Chaos

Jim Mahoney (mahoney@marlboro.edu)

General Info

Time
Tu Thurs 10:00 - 11:30
Place
SciBldg 216
Text
Alligood et al's Chaos: an Introduction to Dynamical Systems
This is an upper-level mathematics course which will look at some fairly recent results in dynamical theory, motivated by computational work, namely chaos, fractals, and all that. We will use the computer to explore and analyze these systems, as well as traditional pencil and paper methods. If you don't already have some familiarity with computers and numerical methods, then you''ll learn as we go along.

Much of my PhD work was done in this field, and it's all stuff that I'm looking forward to doing again. Loads of fun.

Syllabus

My rough plan is to follow the authors suggestion, namely to spend the first half of the term on difference equations, in chapters 1-6, and the second half on differential equations, in chapters 7-9. Although we may start up a bit slowly to get everyone up to speed on the computers, we will be covering roughly one chapter per week in this class once we get going.

Since this is a math course, expect to see some assignments with the word "proof" floating around, and discussions of just what a "proof" is - and not just "find the answer" kinds of problems.

Assignments

  1. for Tue Jan 27
    • Read chapter 1, on one-dimensional maps and the Logistic Map
    • Check out your computer/numerical skills with the following exercises:
      1. Plot a graph of the function y = sin(x)/x for -10 < x < 10.
      2. Solve the following equation: tan(x) = 0.3 x^2
      3. Find and sketch the intersection of the surface z=sin(x^2+y^2) and the plane x+y+z=0.
      4. Find the 100'th Fibonnaci number.
      5. Find the 100'th prime number.
      (You can see a NumberCrunch version of these on the Mac file server at Server:Messages:Jim's Stuff:assign_1 )
  2. for Tue Feb 2
    • Finish your study of chapter 1.
    • Do problems T1.3, Experiment 1.2, T1.6, T1.9, T1.16, 1.3, and either 1.13 or 1.8.
  3. for Tue Feb 9
    • Finish reading chapter 2, on two dimensional maps.
    • Look at T2.4, work through Example 2.13 and try to replicate what they do in the text for yourself, and take a whack at T2.5 and T2.7. We'll talk more about the Henon map and forced, damped pendulum on Tuesday.
  4. for Tue Feb 16
    • Really finish reading chapter 2, on two dimensional maps.
    • Play around with Computer Experiment 2.2
      (Here's Jim's C program.)
    • Try exercises 2.1, 2.5, 2.8
    • Start reading Chap 3, especially 3.1 on Lyaponov exponents.
    • Do T3.2 & Computer Exp. 3.1.
      (See the NumberCrunch file Lyaponov_Chap3.1 in "Jim's Stuff")
  5. for Tue Feb 24
    • Finish chapter 3.
    • Try T3.13, 3.1, 3.4, 3.7, and
    • explain the argument on pg 123 that proves the existance of dense chaotic orbits.
  6. for Tue Mar 3
    • Project:
      Choose one of the following topics and submit a full, formal solution. This is for a grade, guys.
      1. In a numerical and analytic examination of the Logistic family of 1-dimensional maps, x -> a x (1-x), look at both (a) the period-3 to period-6 bifurcation point between a=3.82 and a=3.86 as well as (b) the a=4.0 map, and show examples of orbits which behave as (1) a source, (2) a sink (and describe its basin), (3) chaotically, and (4) eventually periodic, and (5) a period doubling transition (i.e. a 3-orbit which changes to a 6-orbit). In each case, find the Lyaponov exponent as well as numerically calculated examples and discuss what's going on and how you can tell.
      2. Do the Challange 1 from chapter 1, namely follow the text's recipe to show that a period-3 orbit implies chaotic orbits.
      3. In a numerical and analytic examination of the 2-dimensional Henon map with b=0.4 and a>0.85, find examples of orbits which behave as (1) sources, (2) sinks, and (3) saddles, (4) chaotically, and (5) any other types you know of. In each case calculate the Jacobian and describe its significance, show the orbit explicitly, and discuss their basins and any stable and/or unstable manifolds.
      4. Do the Challange 2 from chapter 2, namely follow the text's recipe to examine the Cat map and show how it is related to the Fibonnacci series.
  7. for Tue Mar 3
    • Read chapter 5, on two-dimensional Lyap exponents and the Horseshoe map.
    • Check out my Lyap program.
  8. chapter 6
    • T6.2, Computer Exp. 6.2, T6.9
  9. chapter 7,8
    • 7.5, 7.8, 8.1, 8.2, 8.12
  10. chapter 9
    • T9.2, Ex 9.1, 9.2, Comp Exp 9.3 (with ode)
  11. Final Project - no later than May 11
      Do one of the following three:
    1. Explain the intuition and proof of the Poincare-Bendixson Theorem (chap 8).
    2. Analyze the following diff eq in these ranges:
      • x'' = -a x - b x' + alpha x y
        y'' = -c y - e y' + beta x^2 y^2
      • (a) For alpha=beta=b=e=0, a=3, c=4, find a periodic orbit and explain the dynamics of the system.
      • (b) For a=3, c=4, b=e=0.2, alpha=beta=0, again explain the dynamics.
      • (c) Find a chaotic trajectory when alpha and beta are not zero. Discuss.
      • (d) Find a Lyap. Function for (a), (b), and discuss briefly.
    3. Write and run a program to calculate Feigenbaum's number in the map X[n+1] = a X[n] (1 - X[n]).

    Resources

    Note that you have a variety of computer tools available on campus, as we will discuss in class, including:
    • NumberCrunch (Macintosh: Server:Applications:Area Programs:Physics), an array oriented graphical programmable calculating tool
    • Maple (Mac: Server:Applications:Area Programs:Math), a symbolic and 3D graphing math program - powerful but awkward
    • MathCad (Mac: Server:Applications:Area Program:Math), an engineering-oriented math display/design/solving tool
    • Your own programs written in c, pascal, or a language of your choice, on akbar (main campus unix machine), big (the one in my office), or the Macs (Applications:Programming) or PCs. Check out my chaos directory for some descriptions on how to do it.
    • a variety of unix tools on big, including
      • dstool (dynamical systems tool), a powerful unix/X-windows system for looking at chaotic systems,
      • ode (ordinary differential equations), a simple environment for generating solutions to initial-value diff eq's,
      • graph and it's family of "graph-X", "graph-ps" and so on which can generate graphs from files of points,
      • gnuplot, another graphing tool, and
      • gcc - a C compiler for writing your own code.
      All of these and more are (or will be soon) also described in /~mahoney/chaos/.

      Don't be shy - stop by and I'll show you how it works.

    Physics & Astronomy page | Jim's Schedule

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