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 upperlevel 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 16, and the second half on differential equations,
in chapters 79. 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
 for Tue Jan 27
 Read chapter 1, on onedimensional maps and the Logistic Map
 Check out your computer/numerical skills with the following exercises:
 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.
(You can see a NumberCrunch version of these on the Mac file server at Server:Messages:Jim's Stuff:assign_1 )
 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
 Project:
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 1dimensional maps, x > a x (1x), look at both (a)
the period3 to period6 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 3orbit which changes to a 6orbit).
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 period3 orbit implies chaotic orbits.
 In a numerical and analytic examination of the 2dimensional
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 twodimensional 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 PoincareBendixson 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]).
Resources
Note that you have a variety of computer tools available on campus,
as we will discuss in class, including:
Physics & Astronomy page 
Jim's Schedule
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