1. Natural Fractal Contest Results:
Drum Roll Please…

8. Getting into some Mathematical Spiciness
CHAOS
Getting into some
Mathematical Spiciness

9. The Vernacular Chaos cha·os –noun
1. a state of utter confusion or disorder; a total lack of organization or order.
2. any confused, disorderly mass
3. the infinity of space or formless matter supposed to have preceded the existence of the ordered universe.

10. An Almost Rigorous Definition of Chaos
sensitivity to initial conditions:
arbitrarily close to every state S1 of the system, there is a state S2 whose future eventually is significantly different from that of S1. That is "Even the tiniest change can alter the future in ways you can't imagine.".
dense periodic points:
arbitrarily close to every state S1 of the system, there is a state S2 whose future behavior eventually returns exactly to S2.
mixing:
given any two states S1 and S2, the futures of some states near S1 eventually become near S2.

11. The differences between Vernacular Chaos and Mathematical Chaos

12. Population Growth Dynamics
Unrestrained
Restrained
AGAIN:

13. Unrestrained = Exponential Growth
X(t) = X0 bt
Jalapeños: Calculus

14. Step back: Linear Growth
What is the function X(t)
What is dX/dt

15. Restrained = Logistic Growth
r : growth rate
K : carrying capacity
Jalapeños:

16. To Iterate with a Function
Iterating a linear function f(x)=2x :: y=2x
Consider that graphs organize functions by their coordinate points (x, y) or (x, f(x)) y
1) Choose Initial x : X0
2) Find f(X0)
3) Find f(f(X0) …
X0 = 4
F(4) = 6
F(6) = 7
F(7) = 7.5
F(7.5) = 7.75
F(7.75) = 7.875

17. Question: does the initial value X0, affect the long term behavior of the iterative sequence?
Applet:

18. Iteration webs of more complex Functions:
If iteration of a linear equation models exponential growth, what function should be iterated to model logarithmic growth?

19. The Iteration Game (Target)
Just to make sure you’ve got the hang of it:

20. Fixed Points, Cycles, and Stability
Two of the most interesting behaviors of some iterative paths are limiting towards fixed points and Cycling.

21. Fixed points are of three kinds:
fixed point equation
f(x*) = x*.
In general, a fixed point of a function f(x) is a point x* satisfying the
Fixed points are of three kinds:
                                                         
Stable fixed points signal long-term predictability. If the system winds up near enough to the fixed point, its future behavior is easy to predict: it will approach ever nearer the fixed point. Imagine a marble rolling into a bowl. It will wind up at the bottom of the bowl. Push the marble a little away from the bottom of the bowl, and it will roll back to the bottom. This is the essence of stability: small perturbations fade away.
Unstable fixed points behave in the opposite way. Placed EXACTLY at the fixed point, there you will stay. But stray even the slightest molecule away and you depart rapidly. Invert the bowl and you have the right picture.
Indifferent fixed points are the none of the above case. nearby points either do not move at all, or some move nearer while others move farther away.
Descriptions from:

22. Cycles
What happens if f(n) = some value already in the sequence of iteration?
(hint: look at the title of the slide)
Cycles also can have the same kinds of stability as fixed points in terms of what will happen if you use a value close to the cycle:
The path spirals it toward the fixed point.
The path does not approach the fixed point.
                                                                 
A natural question is
"for an n-cycle, can one of the corresponding fixed points of fn(x) be stable and the other unstable?"
The answer is "No," but the proof requires some calculus.

23. This Presentation, to be continued next week…
That’s right, it’s a cliffhanger…

24. Pseudo assignment:
Master of the Games
Your class presentations are next week, so your only assignment is to continue to work on those…
But, if you need a break now and then, try to become a master of the Target and Cycle Games on
IF YOU HAVEN’T, TAKE A LOOK AT YOUR TOPICS SOON!!! AND ME ANY QUESTIONS