博碩士論文 107221602 詳細資訊

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姓名 黎阮翠雯(Le Nguyen Thuy Van)  查詢紙本館藏   畢業系所 數學系
論文名稱 微分方程中解結構之探究:(1)雲山蚜蟲-鳥類模型解之動態行為(2)球面上橢圓偏微分方程其雙極奇異解之分類與結構性
★ 薛丁格方程式上直立波解的分類。★ Conformality of Planar Parameterization for Single Boundary Triangulated Surface Mesh
★ 一些線性矩陣方程其平滑及週期的最小 l_2-解之探討★ 關於漢米爾頓矩陣的某些平滑性分解
★ 在N維實數域之雙調和微分方程★ 一維動態系統其週期解之研究
★ 一些延滯方程其週期解之探討★ On the Blow-up solutions of Biharmonic Equation on a ball
★ 雙調和微分方程其正整域解的存在性與不存在性之探討★ 高階橢圓偏微分方程解的存在性及其行為之研究
★ 有絲分裂中染色體運動之動態分析★ 非線性橢圓方程及系統中解的唯一性和結構性之探討
★ On the Positive Solution for Grad-Shafranov Equation★ 關於三物種間之高流動性Lotka-Vollterra競爭擴散系統的波形極限行為
★ 非線性橢圓型偏微分方程系統之解結構分析★ On the study of the Golden-Thompson inequality
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摘要(中) 微分方程可以用來描述我們周圍的世界。 它們被應用於生物學,經濟學,物理學,化學和工程學等廣泛領域。

例如,微分方程模型可以顯示物種的種群增長。 在這裡,我們研究了雲杉芽蟲系統在鳥類捕食和殺蟲劑作用下的動力學。 在一定條件下,還表明了動力系統極限環的存在。 此外,我們找出了在特定情況下產生極限環唯一性的一些條件。 如果殺蟲劑曾經用於控制芽蟲的種群,即$n = 1$,
如果由於殺蟲劑的作用而導致的最大芽蟲死亡率低於某個特定值,我們將獲得極限環的唯一性。 同樣,在兩次噴灑殺蟲劑的情況下(即 $ n = 2 $)的情况下,我们也可以获得周期轨道的唯一性。

在後面的部分中,我們考慮 $Delta u + V_{1}( heta) u + V_{2}( heta) |u|^{p-1}u=0$ 上 $mathbb{S}^n$
並僅根據方位角研究正解的結構 $ heta$.
$p$ 和潛在功能 $V_{1}, V_{2}$.
在臨界情況下,顯示了非振盪奇異解的唯一性,圍繞該奇異解,存在無窮多個奇異解。 我們還展示了當非線性指數越過臨界值時奇異解的結構變化 $p_S$.
摘要(英) Differential equations can be used to describe the world around us. They are applied into a wide range of fields, from biology, economics, physics, chemistry and engineering.

For example, differential equations model can present the population growth of species. Here we study the dynamics of the spruce budworm system with the effect of bird predation and insecticide. The existence of limit cycle of dynamical system is also shown under certain conditions. Moreover, we figured out some conditions which yield the uniqueness of limit cycle in specific cases. In case insecticide is used once to control budworm population, i.e, $n=1$, if the maximum budworm mortality rate due to the action of insecticide is less than a particular value, we obtain the uniqueness of limit cycle. Similarly, we also achieve the uniqueness of periodic orbit in case insecticide is sprayed twice, i.e, $n=2$.

In the later part, we consider $Delta u + V_{1}( heta) u + V_{2}( heta) |u|^{p-1}u=0$ on $mathbb{S}^n$ and investigate the structures of positive solutions depending only on the azimuthal angle $ heta$. We shall give the existence of all %positive singular-
singular solutions in all nonlinear cases of $p$ and potential functions $V_{1}, V_{2}$. In critical case, the uniqueness of non-oscillatory singular solution is shown, around which infinitely many singular solutions are oscillating. We also show the change of structure of singular solutions when the exponent of the nonlinearity crosses the critical value $p_s$.
關鍵字(中) ★ 雲山蚜蟲-鳥類 模型解之動態行為
★ 球面上橢圓偏微分方程其雙極奇異解之分類與結構性
關鍵字(英) ★ Spruce budworm-birds interaction
★ Elliptic equations on the unit sphere Sn
論文目次 摘要 i Abstract ii Acknowledgement iii
1 Introduction to stability of non-linear systems and some Biological models 1
1.1 Stabilityoflinearandnon-linearsystems..................... 2
1.2 OrbitalstabilityandLyapunovfunctionsmethod . . . . . . . . . . . . . . . . 3 1.2.1 Orbitalstability.............................. 3 1.2.2 Lyapunovfunctions............................ 5
2 Introduction to Spruce budworm - bird interactions 7
2.1 Stabilityanalysis.................................. 10 2.2 Uniquenessofperiodicorbits........................... 13 2.3 Conclusion .................................... 18
3 Singular solutions for non-linear elliptic equations on Sn 19 3.1 Preliminaries ................................... 22 3.2 Non-oscillatorysingularsolutions......................... 25 3.3 Uniquenessofnon-oscillatorysingularsolution . . . . . . . . . . . . . . . . . 28 3.4 Oscillatorysingularsolutions ........................... 30
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指導教授 陳建隆(Jann-Long Chern) 審核日期 2020-7-21
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