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研究生:黃千豪
研究生(外文):Chain-Hau Huang
論文名稱:研究區間算法如何解決勞倫茲吸子存在性之問題
論文名稱(外文):Study of interval arithmeticThe problem of existence of Lorenz attractor
指導教授:田光復
指導教授(外文):Kuang-Fu Tien
學位類別:碩士
校院名稱:國立臺灣大學
系所名稱: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:2009
畢業學年度:97
語文別:英文
論文頁數:47
中文關鍵詞:Lorenz吸子區間算法
外文關鍵詞:Lorenzattractorinterval arithmetic
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The basis of this thesis is to study intensively what is Tucker’s idea, mathematical theoretical basis, rigorous computation in his article and prove to myself what is not very clearly proved, and hopefully apply this new method to establish answers to the existence of attractors of other system.
Contents

Chapter 1. Basics of Lorenz equation 1

1.1 What is Lorenz equation and their properties 1
1.2 Dynamics of Lorenz equation 6
1.3 Does Lorenz attractor exist? 9

Chapter 2. Interval arithmetic and its application 11

2.1 Linear change of variables of the Lorenz equations 11
2.2 Good Pick for return plane 13
2.3 Directed rounding 13
2.4 Interval arithmetic 14
2.5 Local Euler Poincare ́ box and local Euler Poincare ́ map 17
2.6 Search for global Poincare ́ map 24
2.7 Bisection process 28

Chapter 3. A typical one-dimensional chaotic map 30

3.1 One-dimensional map with topological transitivity 30
3.2 Estimation for evolution of cone 31
3.3 Estimation for evolution of Expansion 33
3.4 Existence for forward invariant cone field 36
3.5 Information for expansions of tangent vectors in cone 38

Chapter 4. Dynamics near the origin 40

4.1 Local change of coordinates 40
4.2 Estimation of normal form flow 42

References 47
References.

[1]J. Guckenheimer, A strange, strange attractor, in: The Hopf Bifurcation and its Applications (J. E. Marsden and M. McCracken, eds.), Springer-Verlag, New York, 1976.
[2]J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York, 1983.
[3]D. Gulick, Encounters with Chaos, McGraw-Hill, New York, 1992.
[4]E. N. Lorenz, Deterministic non-periodic flow, J. Atmospheric Sci. 20 (1963), 130–141.
[5]R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966.
[6]R. E. Moore, Methods and Applications of Interval Analysis, Studies in Applied Mathematics, SIAM, Philadelphia, 1979.
[7]J. Murdock, Normal forms and unfoldings for local dynamical systems, Springer-Verlag, New York, 2003.
[8]C. Robinson, Dynamical Systems, 2nd ed., CRC Press, New York, 1995.
[9]S. Smale, Mathematical problems for the next century, Math. Intelligencer 20, 2 (1998), 7–15.
[10]C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, New York, 1982.
[11]M. Viana, What''s New on Lorenz Strange Attractors? Math. Intell. 22, 6-19.
[12]W. Tucker, The Lorenz attractor exists, C.R. Acad. Sci. Paris, Part 328, Sér. I (1999) 1197–1202.
[13]W. Tucker, A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (1) (2002) 53–117.
[14]S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 2003.
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