Lecture Notes for Math 170: Ideas in Mathematics (Spring 2007) - Topics in Chaos Theory - , Study notes of Mathematics

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Lecture Notes for Math 170: Ideas in

Mathematics (Spring 2007)

by Nathanael Leedom Ackerman January 19, 2007

1

1 TALK SLOWLY AND WRITE NEATLY AND BIG!! 2

1 TALK SLOWLY AND WRITE NEATLY AND BIG!!

2 Review

Recall from last time a discrete dynamical system is a way to model some behavior. It is done by finding a series of numbers P 0 , P 1 ,... related by a formula of the form Pn = f (Pn− 1 ). Further

Definition 2.0.1. Let Pn+1 = f (Pn). We then say a state a is an equilibrium if f (a) = a.

and

Definition 2.0.2. We say a is an unstable equilibrium of Pn = f (Pn− 1 ) if f (a) = a and for all sufficiently small ≤ P 0 = a + ≤ implies limn→∞ Pn 6 = a

3 CHAOS 3

3 Chaos

Definition 3.0.3. In mathematics chaos is when a small change in initial conditions leads to a large change in outcome.

Describe the Butterfly effect.

• Term coined by Edward Lorenz in 1963
• Phrase was “One meteorologist remarked that if the theory were correct, one flap of a seagull’s wings could change the course of weather forever”

Example

Discuss errors and repeated calculations using a calcula- tor.

Population Formula

Give population formula Pn+1 = λx(1−x) (explain about predators)

3 CHAOS 4

Say we want to consider maps where we stay in the in- terval [0, 1] so we can consider it as a % of the maximum possible population.

We want to discuss this over time. One way is to simply write a computer program which spits out values. How- ever we want a pictorial representation of what is going on.

Functions

• Define what a function is.
• Define it as a black box.
• Make connection to black box of probability
• Range of a function/Domain of a function

Graphing Functions

3 CHAOS 5

• Define what a graph of a function.
• Go through an example.

Web Diagrams

• Explain the relation of the graph to the dynamical system.
• Draw the x = y line.
• Go step by step though the case of the exponential
• Go step by step through a case where λ ≤ 1 (so everything is below x = y. Observe that if λ ≤ 4 then everything.

Observe period points (if λ > 3 .3)

If Time

Go through the isosolese triangle and cycles (with a web

diagram) Give Quiz