Part E: Plotting (continued) and Visualization

Exercises

The dynamics homework assigned from NDAC this week asks you to explore a number of one-dimensional maps and the behaviors they produce. If appropriate, modify the programs presented in lab (and available from the course's Python Programming webpage). Whether you're modifying an existing program or are asked to write a new one, give the working Python code the names listed below for each problem.
  1. NDAC Problem 10.2.6 (Cosine map): Program name should be CosineMap.py.
  2. NDAC Problem 10.2.8 (Cubic map): Program name should be CubicMap.py.
  3. NDAC Problem 10.4.4 (Superstable period-3 orbit): Program name should be SuperstableP3.py.
  4. NDAC Problem 10.4.5 (Band-merging and crisis of periodic windows): Program name should be BandMerge.py.
  5. NDAC Problem 10.5.5 (Tent map): Program name should be TentMap.py.
  6. A one-dimensional map bifurcation tool:
    1. Make a collection of one-dimensional map functions that can be used by any of the programs. Do this by making a Python library module OneDMaps.py that includes all of the one-dimensional map functions we've seen so far: logistic, cubic, cosine, exponential, tent, and cusp maps.
    2. The specification of each map in OneDMaps.py should give additional information that is helpful to a parent program that uses the library. In particular, define a function that returns an example initial condition. That is, define a function LogisticMapInitialCondition() that a parent program can call to get an interesting initial condition—such as one that leads to a chaotic orbit. Do the same for a default initial parameter value: e.g., LogisticMapInitialParameter(). Include two more functions that return valid minimum and maximum x values. These will be helpful, for example, for graphics display or scaling in the parent program. For the Logistic map these values would be 0.0 and 1.0, but for other maps they could be different. Finally, include a function LogisticMapNameString() that returns the map's name “Logistic Map, f(x) = r x (1 - x)”. This string can be used to label the plot. Of course, each map will require its own unique functions.
    3. Starting from an existing bifurcation diagram program, write a Python program OneDMapBifn.py that loads this library, just like any Python module.
    4. Modify it so that where it called the logistic map function, it instead calls a generic OneDMap() function with the appropriate number of parameters.
    5. When the program first starts up, have it print a list of available maps. Have it then ask the user which map to use and which parameter range to explore. (Use raw_input() and not input().) The program should use the map's specified default initial condition.
    6. The program should plot the bifurcation diagram with correct and informative labels.
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