本文關(guān)鍵詞：基于Chebyshev正交多項式逼近理論的隨機(jī)Hopf分岔的研究　出處：《蘭州交通大學(xué)》2016年碩士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 二維混沌系統(tǒng) 電機(jī)系統(tǒng) 金融系統(tǒng) 穩(wěn)定性 隨機(jī)Hopf分岔 切比雪夫多項式逼近

【摘要】：在過去的幾十年里,隨機(jī)分岔是動力學(xué)領(lǐng)域中的一個熱門話題�；陔S機(jī)結(jié)構(gòu)和隨機(jī)動力系統(tǒng)理論,借助切比雪夫多項式逼近法探索非線性隨機(jī)動力系統(tǒng)的響應(yīng),分岔和混沌現(xiàn)象。本文主要對含有隨機(jī)參數(shù)的隨機(jī)非線性動力系統(tǒng)的Hopf分岔進(jìn)行了研究。主要內(nèi)容如下:首先,我們構(gòu)造了一個含有隨機(jī)參數(shù)的新的二維混沌系統(tǒng),一個含有隨機(jī)參數(shù)的電機(jī)系統(tǒng)和一個含有隨機(jī)參數(shù)的金融系統(tǒng)。通過選擇適當(dāng)?shù)姆植韰?shù)分析了系統(tǒng)在平衡點處的穩(wěn)定性,存在性和發(fā)生Hopf分岔的條件。更準(zhǔn)確地說,在接下來要研究的系統(tǒng)中我們分別選擇參數(shù)a,b為分岔參數(shù),當(dāng)分岔參數(shù)a,b穿過臨界值0a,0b時,系統(tǒng)就發(fā)生Hopf分岔。為了研究這類系統(tǒng)的動力學(xué)行為,我們首先借助切比雪夫多項式逼近法將其轉(zhuǎn)換成等價的確定性系統(tǒng)。然后通過第一Lyapunov系數(shù)法獲得確保這類系統(tǒng)發(fā)生Hopf分岔的參數(shù)條件。在數(shù)值計算過程中借助Maple,Matlab等數(shù)學(xué)軟件得到轉(zhuǎn)化后的高維確定性系統(tǒng)發(fā)生Hopf分岔的一些重要結(jié)論。并且分析了系統(tǒng)是發(fā)生超臨界Hopf分岔還是亞臨界Hopf分岔,以及發(fā)生超臨界Hopf分岔時如滿足一定的條件,系統(tǒng)從一個不穩(wěn)定狀態(tài)變成一個穩(wěn)定狀態(tài)。我們可以根據(jù)需要去適當(dāng)?shù)母淖兿到y(tǒng)的參數(shù)來避免劇烈波動并且可以解釋和預(yù)測一些實際問題。其次,借助確定性系統(tǒng)理論對隨機(jī)系統(tǒng)進(jìn)行研究,發(fā)現(xiàn)其除了具有與確定性系統(tǒng)相似的一些特征外,還表現(xiàn)出一些隨機(jī)系統(tǒng)特有的特征。與確定性系統(tǒng)不同,隨機(jī)Hopf分岔臨界值的確定不僅取決于隨機(jī)系統(tǒng)中的隨機(jī)參數(shù),而且與隨機(jī)參數(shù)的強(qiáng)度有關(guān)。當(dāng)隨機(jī)參數(shù)的強(qiáng)度改變時,隨機(jī)Hopf分岔的臨界值也會隨之發(fā)生一定的變化。最后,數(shù)值模擬的結(jié)果證明了本文理論結(jié)果是正確有效的。顯然,關(guān)于這類系統(tǒng)還存在更多有趣的問題比如復(fù)雜性,控制,和同步,這些都值得進(jìn)一步去研究。
[Abstract]:In the past few decades, stochastic bifurcation has been a hot topic in the field of dynamics, based on stochastic structure and stochastic dynamical system theory. The response of nonlinear stochastic dynamical systems is investigated by Chebyshev polynomial approximation. Bifurcation and chaos phenomena. This paper mainly studies the Hopf bifurcation of stochastic nonlinear dynamical systems with random parameters. The main contents are as follows: first. We construct a new two-dimensional chaotic system with random parameters. A motor system with random parameters and a financial system with random parameters are used to analyze the stability of the system at the equilibrium point by selecting proper bifurcation parameters. More accurately, in the system to be studied, we select the parameter ab as the bifurcation parameter respectively, when the bifurcation parameter ab passes through the critical value of 0 a / 0 b. In order to study the dynamic behavior of the system, Hopf bifurcation occurs in the system. We first convert it to an equivalent deterministic system by means of the Chebyshev polynomial approximation method. Then we obtain the parameter conditions by the first Lyapunov coefficient method to ensure the Hopf bifurcation of this kind of system. In the course of numerical calculation, Maple is used. Some important conclusions on the Hopf bifurcation of the transformed high-dimensional deterministic system are obtained by using Matlab and other mathematical software. The supercritical Hopf bifurcation or subcritical Hopf bifurcation is analyzed. . And if the supercritical Hopf bifurcation occurs, some conditions are satisfied. System from an unstable state to a stable state. We can appropriately change the system parameters as necessary to avoid violent fluctuations and can explain and predict some practical problems. Secondly. Based on the theory of deterministic system, it is found that the stochastic system not only has some characteristics similar to the deterministic system, but also shows some unique characteristics of the stochastic system, which is different from the deterministic system. The determination of the critical value of stochastic Hopf bifurcation depends not only on the random parameters in the stochastic system, but also on the strength of the random parameters. The critical value of stochastic Hopf bifurcation will also change with it. Finally, the numerical simulation results show that the theoretical results in this paper are correct and effective. There are many more interesting questions about such systems, such as complexity, control, and synchronization, which deserve further study.
【學(xué)位授予單位】：蘭州交通大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：O175

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