本文選題：非光滑 + 多時間尺度��； 參考：《江蘇大學》2017年碩士論文

【摘要】：多時間尺度問題涉及到科學和工程技術等多個領域,具有廣泛的應用背景,多尺度系統(tǒng)存在的復雜簇發(fā)振蕩及其產(chǎn)生的分岔機制是非線性科學的前沿研究熱點之一。迄今為止,對光滑動力系統(tǒng)的研究已形成了一套較為完整的描述其分岔行為的方法理論。然而,工程實際中有許多的動力系統(tǒng)含有非光滑的因素,如力學系統(tǒng)中的碰撞運動,考慮摩擦因素時的粘滑振動,以及電路中的開關等等,故而對非光滑動力系統(tǒng)的進一步探究有著深遠的意義�；谝陨媳尘�,本文考慮一類多時間尺度下的非光滑動力系統(tǒng),并構建了周期外激勵下的非自治Duffing系統(tǒng),當周期激勵系統(tǒng)中的激勵頻率與系統(tǒng)的固有頻率之間存在量級差距時,則可觀察到一種表現(xiàn)為大幅振蕩和微幅振蕩組合的簇發(fā)振蕩現(xiàn)象。選取適當?shù)膮?shù)值,分別研究了三組不同參數(shù)條件下系統(tǒng)具體的簇發(fā)現(xiàn)象,即雙渦卷情形、三渦卷情形和四渦卷情形�；谵D換相圖,并考慮到非光滑因素的影響,討論了相應快子系統(tǒng)的分岔模式,揭示了該類非光滑系統(tǒng)中不同簇發(fā)振蕩的產(chǎn)生及其沉寂態(tài)與激發(fā)態(tài)之間相互轉遷的分岔機理。另外,針對于該類非光滑系統(tǒng)中的多平衡態(tài)共存現(xiàn)象,還考慮到了吸引子自身的演化行為,從而進一步揭示了在非光滑因素影響下系統(tǒng)產(chǎn)生特殊振蕩現(xiàn)象的原因。同樣以Duffing振子為原型,并引入一個參數(shù)激勵項和一個周期外激勵項,建立了一個參、外聯(lián)合激勵下的非光滑動力系統(tǒng)模型。利用deMoivre公式將系統(tǒng)中存在的兩個慢變量等價轉換為一個慢變量,從而可以直接應用傳統(tǒng)的快慢分析法來討論系統(tǒng)中存在的簇發(fā)振蕩,并揭示不同類型簇發(fā)振蕩的產(chǎn)生機理。文中選取了具有代表性的六組激勵頻率進行對比分析,在參數(shù)激勵頻率取定為Ω_1 = 0.01時,結果表明:隨著Ω_2/Ω_1成倍增加,系統(tǒng)振蕩結構越來越復雜,并且系統(tǒng)軌跡經(jīng)歷沉寂態(tài)與激發(fā)態(tài)之間相互轉遷的次數(shù)也是成倍增加的;而當參數(shù)激勵頻率固定為Ω_1=0.02時,得到一般性的結論:在引入慢變量的過程中,對于所構造的兩個函數(shù)f_1(x)= f_m~*(x)和f_2(x)= f_n~*(x),若m和n都為偶數(shù),則與系統(tǒng)對應的轉換相圖一定是軸對稱的,且與原系統(tǒng)的對稱性無關。
[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.
【學位授予單位】：江蘇大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O19

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