Make a file logistic.py and test_logistic.py in the same folder as this readme and the plot_logfun.py file. Implement the code for the logistic map in the logistic.py file:

a) Implement the logistic map f(𝑥)=𝑟∗𝑥∗(1−𝑥) . Import pytest and use @pytest.mark.parametrize to test the function for the following cases:

  x=0.1, r=2.2 => f(x, r)=0.198
  x=0.2, r=3.4 => f(x, r)=0.544
  x=0.75, r=1.7 => f(x, r)=0.31875

b) Implement the function iterate_f that runs f for it iterations, each time passing the result back into f. Use @pytest.mark.parametrize to test the function for the following cases:

  x=0.1, r=2.2, it=1 => iterate_f(it, x, r)=[0.198]
  x=0.2, r=3.4, it=4 => iterate_f(it, x, r)=[0.544, 0.843418, 0.449019, 0.841163]
  x=0.75, r=1.7, it=2 => iterate_f(it, x, r)=[0.31875, 0.369152]

c) Import and call the plot_trajectory function from the plot_logfun module to look at the trajectories generated by your code. The plot_logfun imports and uses your logistic.py code. Import the module and call the function in a new plot_script.py file.

Try with values r<3, r>4 and 3<r<4 to get a feeling for how the function behaves differently with different parameters. Note that your input x0 should be between 0 and 1.

a) Write a randomized test that checks that, for r=1.5, all starting points x0 converge to the same value (attractor) f(x, r) = 1/3.

b) Use pytest.mark to mark the tests from the previous exercise with one mark (they relate to the correct implementation of the logistic map) and the test from this exercise with another (relates to the behavior of the logistic map). Try executing first the first set of tests and then the second set of tests separately.

Some r values for 3<r<4 have some interesting properties. A chaotic trajectory doesn't diverge but also doesn't converge.

a) Use the plot_trajectory function from the plot_logfun module using your implementation of f and iterate_f to look at the bifurcation diagram.

The script generates an output image, bifurcation_diagram.png.

b) Write a test that checks for chaotic behavior when r=3.8. Run the logistic map for 100000 iterations and verify the conditions for chaotic behavior:

1. The function is deterministic: this does not need to be tested in this case
2. Orbits must be bounded: check that all values are between 0 and 1
3. Orbits must be aperiodic: check that the last 1000 values are all different
4. Sensitive dependence on initial conditions: this is the bonus exercise below

The test should check conditions 2) and 3)!

For the same value of r, test the sensitive dependence on initial conditions, a.k.a. the butterfly effect. Use the following definition of SDIC.

f is a function and x0 and y0 are two possible seeds. If f has SDIC then: there is a number delta such that for any x0 there is a y0 that is not more than init_error away from x0, where the initial condition y0 has the property that there is some integer n such that after n iterations, the orbit is more than delta away from the orbit of x0. That is |xn-yn| > delta