Part C: Dictionaries, Arrays, Functions, and Modules

Exercises

1. A simple simulation engine: Write a Python function IterateLogistic() that takes three parameters: two real numbers and an integer. Interpret the first two as the current state and the control parameter value for the one-dimensional map: f(x) = r x (1 - x). The function should return the state after the given number of iterations, which is the third parameter. Test your function interactively.
2. Experimental data: Write a Python program LogisticData.py that generates 10 data files, each with 50 logistic map iterates, at map parameter r = 4.0. Start each program run with a different initial condition. The initial conditions should be close: Use x_0 = 0.3 + delta_i, where delta_i = i * .005, i = 0, 1, 2, 3, ..., 9. Name the 10 files "data1" to "data10".
   Hint: Use the program you wrote for the Part B exercises to generate the data files, adding the iteration function you wrote above.
   
3. Write a Python module MatrixIO.py containing two functions. One function should read a file containing a text representation of a matrix; that is, a certain number of lines each containing the same number of numbers separated by spaces. The function should return an array containing this data. Note that the function should work for matrices of any size. The second function should do the inverse: write a matrix of any size to a text file.
4. Processing experimental data: Recall that each line in the data files you just generated has an iteration number and then a state value. Assume that all files have the same length and that these time values agree line by line. Write a Python program MaxSeparation.py that produces another time series consisting of lines with iteration number and d_max, where d_max is the largest distance between all pairs of states taken from the different files at the same iteration number. Store this new times series in a file. What do you see? What is your interpretation of the d_max values as a function of time?
   Hint: Use the matrix module written above by importing it.
   
5. Ordinary differential equation simulation:
   1. Write a Python program OneDEuler.py that integrates one-dimensional ordinary differential equations using the Euler method. (See this week's Lecture Notes for the Euler integration method.) Write the integrator so that you can pass in any function that specifies a one-dimensional ODE. Have the integrator take the current state and time step size and return the state after one time step.
   2. Write a function Pitchfork() that can be passed to the integrator to solve the ODE for pitchfork bifurcations: dx/dt = r x - x³.
   3. Using a time step of dt = 0.03, run from various initial conditions and parameter settings: r = -1.0, 0.0, 1.0 and x₀ = 2.0, 0.1, -2.0. Print out a time series of 20 steps for each of the 9 possible combinations of parameter r and initial condition x₀.