两尺度下一类Filippov系统的非光滑分岔分析

发布时间：2018-05-12 04:59

本文选题：Filippov系统 + 两尺度　；参考：《江苏大学》2017年硕士论文

【摘要】：Filippov系统在其非光滑分界面上的向量场存在跳跃,导致系统产生一些特殊的振荡行为,如滑动、擦边运动等。同时,频域上的不同尺度耦合使得系统会出现不同模式的簇发振荡,探讨该类系统的不同尺度效应是当前国内外的热点和前沿课题之一。本文主要以两BVP振子耦合模型的修正系统为例,选择适当参数,建立了存在频域两尺度的Filippov系统。运用非线性分岔理论、快慢分析法以及数值模拟等方法,揭示了不同平衡态下一类Filippov系统的典型的复杂簇发振荡及其产生机制。首先,借助于含单非光滑分界面的耦合BVP电路系统,选取适当参数使得周期激励频率与系统固有频率之间存在量级差异,构建了两频域尺度的Filippov系统,考虑单平衡态下一类Filippov系统的簇发振荡及其分岔机理。运用非线性动力学的相关理论对两个光滑子系统分别进行平衡点的稳定性分析和常规分岔分析,采用快慢分析法,将平衡点分岔图与系统相图相叠加,探讨不同簇发振荡的产生机理。同时,利用微分包含理论研究非光滑分界面处系统可能出现的非常规分岔及其存在条件。分析发现随着参数的变化,不同簇发现象中沉寂态与激发态的相互转迁主要由非光滑因素导致的。对于三种典型的外激励振幅情形,给出了系统具有滑动结构的典型周期簇发振荡模式,揭示了系统轨迹与非光滑分界面未接触、接触而未穿过、接触并滑动再穿过分界面时,激发态与沉寂态相互转迁的动力学机制。其次,对同一电路模型,选取适当参数使系统呈现出多平衡态,结合多尺度因素,进一步探究多平衡态下一类Filippov系统的簇发振荡及其机理。将整个周期激励项视为慢变参数,得到不同区域中两子系统的平衡曲线,分析了其中的分岔行为,进而考察激励幅值对系统振荡行为的影响。选取两种典型的外激励振幅情形,分别给出其相应的簇发振荡模式,采用快慢分析法,借助非光滑分界面两侧向量场的动力学特性,并基于转换相图,给出了各自振荡的产生机制。研究发现多平衡态下,系统可能呈现出更为复杂的簇发振荡。另外,在平衡曲线的某些特殊点处会产生激发振荡,而随着外激励幅值的增加,当其相应的平衡曲线穿越这些特殊点时会产生簇发振荡。与光滑系统不同,Filippov系统中出现的激发态表现为滑动与大幅振荡的交替组合,根据平衡曲线的特性,结合向量场的变化特性,揭示了这种激发态模式的产生机制。最后,对本文的主要研究内容进行了适当的概括总结,同时指出了本文的一些欠缺之处,并对接下来的研究工作进行了展望。
[Abstract]:There is a jump in the vector field of the Filippov system on its non-smooth boundary surface, which results in some special oscillatory behaviors, such as sliding, edge-scrubbing motion and so on. At the same time, different scale coupling in frequency domain leads to cluster oscillation of different modes. It is one of the hot and frontier topics to study the different scale effects of this kind of systems at home and abroad. In this paper, the modified system of coupling model of two BVP oscillators is taken as an example, and the Filippov system with two scales in frequency domain is established by selecting appropriate parameters. By means of nonlinear bifurcation theory, fast and slow analysis and numerical simulation, the typical complex cluster oscillation and its generation mechanism of a class of Filippov systems with different equilibrium states are revealed. Firstly, with the help of the coupled BVP circuit system with a single non-smooth boundary surface, the order of magnitude difference between the periodic excitation frequency and the system natural frequency is obtained by selecting appropriate parameters, and a two-frequency domain scale Filippov system is constructed. The cluster oscillation and bifurcation mechanism of a class of Filippov systems in a single equilibrium state are considered. The stability analysis of the equilibrium point and the normal bifurcation analysis of the two smooth subsystems are carried out by using the related theory of nonlinear dynamics. The bifurcation diagram of the equilibrium point is superposed with the phase diagram of the system by using the fast and slow analysis method. The mechanism of different cluster oscillations is discussed. At the same time, the differential inclusion theory is used to study the possible unconventional bifurcation and its existence condition of the system at the non-smooth interface. It is found that, with the change of parameters, the intertransformation of silent and excited states in different cluster phenomena is mainly caused by non-smooth factors. For three typical external excitation amplitudes, a typical periodic cluster oscillation mode with sliding structure is given. It is revealed that the system trajectory is not in contact with the non-smooth interface, but not passing through, and when the system is in contact with and sliding through the sub-interface, The dynamic mechanism of the interaction between excited and silent states. Secondly, for the same circuit model, the cluster oscillation and its mechanism of a class of Filippov systems under multi-equilibrium state are further explored by selecting appropriate parameters to make the system appear multi-equilibrium state. The whole periodic excitation term is regarded as a slowly varying parameter and the equilibrium curves of the two subsystems in different regions are obtained. The bifurcation behavior is analyzed and the effect of the excitation amplitude on the oscillation behavior of the system is investigated. Two typical external excitation amplitudes are selected, and their corresponding cluster oscillation modes are given respectively. The fast and slow analysis method is used to analyze the dynamics of vector fields on both sides of the non-smooth interface, and based on the transformation phase diagram, The generation mechanism of their oscillations is given. It is found that the system may exhibit more complex cluster oscillations in the multi-equilibrium state. In addition, excitation oscillations occur at some special points of the equilibrium curve, and cluster oscillations occur when the corresponding equilibrium curves cross these special points with the increase of the external excitation amplitude. Different from smooth system, the excited state in Filippov system is alternately composed of sliding and large oscillation. According to the characteristics of equilibrium curve and the variation of vector field, the mechanism of the excited state mode is revealed. Finally, the main research contents of this paper are summarized, and some shortcomings of this paper are pointed out, and the future research work is prospected.
【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】：O175

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