Dynamical Systems (Spring 2008)

Math 211B (Topics in Real Analysis)

     Course Code: 45015

 

MWF 1:00 – 1:50 // 114 MSTB

Final Exam: Wednesday, June 11, 1:30-3:30pm  

Instructor: Anton Gorodetski
        Email: asgor@uci.edu
        Phone: (949) 824-1381
        Office Location: 219 MSTB (Multipurpose Science & Technology Building)
        Office Hours: Monday 1-3pm or by appointment

Dynamical systems is the study of the long-term behavior of evolving systems. The modern theory of dynamical systems originated at the end of the 19th century with fundamental question concerning the stability and evolution of the solar system. Attempts to answer those questions led to the development of a rich and powerful field with applications to physics, biology, meteorology, astronomy, economics, and other areas. The mathematical core of the theory is the study of the global orbit structure of maps and flows with emphasis on properties invariant under coordinate changes.

This introductory course is aimed at advanced undergraduates, graduate students, physicists and other non-experts who may want to gain a basic understanding of the subject.

The following topics will be covered:

  • Basic notions and examples of topological and smooth dynamics: equivalence, classification, invariants, conjugacy, structural stability.
  • Low-dimensional dynamics. Sharkovsky Theorem. Rotation number.
  • Symbolic dynamics, coding, horseshoes, attractors.
  • Hyperbolicity. The Hadamard-Perron Theorem. The Hartman-Grobman Theorem. Hyperbolic sets and attractors.
  • Chaos and fractals.
  • Applications to Number Theory. Van der Waerden Theorem.
  • Last one or two lectures will be devoted to an overview of some recent developments at a theory of dynamical systems, including the discussion of several most challenging open problems. 

Recommended Texts:

  • A.Katok, B.Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, any edition.
  • M.Brin, G.Stuck,  Introduction to Dynamical Systems, Cambridge University Press, 2002.

Additional references will be given for a few topics not covered by these books.

 

 

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