【摘要】：動(dòng)力系統(tǒng)是非線性科學(xué)領(lǐng)域重要的研究?jī)?nèi)容.歷經(jīng)龐家萊、李雅譜諾夫等大量學(xué)者的的研究、探索、發(fā)展和完善,動(dòng)力系統(tǒng)已成為現(xiàn)代數(shù)學(xué)中的一個(gè)獨(dú)立的、有趣的、具有很好的科研價(jià)值和應(yīng)用前景的研究方向.其中,穩(wěn)定性和分支是動(dòng)力系統(tǒng)的動(dòng)態(tài)行為研究領(lǐng)域里主要的的研究對(duì)象.穩(wěn)定性是動(dòng)力系統(tǒng)拓?fù)浣Y(jié)構(gòu)平衡性的一種體現(xiàn),對(duì)它的深入探索能更好的展現(xiàn)豐富、復(fù)雜的系統(tǒng)動(dòng)力學(xué)特征.分支是指系統(tǒng)的一些特性發(fā)生突變的現(xiàn)象伴隨參數(shù)發(fā)生變化而經(jīng)過臨界值得過程中.本文借助規(guī)范型理論、中心流行定理、分支理論等全面的展示了兩類離散和連續(xù)動(dòng)力系統(tǒng)隨參數(shù)變化時(shí)豐富的動(dòng)力學(xué)行為.對(duì)離散時(shí)間動(dòng)力系統(tǒng)的動(dòng)態(tài)特征結(jié)構(gòu)的研究,利用向前歐拉離散法詳細(xì)的研究了一個(gè)離散的傳染病模型的分支和混沌現(xiàn)象,通過中心流形定理和分支理論推理和證明了離散系統(tǒng)不動(dòng)點(diǎn)的穩(wěn)定性和在一定的參數(shù)條件下從不動(dòng)處產(chǎn)生分支及分支閉軌的穩(wěn)定性.最后,并借助數(shù)值模擬清晰地觀察到系統(tǒng)出現(xiàn)了穩(wěn)定的周期窗口、倍周期重疊、周期到混沌和混沌到穩(wěn)定的周期窗口的動(dòng)態(tài)行為的跳躍變化.對(duì)連續(xù)系統(tǒng)的研究,本文探究了一個(gè)延遲反饋控制混沌動(dòng)力系統(tǒng)隨分支參數(shù)的變化而產(chǎn)生的系統(tǒng)動(dòng)力學(xué)行為的變化規(guī)律及反饋控制器對(duì)控制混沌的有效果性.利用中心流形、規(guī)范型和分支定理研究了參數(shù)變化時(shí)系統(tǒng)的穩(wěn)定性及分支參數(shù)通過某一臨界值時(shí)系統(tǒng)產(chǎn)生Hopf分支周期解和分支周期解的性質(zhì)(穩(wěn)定性、方向和振幅).利用數(shù)值方法進(jìn)一步的得出混沌是可以控制的,延遲反饋控制可以誘發(fā)穩(wěn)定的周期閉軌.
[Abstract]:Dynamic system is an important research content in the field of nonlinear science. After a lot of research, exploration, development and improvement, dynamic system has become an independent and interesting research direction with good research value and application prospect in modern mathematics. Among them, stability and bifurcation are the main research objects in the field of dynamic behavior of dynamic systems. Stability is a reflection of the equilibrium of dynamical system topological structure, and its deep exploration can better show the rich and complex dynamic characteristics of the system. Bifurcation refers to the phenomenon that some characteristics of the system change with the change of the parameters and pass through the critical value in the process of obtaining the critical value. In this paper, two kinds of discrete and continuous dynamical systems are presented with the help of normal form theory, central popular theorem, bifurcation theory and so on. In this paper, the bifurcation and chaos of a discrete infectious disease model are studied in detail by using forward Euler discrete method, which is used to study the dynamic characteristic structure of discrete time dynamic system. By means of the central manifold theorem and bifurcation theory, the stability of fixed point of discrete system and the stability of bifurcation and closed orbit generated from fixed point under certain parameter conditions are proved. Finally, with the help of numerical simulation, it is clearly observed that the system has stable periodic window, double period overlapping, period to chaos and chaotic to stable periodic window dynamic behavior jump change. For the study of continuous systems, the dynamic behavior of a delayed feedback chaotic dynamic system with the variation of bifurcation parameters and the effectiveness of the feedback controller in controlling chaos are studied in this paper. By using the central manifold, normal form and bifurcation theorem, we study the stability of the system and the properties (stability, direction and amplitude) of the Hopf bifurcation periodic solution and the bifurcation periodic solution (stability, direction and amplitude) when the bifurcation parameter passes through a certain critical value. By using numerical method, it is further concluded that chaos is controllable, and delay feedback control can induce stable periodic closed orbits.
【學(xué)位授予單位】：北方民族大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O19

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相關(guān)期刊論文 前3條

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本文編號(hào)：2337426