- Exams Returned
- Again: Add 5 to your raw score, then consider your score out of 80.
- New assignment:
Wed 3/24 More Chaos - Make sure that you've read section 6.5
- Reminder: Homework pp. 498, #19-24; due Mon, 3/29
- Chaos is roughly defined as
sensitive dependence on initial conditions, but what this means is that if you start at two different points in a system (for example, in weather you might be thinking of a state of three variables, temperature, barometric pressure, and wind speed), very close to each other but distinct, then eventually the system will produce wildly different behaviors (even those for which the starting conditions were extremely close). - Last time we had a look at the calculator exercises in the text:
- A simple rule,
- repeated over and over,
- leads to complicated dynamics.
- We also looked at an important real world example: The Lorenz Equations
- The Lorenz attractor in 3-D
(here's a local copy)
- Make your own butterfly! (here's a local copy)
- The Lorenz attractor in 3-D
(here's a local copy)
- Let's talk about the logistic map:
(our authors use this in their discussion of the Verhulst Model for Population growth). This will lead us back to fractals, as well as to the new idea of "cobwebbing". Here's a nice animation.
- Cobwebbing - another example (#5 and 6), using a "tent function"
- A gorgeous example of what are called "Julia Sets" (this is the material of section 6.4, which we're not going to cover).