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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.
 ContentsChapter 1. Basics of Lorenz equation 11.1 What is Lorenz equation and their properties 11.2 Dynamics of Lorenz equation 61.3 Does Lorenz attractor exist? 9Chapter 2. Interval arithmetic and its application 112.1 Linear change of variables of the Lorenz equations 112.2 Good Pick for return plane 132.3 Directed rounding 132.4 Interval arithmetic 142.5 Local Euler Poincare ́ box and local Euler Poincare ́ map 172.6 Search for global Poincare ́ map 242.7 Bisection process 28Chapter 3. A typical one-dimensional chaotic map 303.1 One-dimensional map with topological transitivity 303.2 Estimation for evolution of cone 313.3 Estimation for evolution of Expansion 333.4 Existence for forward invariant cone field 363.5 Information for expansions of tangent vectors in cone 38Chapter 4. Dynamics near the origin 404.1 Local change of coordinates 404.2 Estimation of normal form flow 42References 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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