My office phone, 523-6878, goes straight to voice mail.
You can send me e-mail at
Jim.Swift@nau.edu.
Here is my weekly schedule.
Please feel free to contact me any time via e-mail with any
questions about the math, or with any feedback about the class.
You can always come to my office, AMB 110, during these times:
M: 1:45-4:00
W: 1:45-4:00
You can always send me e-mail, drop in to my office,
or make an appointment if these times aren't convenient.
Darryl Nester's Slope Field Applet. Here is the Slope Field App, Old Interface
Chart of letters of the Greek alphabet.
One-Dimensional Maps (March 5 - April 28)
Read this seminal article:
Simple mathematical models with very complicated dynamics, by Robert May.
With a focus on the Logistic map.
Takashi Kanamaru's Logistic Map Time Series.
This shows the cobweb diagram and the time series.
You *might* be able to run the Java program after saving it to disk.
Here is a
Bifurcation Diagram from the wikipedia page on the logistic map.
Here is my bifurcation diagram and skeleton
as printed on the LMC wall.
The bottom shows the first few iterates of x0 = 0.5.
Unfortunately, the detail in the upper right corner
was not printed. A small version of the proper diagram is posted in the AMB near
the mail room.
This page of the Logistic Map has another version of the bifurcation diagram for the LMC wall.
Mark Haferkamp's cobweb diagram generator.
This was a project when Mark took MAT 667 in 2017.
Takashi Kanamaru's Logistic Map Bifurcation Diagram.
You can zoom in to observe Feigenbaum's universal constants, \(\delta \approx 4.669\) and \(\alpha \approx -2.503\).
These are similar I made:
Here is an animation with 3 bifurcation diagrams with \( 3 \leq a \leq 4\) (level 1),
then level 3 (period doubing 4 to 8 at left, 4 to 2 band merging at right) and level 5 (period doubing 16 to 32 at left, 16 to 8 band merging at right)
Here is an animation with 2 bifurcation diagrams (levels 3 and 5).
Here is an animation with 200 frames (zooming from level 3 to level 5).
My Mathematica notebook, iteratedMap1D.nb
Sharkovskii ordering and Sharkovskii's Theorem.
Here is an article on A Simple Proof of Sharkovsky's theorem (Revisited).
A proof that period 3 implies all periods
, that I followed in class on April 23.
Feigenbaum Constants
The Period Doubling Operator. This is my notation, which is not common. Most people define the domain of \(\mathcal T\)
to be a subset of all functions with \(f(0) = 1\).
My domain has \(f(-1) = f(1) = -1\).
Mathematica notebood iteratedMapForPeriodDoubling.nb.
Figure of logistic map \(f_a\) with superstable period 8 (red), \(\mathcal{T}f\) (green), \(\mathcal{T}^2 f\) (blue)
and \(\mathcal{T}^3 f\) (black).
Figure of logistic map \(f_a\) at accumulation of period doubling (red), \(\mathcal{T}f\) (green), \(\mathcal{T}^2 f\) (blue, hidden)
and \(\mathcal{T}^3 f\) (black).
This figure explains Feigenbaum's universal constants
\(\delta = 4.669\ldots\) and \(\alpha = 2.5029\ldots\).
Here is an animation with 3 bifurcation diagrams with \( 3 \leq a \leq 4\) (level 1),
then level 3 (period doubing 4 to 8 at left, 4 to 2 band merging at right) and level 5 (period doubing 16 to 32 at left, 16 to 8 band merging at right)
Here is an animation with 2 bifurcation diagrams (levels 3 and 5).
Here is an animation with 200 frames (zooming from level 3 to level 5).
The Lorenz Equations (January 31 - March 3)
Homework 2 is due at the beginning of class on Monday, March 24.
Here is a link to the 1963 paper by Lorenz.
My page on the
Lorenz attractor.
Mark McClure visualization Shows convection cells and has the round button (rounds \(\pi\) to 3.142, etc.)
Hendrik Wernecke Simulation. Simple app with only STOP, Reset, and a slider for \(\rho\). This uses Euler's method which is very inaccurate.
Malin Christersson simulation. Swarms of butterflies!
Takashi Kanamaru's Lorenz Equation simulation. This is a good one, but does not run on all machines.
All of those programs have their drawbacks. Twenty years ago we had better web-based demonstrations
written by dynamical systems experts, but most don't run today. Peter Scott's home page has a link to a page on the Lorenz equations. He has a C++ program that integrates the Lorenz equations that runs in linux.
A possible program project for this course would be to get this program to run on
your laptop or the classroom computer,
and demonstrate the program for the class.
Newton's Law of Cooling, Scaling and other Dynamical Systems (January 22-29)
Homework 1 is due at the beginning of class on Monday, January 27.
You can send a pdf before class starts, or bring hardcopy to class.
Here are some comments about the dead body cooling problem.
Introduction (January 13-17)
Dynamical systems: PDEs, ODEs and iterated maps.
Monday: Each of these can be conservative or dissipative. This allows a
classification
of dynamical systems.
We use Darryl Nester's
Slope Field App, Old Interface
to investigate these ODEs with parameters
Newon's Law of Cooling,
\(\displaystyle \frac{du}{dt} = -k (u-A)\),
the simple harmonic oscillator, \(\displaystyle m\frac {d^2 u}{dt^2} = - k u\),
the simple pendulum, \(\displaystyle m\frac {d^2 \theta}{dt^2} = -\frac{mg}{L} \sin(\theta)\),
and
the van der Pol oscillator, \(\displaystyle \frac {d^2 y}{dt^2} = - y + \varepsilon (1-y^2) \frac{dy}{dt}\).
