MTH455/555 
Introduction to Chaotic Dynamical Systems

 
Tuesday - Thursday: 11:00 AM - 12:15PM
Lippitt Hall Room 201

Instructor    Araceli Bonifant  
Office: Lippitt Hall 202 G
Phone: 874-4394
Email: bonifant@uri.edu

Office Hours:

Book: Chaos and Introduction to Dynamical Systems by K. T. Alligood, T. D. Sauer and J. A. Yorke

Course Description. Chaotic phenomena flourish in nature. They often originate in systems whose components are governed by simple laws, but whose overall behavior is very complex. The objective of this course is to give an introduction to the theory of chaotic systems.

Prerequisites:MTH 243 or permission of instructor.


Tentative List of Topics:

  • One-dimensional maps
    -- One-dimensional maps
    -- Cobweb Plot: Graphical representation of an Orbit
    -- Stability of fixed points
    -- Periodic points
    -- The family of logistic maps
    -- The logistic map g(x)=4x(1-x)
    -- Sensitive dependence on initial conditions
    -- Itineraries
    -- Period 3 implies chaos


  • Two-dimensional maps
    -- Sinks, sources and saddles
    -- Linear maps
    -- Coordinate changes
    -- Nonlinear maps and the Jacobian matrix
    -- Stable and unstable manifolds
    -- Counting the periodic orbits of linear maps on a torus


  • Chaos
    -- Lyapunov exponents
    -- Chaotic orbits
    -- Conjugacy and the logistic map
    -- Transition graphs and fixed points
    -- Basins of attraction
    -- Sharkovskii's theorem


  • Fractals
    -- Cantor sets
    -- Probabilistic construction of fractals
    -- Fractals from deterministic systems
    -- Fractal basin boundaries
    -- Fractal dimension
    -- Computing the box counting dimension


    • Evaluation Policy:

  • Homework sets:         60%
  • Project :                     40%

    • Suggested Reading:

  • Dynamical Systems:Stability, Symbolic Dynamics and Chaos, by Clark Robinson.
  • An Introduction to Chaotic Dynamical Systems by Robert Devaney.
  • Any book on Dynamical Systems on R.

    Standards of behaviour: Students are responsible for being familiar with and adhering to the published "Community Standards of Behavior: University Policies and Regulations" which can be accessed in the University Student Handbook. If you must come in late, please do not disrupt the class. Please turn off all cell phones, pagers, or any electronic devices.

    Special Accommodations: Any student with a documented disability is welcome to contact me as early in the semester as possible so that we may arrange reasonable accommodations. As part of this process, please be in touch with Disability Services for Students Office at 330 Memorial Union, 401-874-2098.

    About your submitted work: All submitted work must be your own. If you consult other sources (class readings, articles or books from the library, articles available through internet databases, or websites) these MUST be properly documented, or you will be charged with plagiarism and will receive an F for the paper. In some cases, this may result in a failure of the course as well. In addition, the charge of academic dishonesty will go on your record in the Office of Student Life. If you have any doubt about what constitutes plagiarism, visit the following website: http://www.lib.usm.edu/legacy/plag/whatisplag.php, the URI Student Handbook, and UNIVERSITY MANUAL sections on Plagiarism and Cheating at http://web.uri.edu/studentconduct/files/2015-2017-Student-Handbook.pdf - Academic Honesty Procedures. Any good writer's handbook as well as reputable online resources will offer help on matters of plagiarism and instruct you on how to acknowledge source material. If you need more help understanding when to cite something or how to indicate your references, PLEASE ASK.