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NAVIGATIONAL LINKS:

Course Web Site

Computer Exercises

RELATED EXERCISES:

Logistic Map

Chaos and Lyapunov

Exponents

Invariant Measures

Fractal Dimensions

Period Doubling Routes

to Chaos

ILES

Lecture on Chaos and

Lyapunov Exponents /

Invariant Measure

Python:

Logistic iteration hints

Logistic iteration answer and demo

INKS

Python nanotutorials

Starting up a new Python

project under Linux

Python setup info

Course Web Page

Robert M. May,

"Simple mathem atical

mod els with very

complicated dynamics"

, Nature 261, 459-467

(1976).

Physics 7682 / Computing & Information Sciences 6229 - Fall 2012

INSTRUCTOR: ERICH MUELLER

MONDAYS & FRIDAYS 1:30-3:30, ROCKEFELLER B3 (DIRECTIONS)

http://www.physics.cornell.edu/~myers/teaching/ComputationalMethods/

BackgroundWe have five exercises on discrete maps. This one develops the main tools, and should be done

before the other four.

Often models of nature have a discrete deterministic structure. A classic example is in population

dynamics, where one hypothesises that the number of rabbits in year is simply a function of

the number in the previous year:

If is linear one has the logistic equation, and the number of rabits grows exponentially. This

is a reasonable model if there are infite resources no preditors. The next most sophisticated model

would be to add a nonlinear term representing either the finite resources or the role of preditors.

This gives the logistic map

which is a map of the interval onto itself. There is one parameter , which will change the

behavior of the map. For example, at , this function precisely folds the unit interval in half

and stretches it (nonuniformly) to cover the original domain. It turns out that this simple map

displays some complicated behavior.

Learning Goals

Science: You will learn about Chaos and discrete maps.

Computation: You will learn about iterative algorithms.

Procedure

xn

=f( ).xn+1 xn

f(x)

f(x) = 4x(1 x),

(0, 1) = 1


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1. Create a new directory

2. Download Logistic iteration hints,and rename it IterateLogistic.py.

3. Open this file in a text editor (kate or emacs -- we recommend against us ing kedit) or

load it into the IPython dashboard (started with ipython notebook --pylab inline. In

this file, you will notice Python code that has already been written, but it mostly consists

of hints to help you flesh out the code, i.e., comments (lines that begin with #) and

documentation strings (material enclosed in triple quotes """ that document what each

module, class, and function is about and can be queried with the Python help()

function). You will also notice the keyword pass embedded within each function body:

pass is a legal Python expression that means "do nothing". The first thing you will do

when you start to define a function is to remove the pass statement and replace it with

new code.

4. Open a terminal window, move to the correct directory and start ipython, or click on the

notebook in the dashboard. You will find it convenient to start python with the --pylab

flag -- ie type ipython --pylab or ipython notebook --pylab.

5. define a function f(x,mu) which when called with and returns .

6. Use pylab commands to plot f as a function of x for different values of mu.

7. Write a function Iterate(g, x0, N, mu)which is passed a function g, an initial point x0

and integer N and a parameter mu which returns g(g(...(g(x0,mu),mu)...). That is it

iterates the function Ntimes, starting at x=x0.

8. Test this function with f. Are there any other functions you can write to test it more

easily?

9. Write a function IterateList(g, x0, N, mu)which acts just like iterate, but returns a

list of the sequential values of x. (A more verbose description is in the hints file.)

10. Play a bit with this function.

Type ylim((0,1)) then plot(IterateList(f,0.5,100,0.1)). This should

make a graph of the iterates of x, starting with x0=0.5, using mu=0.1. As

you can see, the bunnies don't do so well for small mu.

Try different values of mu. You should find that mu=0.25 is special. How does

the behavior of the iterates change when mu>0.25?

How does the behavior change when mu>0.75?

To explore this structure in more detail, see

Period Doubling Routes to Chaos

this structure can be compactly encoded in a "bifurcation diagram"

11. Write a function BifurcationDiagram(g, x0, nTransient, nCycle, muArray,

showPlot)which takes as arguments:

g -- a function

x0 -- the starting value for the iterates

nTransients -- an integer

nCycle -- another integer

muArray -- a list of mu values

showPlot -- True/False

For each value of mu in muArray, this function should iterate g, starting a x0. It throwsaway the first nTransient iteratates. It then stores the next nCycle iterates. It then

makes a plot where the x-axis is the values of mu, and the y-axis are the values of x.

12. Try the following parameters:

BifurcationDiagram(f, 0.1, 500, 500, scipy.linspace(0., 1.00001, 200))

BifurcationDiagram(f, 0.1, 500, 500, scipy.linspace(0.85, 1.00001, 300))

BifurcationDiagram(f, 0.1, 500, 500, scipy.linspace(0.96, 0.97, 400))

13. To explore this structure in more detail see Fractal Dimensions

James P. Sethna, Christopher R. Myers, Erich J. Mueller.

Last modified: July 2012

x 4x(1 x)


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Statistical Mechanics: Entropy, Order Para meters, and C omplexity, now a vai lable at

Oxford University Press (USA, Europe).

logistic period doub

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