Emily and Me

Make your own bifurcation diagram of the logistic map

Coarse logistic map bifurcation diagram

I made this image with a javascript-based app. Basically I ran the app three times.
1. N=10000, n=8, x0=0.2, r=0:0.05:3
2. N=10000, n=8, x0=0.2, r=3.02:0.02:3.54
3. N=20000, n=200, x=0.2, r=3.55:0.01:4
The last run is a bit long and may tell you that a script is not responding. Just let it finish.

Notre Dame AMS Sectional Meeting

This weekend is the AMS Sectional Meeting at Notre Dame. We are organizing a special session titled “Undergraduate Mathematics Education: A Vision for the 21st Century.” I am very much looking forward to this forum for discussing how the changing social context will affect students, faculty, mathematics and society. I’ll be talking on Saturday morning in this session. On Friday afternoon, I’ll give a talk in the applications of differential geometry special session. It should be a profitable weekend.

Fall Break 2010

A long weekend in New England with the family added some serenity to the mid-semester.

RET @ ND 2010

The RET math program at Notre Dame finished on July 30, with the presentation of several projects at the RET Symposium. The topic for summer in mathematics was Dynamical Systems and MATLAB. Projects included Polygonal Billiards, Coupled Difference Equations, Image Processing for Puzzle Solving, Random Walks with Random Velocity, and Game Simulation.

Cobweb diagram for the logistic equation

Cobweb diagram for the logistic equation with r=3.9

The discrete logistic equation is given by the difference equation .  It is easy to see that under ordinary assumptions, the logistic equation has a fixed point at .  However, depending on the initial condition and the value of , a number of different types of behaviors are possible.  The fixed point may be attractive (meaning that applying the logistic equation over and over pulls successive values closer to it) or repulsive (once the x value gets close to the fixed point, it is pushed away), or iterations of the logistic equations may orbit the fixed point retracing its steps in a periodic orbit.

Here is a link (http://http://sites.saintmarys.edu/~sbroad/example-logistic-cobweb.html) to a javascript implementation of a cobweb diagram for the logistic equation.  This diagram is generated based on the user’s choice of initial condition, parameter, and number of iterations.  The coordinate of the point of intersection between the parabola and the line is the fixed point of the logistic equation.  The diagram allows the user to see how much the behavior of the logistic equation is affected by the choice of the parameter.  It also shows the sensitive dependence on initial conditions for certain values of which are a hallmark of a chaotic dynamical system.

Lorenz Attractor and Runge-Kutta

A picture of the Lorenz attractor with sigma=10, rho=28, beta=8/3

We will talk about the Runge-Kutta method for numerically solving differential equations. Click the link http://sites.saintmarys.edu/~sbroad/example-runge-kutta.html for a Javascript Runge-Kutta solution example, where you can see how accuracy is affected by decreasing the step size.

Happy New Year!

It’s 2010!  I have always preferred even numbered years to odd numbered.  Despite some technical difficulties, the new year is starting well and we had a very pleasant  trip to Houston for Christmas.  It’s amazing how warm 55 degrees can feel after acclimating to 15.  We even spent a day at Sea World in San Antonio where the temperature was in the mid 70s.

Also, I recently learned that I am the recipient of a Fulbright grant to visit the University of Sao Paulo where my friend Jorge Sotomayor is a professor in the applied mathematics department of the Instituto de Matemática e Estatística.  It is a great opportunity for me to do some research and experience life in Brazil.

Wishing you the best of 2010!

Witch and Dinosaur Halloween

Emily and Aiden collected a trick-or-treating hatrick disguised as a witch and a dinosaur.  They started Sunday at the Lil Zoo Boo at the Potawatomi Zoo.  Then they traversed the Haunted Halls of Saint Mary’s College on Thursday.  They finished the trifecta with a circuit through the neighborhood on Saturday.

Aiden as a T. Rex at the Lil Zoo Boo

Emily as a witch at the Lil Zoo Boo

Publication of OCW Course

Professor Alex Himonas‘ Elements of Calculus I OpenCourseWare course was published on the Notre Dame OCW site.  I am grateful for the experience of producing the course.  This course is truly “Calculus for Everyone” as Prof. Himonas likes to say, and the presentation of this course is rich, interesting and it makes it possible for students anywhere in the world to learn calculus.

This OCW course has received a lot of attention and has been viewed by people in more than 33 countries.