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研究生:鄭凱仁
研究生(外文):Kai-Jen Cheng
論文名稱:區間函數上週期3和混沌現象的關聯
論文名稱(外文):Period Three and Chaos in Interval Maps
指導教授:彭栢堅吳裕振吳裕振引用關係
指導教授(外文):Kenneth PalmerYuh-Jenn Wu
學位類別:博士
校院名稱:中原大學
系所名稱:應用數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2014
畢業學年度:102
語文別:英文
論文頁數:50
中文關鍵詞:週期3混沌動態系統
外文關鍵詞:dynamical systemchaosperiod 3
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動態系統主要考慮系統中,軌道的終極行為.而其中最重要的研究課題是尋找系統中的吸子,並判斷吸子附近系統的行為.
在動態系統中有下列幾種常見的吸子:
1固定點
2週期軌道
3混沌不變集
在這一篇論文中,我們研究了下列兩個一維離散動態系統的問題,在這兩個問題中,我們將對上述的吸子做嚴格的討論.
1單峰函數上週期3軌道和混沌現象的關聯:
我們在這部分的研究中將把 logistic map 所具有的性質,擴展到一般的單峰函數上.
2一個生物模型上的混沌行為:
我們在這一部分的研究中,將仔細的研究 generalized resource budget map 上的混沌現象.

In the subject of dynamical systems, we consider the asymptotic behavior of the orbits. And the most important object of dynamical systems is to find the attractors and describe the behavior on the attractors.
There are several kinds of attractors:
1.fixed points
2.period orbits
3.chaotic invariant set

In this dissertation, we study two questions regarding one dimensional dynamical systems and in
these two questions we will consider the attractors listed above.
1.Period 3 and Chaos for Unimodal Maps:
We extend the property of logistic map, to a large class of maps, called unimodal maps.

2.Chaos in a Model for Masting:
We study the dynamics of the generalized resource budget map.

目錄
摘要 I
Abstract II
謝辭 III
目錄
IV
圖目錄
V
1 Introduction 1
2 Preliminaries 3
2.1 Bifurcation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Sharkovsky’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .12
2.3 Singer’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
2.4 Devaney’s Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Period 3 and chaos for unimodal maps 22
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
3.2 Ordering of the critical points of f3 relative to a period 3 orbit for a
unimodal map f. . ..................................................... . 23
3.3 Ordering of the critical and fixed points of f3 for a unimodal map with a stable and an unstable period 3 orbit . . . . . . . . . . . . . . . . . . . . 25
3.4 Adding the assumption of negative Schwarzian derivative . . . . . . . . . 26
3.5 Semi-Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30
3.6 Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4 Chaos in a model for masting 34
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
4.2 Dynamics of the map when 0 < k ≤ 1 and k > (1 +sqrt(5))/2 . . . . . . . . 37
4.3 Form of iterates of g2 when 1 < k < 2 . . . . . . . . . . . . . . . . . . 38
4.4 Properties of g^(2^p) on [Bp, 1] . . . . . . . . . . . . . . . . . . . . .44
4.5 Attracting Set of Intervals . . . . . . . . . . . . . . . . . . . . . . . 46
4.6 Snapback repellers . . . . . . . . . . . . . . . . . . . . . . . . . . . .48
Reference 50
[1] B. Aulbach and B. Kieninger, An elementary proof for hyperbolicity and chaos of the logistic maps,J. Difference Eqns. Appl., 10 (2004), 1243–1250.
[2] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney’s Definition of Chaos, Amer.Math. Monthly, 99 (1992), 332–334.
[3] J. Bechhoefer, The Birth of Period 3, Revisited, Math. Magazine, 69 (1996), 115–118.
[4] S. Bassein, The Dynamics of a Family of One-Dimensional Maps, Amer. Math. Monthly, 105(1998), 118–130.
[5] S. M. Chang and H. H. Chen, Applying Snapback Repellers in Resource Budget Models, Chaos, 21(2011), 043126.
[6] R. L. Devaney, “An Introduction to Chaotic Dynamical Systems”, 2nd Edition, Addison-Wesley,Redwood City, 1989.
[7] R. L. Devaney, “A First Course in Dynamical Systems: Theory and Experiment”, Westview Press,1992.
[8] S. N. Elaydi, Discrete Chaos, 2nd Edition, Chapman and Hall/CRC, 2007.
[9] S. N. Elaydi, Discrete Chaos, Chapman and Hall/CRC, 1999.
[10] W.B. Gordon, Period Three Trajectories of the Logistic Map, Math. Magazine, 69 (1996), 118–119.
[11] Y. Isagi, K. Sugimura, A. Sumida, and H. Ito, How Does Masting Happen and Synchronize, J. Theor. Biol, 187 (1997), 231–239.
[12] R. L. Kraft, Chaos, Cantor sets, and hyperbolicity for the logistic maps, Amer. Math. Monthly, 106(1999), 400–408.
[13] T.-Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985–992.
[14] F. R. Marotto, Snap-back repellers imply chaos in Rn, J. Math. Anal. Appl., 63 (1978), 199–223.
[15] H. O. Peitgen, H. Jurgens, and D. Saupe “Chaos and Fractals: New Frontiers of Science”, 2nd Edition, Springer, 2004.
[16] C. Robinson, “Dynamical Systems: Stability, Symbolic Dynamics, and Chaos”, 2nd Edition, CRC Press, 1998.
[17] A. Satake and Y. Iwasa, Pollen-coupling of Forest Trees: Forming Synchronized and Periodic Reproduction out of Chaos, J. Theor. Biol, 203 (2000), 63–84.
[18] A. N. Sharkovsky, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Math. J., 16 (1964), 61–71.
[19] D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978),260–267.
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