _ 2 / 惟 _ s.Moreover, the number of transitions between quiet and excited states is increased exponentially, and when the excitation frequency is fixed at 惟 1 / 0.02, a general conclusion is drawn: in the process of introducing slow variables,For the two functions constructed in this paper, f _ S _ 1T _ x _ n = f _ S _ m _ n _ (x) and f _ S _ 2N _ x = f _ S _ n, if m and n are both even numbers, the corresponding transformation phase diagram of the system must be axisymmetric and independent of the symmetry of the original system.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O19

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