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.
- Tu Thurs 10:00 - 11:30
- SciBldg 216
- Alligood et al's
Chaos: an Introduction to Dynamical Systems
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.
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
- 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:
(You can see a NumberCrunch version of these on the Mac file server at Server:Messages:Jim's Stuff:assign_1 )
- Plot a graph of the function y = sin(x)/x for -10 < x < 10.
- Solve the following equation: tan(x) = 0.3 x^2
- Find and sketch the intersection of the surface z=sin(x^2+y^2) and the plane x+y+z=0.
- Find the 100'th Fibonnaci number.
- Find the 100'th prime number.
- 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.
- 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.
- 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")
- 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.
- for Tue Mar 3
Choose one of the following topics and submit a full, formal
solution. This is for a grade, guys.
- 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.
- Do the Challange 1 from chapter 1, namely follow the text's
recipe to show that a period-3 orbit implies chaotic orbits.
- 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.
- 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.
- for Tue Mar 3
- Read chapter 5, on two-dimensional Lyap exponents and the Horseshoe map.
- Check out my Lyap program.
- chapter 6
- T6.2, Computer Exp. 6.2, T6.9
- chapter 7,8
- chapter 9
- T9.2, Ex 9.1, 9.2, Comp Exp 9.3 (with ode)
- Final Project - no later than May 11
Do one of the following three:
- Explain the intuition and proof of the Poincare-Bendixson Theorem (chap 8).
- 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.
- Write and run a program to calculate Feigenbaum's number in the
map X[n+1] = a X[n] (1 - X[n]).
Note that you have a variety of computer tools available on campus,
as we will discuss in class, including:
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