發(fā)布時間：2018-05-12 04:59

  本文選題：Filippov系統(tǒng) + 兩尺度��； 參考：《江蘇大學(xué)》2017年碩士論文

【摘要】：Filippov系統(tǒng)在其非光滑分界面上的向量場存在跳躍,導(dǎo)致系統(tǒng)產(chǎn)生一些特殊的振蕩行為,如滑動、擦邊運動等。同時,頻域上的不同尺度耦合使得系統(tǒng)會出現(xiàn)不同模式的簇發(fā)振蕩,探討該類系統(tǒng)的不同尺度效應(yīng)是當前國內(nèi)外的熱點和前沿課題之一。本文主要以兩BVP振子耦合模型的修正系統(tǒng)為例,選擇適當參數(shù),建立了存在頻域兩尺度的Filippov系統(tǒng)。運用非線性分岔理論、快慢分析法以及數(shù)值模擬等方法,揭示了不同平衡態(tài)下一類Filippov系統(tǒng)的典型的復(fù)雜簇發(fā)振蕩及其產(chǎn)生機制。首先,借助于含單非光滑分界面的耦合BVP電路系統(tǒng),選取適當參數(shù)使得周期激勵頻率與系統(tǒng)固有頻率之間存在量級差異,構(gòu)建了兩頻域尺度的Filippov系統(tǒng),考慮單平衡態(tài)下一類Filippov系統(tǒng)的簇發(fā)振蕩及其分岔機理。運用非線性動力學(xué)的相關(guān)理論對兩個光滑子系統(tǒng)分別進行平衡點的穩(wěn)定性分析和常規(guī)分岔分析,采用快慢分析法,將平衡點分岔圖與系統(tǒng)相圖相疊加,探討不同簇發(fā)振蕩的產(chǎn)生機理。同時,利用微分包含理論研究非光滑分界面處系統(tǒng)可能出現(xiàn)的非常規(guī)分岔及其存在條件。分析發(fā)現(xiàn)隨著參數(shù)的變化,不同簇發(fā)現(xiàn)象中沉寂態(tài)與激發(fā)態(tài)的相互轉(zhuǎn)遷主要由非光滑因素導(dǎo)致的。對于三種典型的外激勵振幅情形,給出了系統(tǒng)具有滑動結(jié)構(gòu)的典型周期簇發(fā)振蕩模式,揭示了系統(tǒng)軌跡與非光滑分界面未接觸、接觸而未穿過、接觸并滑動再穿過分界面時,激發(fā)態(tài)與沉寂態(tài)相互轉(zhuǎn)遷的動力學(xué)機制。其次,對同一電路模型,選取適當參數(shù)使系統(tǒng)呈現(xiàn)出多平衡態(tài),結(jié)合多尺度因素,進一步探究多平衡態(tài)下一類Filippov系統(tǒng)的簇發(fā)振蕩及其機理。將整個周期激勵項視為慢變參數(shù),得到不同區(qū)域中兩子系統(tǒng)的平衡曲線,分析了其中的分岔行為,進而考察激勵幅值對系統(tǒng)振蕩行為的影響。選取兩種典型的外激勵振幅情形,分別給出其相應(yīng)的簇發(fā)振蕩模式,采用快慢分析法,借助非光滑分界面兩側(cè)向量場的動力學(xué)特性,并基于轉(zhuǎn)換相圖,給出了各自振蕩的產(chǎn)生機制。研究發(fā)現(xiàn)多平衡態(tài)下,系統(tǒng)可能呈現(xiàn)出更為復(fù)雜的簇發(fā)振蕩。另外,在平衡曲線的某些特殊點處會產(chǎn)生激發(fā)振蕩,而隨著外激勵幅值的增加,當其相應(yīng)的平衡曲線穿越這些特殊點時會產(chǎn)生簇發(fā)振蕩。與光滑系統(tǒng)不同,Filippov系統(tǒng)中出現(xiàn)的激發(fā)態(tài)表現(xiàn)為滑動與大幅振蕩的交替組合,根據(jù)平衡曲線的特性,結(jié)合向量場的變化特性,揭示了這種激發(fā)態(tài)模式的產(chǎn)生機制。最后,對本文的主要研究內(nèi)容進行了適當?shù)母爬ǹ偨Y(jié),同時指出了本文的一些欠缺之處,并對接下來的研究工作進行了展望。
[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.
【學(xué)位授予單位】：江蘇大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175

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