Natural Fractal Contest Results:

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Natural Fractal Contest Results

• Drum Roll Please

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CHAOS

• Getting into some
• Mathematical Spiciness

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The Vernacular Chaos

• chaos  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.

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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.
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The differences between Vernacular Chaos and
Mathematical Chaos
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Population Growth Dynamics

• Unrestrained

Restrained
AGAIN http//www.otherwise.com/population/expo
nent.html
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Unrestrained Exponential Growth
X(t) X0 bt
Jalapeños Calculus
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Step back Linear Growth

• What is the function X(t)
• What is dX/dt

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Restrained Logistic Growth
r growth rate K carrying capacity
Jalapeños
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To Iterate with a Function

• Iterating a linear function f(x)2x y2x
• 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
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• Question does the initial value X0, affect the
  long term behavior of the iterative sequence?

Applet http//math.bu.edu/DYSYS/applets/linear-we
b.html
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Iteration webs of more complex Functions

• If iteration of a linear equation models
  exponential growth, what function should be
  iterated to model logarithmic growth?
• http//math.bu.edu/DYSYS/applets/nonlinear-web.htm
  l

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The Iteration Game (Target)

• Just to make sure youve got the hang of it
• http//math.bu.edu/DYSYS/applets/targetPractice.ht
  ml

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Fixed Points, Cycles, and Stability

• Two of the most interesting behaviors of some
  iterative paths are limiting towards fixed points
  and Cycling.

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Fixed points
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 http//classes.yale.edu/Fract
als/
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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.
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This Presentation, to be continued next week
Thats right, its a cliffhanger
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Master of the Games
Pseudo assignment

• 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
http//math.bu.edu/DYSYS/applets/
IF YOU HAVENT, TAKE A LOOK AT YOUR TOPICS
SOON!!! AND E-MAIL ME ANY QUESTIONS