本文關(guān)鍵詞： 狀態(tài)依賴時(shí)滯 van der Pol模型 捕食-被捕食者模型 慢振蕩解 不動點(diǎn)定理 噴射性 攝動法 超臨界Hopf分岔 次臨界Hopf分岔　出處：《湖南大學(xué)》2016年博士論文　論文類型：學(xué)位論文

【摘要】：本文主要研究狀態(tài)依賴時(shí)滯微分方程的動力學(xué)行為.關(guān)于狀態(tài)依賴時(shí)滯微分方程目前還沒有建立完善的理論體系,這一方面的理論和應(yīng)用著作相對來說還很少.不過,隨著很多學(xué)科如物理、化學(xué)、自動控制、人口增長、神經(jīng)網(wǎng)絡(luò)、細(xì)胞繁殖以及傳染性疾病等領(lǐng)域的發(fā)展,越來越多的狀態(tài)依賴時(shí)滯微分方程被用來描述一些模型的動力學(xué)行為.所以,對狀態(tài)依賴時(shí)滯微分方程的深入研究具有非常重要的現(xiàn)實(shí)指導(dǎo)意義.本文主要研究了兩類狀態(tài)依賴時(shí)滯微分模型的動力學(xué)行為.主要內(nèi)容如下:第一,我們研究了三種狀態(tài)依賴時(shí)滯van der Pol模型的動力學(xué)行為.首先,我們通過分析第一種顯性的時(shí)滯依賴于狀態(tài)的van der Pol方程的結(jié)構(gòu),得到了解的有界性、存在唯一性、以及解對初值的連續(xù)依賴性;通過構(gòu)造一個適當(dāng)?shù)木o集和一個連續(xù)的緊映射,再根據(jù)其平衡態(tài)的噴射性,運(yùn)用不動點(diǎn)定理得到其慢振蕩周期解的存在性.然后,我們研究了第二種由狀態(tài)和時(shí)滯一起決定的微分形式的時(shí)滯van der Pol模型的慢振蕩周期解問題.這個模型和前一個van der Pol模型的區(qū)別在于其時(shí)滯函數(shù)的變化.這種時(shí)滯的變化,使得所構(gòu)造的后繼映射對于時(shí)滯項(xiàng)在平衡態(tài)處是不連續(xù)的.為了能夠利用合適的不動點(diǎn)定理,我們通過尋找適當(dāng)?shù)木o集,并且在其上進(jìn)行對時(shí)間項(xiàng)的單位化,以此來構(gòu)造一個輔助的緊狀態(tài)空間,再在這個狀態(tài)空間上構(gòu)造后繼映射,并為了解決時(shí)滯項(xiàng)在平衡態(tài)的不連續(xù)問題,我們構(gòu)造一個后繼映射的輔助映射,從而使得我們在其上構(gòu)造的系統(tǒng)在輔助緊狀態(tài)空間中保持不變,利用擬噴射不動點(diǎn)研究局部動力學(xué)性質(zhì),證明了其慢振蕩周期解的存在性.最后,我們詳細(xì)討論了第三種狀態(tài)依賴時(shí)滯微分van der Pol方程的動力學(xué)行為:我們研究了其平衡點(diǎn)的局部穩(wěn)定性,解的漸近性,以及其Hopf分岔的存在性;通過攝動過程以及Fredholm正交公式得到了方程的局部分岔是超臨界Hopf分岔還是次臨界Hopf分岔的判定,并得到其分岔出的周期解的穩(wěn)定性;利用三種常見的時(shí)滯函數(shù),用數(shù)值模擬的方法,驗(yàn)證了我們所提供的理論分析.第二,我們研究了一類捕食-被捕食者過程模型的動力學(xué)行為.首先,在對方程中的幾個參數(shù)作出適當(dāng)假設(shè)的情況下,我們研究了模型中正平衡點(diǎn)的局部穩(wěn)定性、解的振蕩性等動力學(xué)性質(zhì),尋找到了解產(chǎn)生Hopf分岔的參數(shù)范圍;通過將原系統(tǒng)在正平衡點(diǎn)處攝動化,把狀態(tài)依賴時(shí)滯微分方程的Hopf分岔問題轉(zhuǎn)化為常時(shí)滯微分方程的Hopf分岔問題;通過Fredholm正交公式得到了方程的局部Hopf分岔定理,從而得到其是超臨界Hopf分岔或次臨界Hopf分岔的判定以及分岔所產(chǎn)生的周期解的穩(wěn)定性;利用三種常見的時(shí)滯函數(shù),用數(shù)值模擬的方法,驗(yàn)證了我們所得到的理論分析.
[Abstract]:In this paper, we mainly study the dynamical behavior of state dependent delay differential equations. There is no perfect theoretical system for state dependent delay differential equations at present, but there are few theoretical and practical works on this aspect. With the development of many fields such as physics, chemistry, automatic control, population growth, neural network, cell reproduction and infectious diseases, More and more state-dependent delay differential equations are used to describe the dynamic behavior of some models. In this paper, the dynamic behavior of two classes of state-dependent delay differential models is studied. The main contents are as follows: first, We study the dynamical behavior of three state-dependent delay van der Pol models. Firstly, we obtain the boundedness and uniqueness of the solution by analyzing the structure of the first explicit delay-dependent van der Pol equation. By constructing a proper compact set and a continuous compact mapping, and according to the ejection of its equilibrium state, the existence of periodic solutions of its slow oscillation is obtained by using the fixed point theorem. In this paper, we study the slow oscillatory periodic solution of the second kind of delay van der Pol model, which is determined by both state and delay. The difference between this model and the previous van der Pol model lies in the variation of the delay function. In order to make use of the suitable fixed point theorem, we find the appropriate compact set and make the time term unit on it. In order to solve the discontinuity problem of the delay term in the equilibrium state, we construct an auxiliary mapping of the successor mapping, in order to construct an auxiliary compact state space, and then construct a successor map on the state space, in order to solve the discontinuity problem of the delay term in the equilibrium state, we construct an auxiliary mapping of the successor mapping. Therefore, the system constructed on it remains invariant in the auxiliary compact state space. The local dynamical properties of the system are studied by using quasi-jet fixed points, and the existence of periodic solutions for its slow oscillation is proved. We discuss the dynamic behavior of the third state-dependent delay differential van der Pol equation in detail. We study the local stability of the equilibrium point, the asymptotic behavior of the solution and the existence of its Hopf bifurcation. By means of perturbation process and Fredholm orthogonal formula, the local bifurcation of the equation is determined by supercritical Hopf bifurcation or subcritical Hopf bifurcation, and the stability of the periodic solution of the bifurcation is obtained. The numerical simulation method is used to verify the theoretical analysis provided by us. Secondly, we study the dynamic behavior of a kind of predator-prey process model. In this paper, we study the local stability and oscillatory properties of the positive equilibrium point in the model, and find out the parameter range of the Hopf bifurcation, by perturbing the original system at the positive equilibrium point. The Hopf bifurcation problem of the state dependent delay differential equation is transformed into the Hopf bifurcation problem of the ordinary delay differential equation, and the local Hopf bifurcation theorem of the equation is obtained by the Fredholm orthogonal formula. The results show that the bifurcation is a supercritical Hopf bifurcation or a subcritical Hopf bifurcation and the stability of the periodic solution generated by the bifurcation is verified by using three common time-delay functions and the numerical simulation method.
【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O175

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本文編號：1503667