區間函數上週期3和混沌現象的關聯

Period Three and Chaos in Interval Maps

鄭凱仁(Kai-Jen Cheng)

指導教授 ： 彭栢堅 ； 吳裕振

中原大學/理學院/應用數學研究所/博士(2014年)

https://doi.org/10.6840/cycu201400445

摘要

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.

關鍵字

週期3 ； 混沌 ； 動態系統

並列摘要

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.

並列關鍵字

dynamical system ； chaos ； period 3

參考文獻

[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.