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hands_on_solutions/logistic_fun/readme.md

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+# Testing Project for ASPP 2023 Mexico
+
+## Exercise 1 -- @parametrize and the logistic map
+
+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) . Use `@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 `@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 => f(x, r)=[0.544, 0.843418, 0.449019, 0.841163]
+  x=0.75, r=1.7, it=2 => f(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.
+
+## Exercise 2 -- Check the convergence of an attractor using fuzzing
+a) Write a numerical fuzzing test that checks that, for `r=1.5`, all
+starting points converge to the 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.
+
+## Exercise 3 -- Chaotic behavior
+Some r values for `3<r<4` have some interesting properties. A chaotic
+trajectory doesn't diverge but also doesn't converge.
+
+## Visualize the bifurcation diagram
+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)!
+
+
+## Bonus Exercise 4 -- The Butterfly Effect
+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
+