本文關(guān)鍵詞： 相圖 Hilbert第16問(wèn)題 Hilbert數(shù) Hamiltonian系統(tǒng) 分片近-Hamiltonian系統(tǒng) Lienard系統(tǒng) 可逆系統(tǒng) 同宿環(huán) 異宿環(huán) 三角形異宿環(huán) 同宿環(huán)軌道穩(wěn)定性 Hopf分支 Poincare分支 同宿分支 異宿分支 環(huán)性數(shù) 首階Melnikov函數(shù)　出處：《上海師范大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：非線性系統(tǒng)在物理、生物等科學(xué)中具有廣泛的應(yīng)用.這些學(xué)科中的許多現(xiàn)象如振動(dòng)、捕食-食餌、物種增長(zhǎng)等常需要用非線性系統(tǒng)所確定的數(shù)學(xué)模型來(lái)描述.因此,通過(guò)對(duì)非線性系統(tǒng)解的相關(guān)性質(zhì)的研究來(lái)分析這些系統(tǒng)的動(dòng)力學(xué)行為,具有重要的理論和實(shí)際意義.本文以幾類非線性系統(tǒng)為研究對(duì)象,對(duì)其相圖、Hopf分支、Poincare分支、同宿分支與異宿分支進(jìn)行深入的研究,獲得了一些有趣的結(jié)果.首先,給出了研究光滑與非光滑近-Hamiltonian系統(tǒng)極限環(huán)個(gè)數(shù)的雙參數(shù)擾動(dòng)方法,對(duì)光滑與非光滑近-Hamiltonian系統(tǒng)引入雙參數(shù),導(dǎo)出相應(yīng)的首階Meilnikov函數(shù)的顯式表達(dá)式,來(lái)研究系統(tǒng)的極限環(huán)個(gè)數(shù).應(yīng)用此方法,我們研究了一類分片二次多項(xiàng)式系統(tǒng)和一類三次多項(xiàng)式系統(tǒng)的極限環(huán)的最大個(gè)數(shù),此問(wèn)題分別被[Llibre and Mereu, J-MAA(2014)]口[Li and Zhao, IJBC(2014)]進(jìn)行研究；與這些結(jié)果相比,用雙參數(shù)擾動(dòng)方法可以多獲得一個(gè)極限環(huán).應(yīng)用此方法,我們研究了含三角形異宿環(huán)的二次多項(xiàng)式系統(tǒng)在二次多項(xiàng)式擾動(dòng)下從三角形異宿環(huán)附近可分支出最大極限環(huán)的個(gè)數(shù)問(wèn)題,又稱三角形異宿環(huán)環(huán)性數(shù),證明了文獻(xiàn)[Wang and Han, JMAA(2015)]中定理5.2的結(jié)論.應(yīng)用此方法并結(jié)合引進(jìn)一些新的思想(比如：屬性Z(n,m, l)),我們研究了一類多項(xiàng)式Lienard系統(tǒng)的Hilbert數(shù)并給出了該數(shù)的下界,改進(jìn)和豐富了已有結(jié)果.近年來(lái),[Han et al, JDE(2009)]獲得了含m-階冪零尖點(diǎn)同宿環(huán)的C∞° Mamiltonian系統(tǒng)在任意C∞系統(tǒng)擾動(dòng)下所產(chǎn)生的首階Melnikov函數(shù)在此環(huán)附近的近似展開(kāi)式,并給出了m=1的部分系數(shù)的計(jì)算公式；之后,[Atabaigi et al.,NATMA(2012)]獲得m=2的部分系數(shù)的計(jì)算公式.本文對(duì)一般的m進(jìn)行探討,給出了一種計(jì)算所有的m≥2的部分系數(shù)表達(dá)式的方法.特別地,利用此方法給出m=3部分系數(shù)的表達(dá)式,并利用這些系數(shù)給出了在同宿環(huán)附近出現(xiàn)極限環(huán)的充分條件,同時(shí)也給出了相應(yīng)的應(yīng)用并改進(jìn)了已有的結(jié)果.顯然,冪零尖點(diǎn)是高次奇點(diǎn).一般而言,高次奇點(diǎn)周圍呈現(xiàn)復(fù)雜的軌線結(jié)構(gòu).進(jìn)一步,本文對(duì)一類含高次奇點(diǎn)的可積系統(tǒng)在高次奇點(diǎn)處的局部相圖進(jìn)行分析,獲得所有可能的相圖；其次當(dāng)出現(xiàn)同宿環(huán)時(shí),擾動(dòng)該系統(tǒng),得到了相應(yīng)的首階Melnikov函數(shù)在同宿環(huán)附近的近似展開(kāi)式,同時(shí)給出部分系數(shù)的表達(dá)式；并且利用這些系數(shù)給出存在極限環(huán)的充分條件.最后,我們對(duì)一類分片多項(xiàng)式系統(tǒng)的極限環(huán)個(gè)數(shù)進(jìn)行了研究.首先,給出未擾動(dòng)系統(tǒng)在出現(xiàn)一簇閉軌族時(shí)所有可能的相圖(共42種),并對(duì)滿足其中一種相圖的系統(tǒng)進(jìn)行分片多項(xiàng)式擾動(dòng),研究了Hopf分支和同宿分支.在非光滑情形下,通過(guò)建立Poincare映射獲得判定同宿環(huán)軌道穩(wěn)定性的判定準(zhǔn)則；建立了改變同宿環(huán)軌道穩(wěn)定性來(lái)研究同宿分支和異宿分支的方法并給出了相應(yīng)的應(yīng)用,發(fā)現(xiàn)了Alien極限環(huán)并給出其一般性的定義.
[Abstract]:Nonlinear systems are widely used in physics, biology and other sciences. Many phenomena in these disciplines such as vibration, predator-prey. Species growth often needs to be described by mathematical models determined by nonlinear systems. Therefore, the dynamic behavior of nonlinear systems is analyzed by studying the related properties of the solutions of these systems. It is of great theoretical and practical significance. In this paper, the phase diagram Hopf bifurcation Poincare bifurcation, homoclinic bifurcation and heteroclinic bifurcation are studied. Some interesting results are obtained. Firstly, a two-parameter perturbation method is given to study the number of limit cycles of smooth and non-smooth near-Hamiltonian systems. Two parameters are introduced to smooth and non-smooth near-Hamiltonian systems, and the explicit expressions of the corresponding first-order Meilnikov functions are derived. Using this method, we study the maximum number of limit cycles for a class of piecewise quadratic polynomial systems and a class of cubic polynomial systems. [Llibre and Mereuu, J-MAAMAA 2014]. [Li and Zhao, IJBC (2014); Compared with these results, a new limit cycle can be obtained by using the two-parameter perturbation method. In this paper, we study the problem of the number of maximal limit cycles of quadratic polynomial systems with triangular heteroclinic rings, which can be branched from the vicinity of triangular heteroclinic rings under the disturbance of quadratic polynomials, also called the number of triangular heteroclinic rings, and prove the literature. [Conclusion of Theorem 5.2 in Wang and Han. JMAA (2015). Applying this method and introducing some new ideas (for example, attribute Zang NM, LU). In this paper, we study the Hilbert number of a class of polynomial Lienard systems and give the lower bound of the number, which improves and enriches the existing results. [Han et al. JDEG 2009). In this paper, we obtain the C 鈭，

本文編號(hào)：1467928