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研究生:林怡佳
研究生(外文):Yi-Chia Lin
論文名稱:一維動態系統之歸返映射
論文名稱(外文):First Return Maps for One-Dimensional Dynamics
指導教授:石至文
指導教授(外文):Chih-Wen Shih
學位類別:碩士
校院名稱:國立交通大學
系所名稱:應用數學系
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:2002
畢業學年度:90
語文別:英文
論文頁數:30
中文關鍵詞:歸返映射動態系統
外文關鍵詞:First Return MapsOne-Dimensional Dynamics
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我們主要探討Schwarzian derivative為負值和有兩個極值點的一維離散型動態系統,並觀察它的疊代行為。同時,在這系統上定義歸返映射,運用歸返映射的許多重要性質與特徵,我們可以得到這個動態系統有拓撲轉移性(topology transitivity)、週期點的稠密性、及對初始條件的靈敏性(sensitively dependent to the initial conditions)等結果。然後,會舉一些三次多項式的例子來印證我們的結論。另一方面,我們也在具短暫混沌性質的神經網路上運用我們的理論,並舉一些例子來說明。

We investigate the iteration of maps of the interval which have negative Schwarzian derivative and two critical points. Using the characteristic of the first return map, we conclude the topological transitivity, dense periodic points, and sensitive dependence on initial conditions for the considered one-dimensional discrete-time dynamical systems. Some cubic polynomials are taken as examples to illustrate the results. We also attempt to apply the theory to the one-dimensional chaotic neural network.

Contents:
1. Introduction.............................................. 1
2. The Dynamics for One-Dimensional Maps.................... 3
2.1 Preliminary......................................... 3
2.2 Bi-Modal Mappings................................... 6
3. The First Return Map..................................... 9
3.1 The Fundamental Properties.......................... 9
3.2 Chaos............................................... 16
3.3 Examples............................................ 18
4. One-Dimensional TCNN..................................... 21
4.1 The Fundamental Properties for TCNN................. 21
4.2 Chaotic Behaviors................................... 27
5. Conclusion............................................... 28

[1] J.Banks, J.Brooks, G,Cairns, G.Davis and P.Stacey,On Devaney's Definition of Chaos, Ameri. Math. Monthly 99, (1992), pp.332-334.
[2] L.Chen and K.Aihara, Chaotic simulated annealing for combinatorial optimization, Neural Networks, 8 (1995), pp.915-930.
[3] L.Chen and K.AiharaChaos, Chaos and asymptotical stability in discrete-time neural networks, Physica D, 104(1997), pp.286-325.
[4] L.Chen and K.AiharaChaos, Global searching ability of chaotic neural networks, IEEE Trans. Circuits Syst. I: Fundamental Theory and Applications, 46(1999), pp.974-993.
[5] R.Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, 1989.
[6] R.Devaney, A First Course in Chaotic Dynamical Systems, Addison-Wesley, 1992.
[7] W.de Melo and S. J.Van Strien, One-Dimensional Dynamics, Springer-Verlag, New York, Heidelberg, Berlin, 1993.
[8] C.Robinson, Dynamical Systems, Stability, Symbolic Dynamics, and Chaos, CRC Press, London, 1995.

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