2-1) Below is a graph of the tent map, defined by the following iteration rule:
Xn+1 = s Xn ..........................if Xn<1/2
Xn+1 = s (1-Xn) ....................if Xn>1/2
a) Show the maximum height of the tent map graph is s/2.
b) Show the nonzero fixed point occurs at Xf = s/(1+s).
c) For s > 1, show the first iterate of s/2 has the value s (1 - s/2 ).
d) Use graphical iteration to locate four points which the tent map takes to the fixed point w of the Figure below.
e) Numerically calculate the four values of x you found when s = 2.
e) Show how to locate infinitely many points which the tent map takes to w.
2-2) LIFE AND NONLINEAR DYNAMICS
Write a parable of your own. Specifically, take an instance from your life where a small choice, perhaps made in an offhand fashion, led eventually to a significant difference in your life. For your parable to be apt to the context of this course, the train of events flowing from your choice has to be deterministic. Try to identify some of the nonlinearities in the "dynamics of your life" that helped cause your choice to initiate the unexpected chain of events it did.
To emphasize the magnitude of the effect of this small choice, construct a plausible scenario of the result of your making a different small choice (under the same "life dynamics").
a) Do some research to find out what do the economists refer to by the "tulips' boom/bust" experience.
b) What do you think is the conexion between market crashes and nonllinear systems?
c) And outbreak of war and nonllinear systems?
d) How about epidemics?
2-4) The Game of Life
a) Determine by hand the next generation of the Life configurations below:
b) Determine by hand the next two generations of the Life configurations below:
Using LifeLab and the random initial condition, observe which stable configurations arise spontaneously from the Life rules. (Allow some time to pass before drawing any conclusions - the transients must be allowed to die down.).
A textbook description is no substitute for the experience of Life. Play with LIFE to get a hint of the Game's many wonders. But here's a warning label for the user: Life is powerfully addictive! Use only when not faced with an impending deadline.
"The common experiences of self-awareness, free will, and feelings can never be understood in terms of physical laws."
"People are mechanistic entities whose behaviors emerge from the same principles that govern all forms of matter."
Here are two opposite views of humanity.
Construct an argument that supports each position.
Which argument do you find more compelling?