Dynamical Systems (Fall 2010)

Math 211A (Topics in Analysis)

     Course Code: 45025

 

MWF 1:00 1:50

Final Exam: Wednesday, Dec 8, 1:30-3:30pm  

Instructor: Anton Gorodetski
        Email: asgor@uci.edu
        Phone: (949) 824-1381
        Office Location: RH 510G
        Office Hours: Monday 2-3pm 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 examples of topological and smooth dynamics: linear maps, translations on the torus, gradient flows, expanding maps, symbolic dynamical systems.
  • Fundamental concepts of dynamical systems: conjugacy, equivalence, classification, invariants, structural stability.
  • Local analysis in smooth dynamics: hyperbolic periodic orbits, Hadamard-Perron theorem, Haryman-Grobman theorem, local structural stability, normal forms.
  • Symbolic dynamics, coding, horseshoes, attractors.
  • Hyperbolic dynamics: horseshoes, Anosov diffeomorphisms, DA maps, Smale-Williams solenoid, general hyperbolic sets, Markov partitions, coding, local product structure, stability, spectral decomposition.
  • Fractals in dynamics. Dynamically defined Cantor sets.
  • Topological entropy. Calculation of a topological entropy for topological Markov shifts, hyperbolic automorphisms of the torus, solenoid.
  • Applications of hyperbolic dynamics to some problems in celestial mechanics (three body problems) and spectral theory (spectral properties of Fibonacci Hamiltonian).

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.


Homework

Homework 1 (due Friday, October 8)

Homework 2 (due Friday, October 15)

Homework 3 (due Friday, October 22)

Homework 4 (due Friday, November 5)

Homework 5 (due Friday, November 19)

Homework 6 (due Wednesday, December 1)

Final (take home, due Friday, December 10)


 

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