82 lines
3.3 KiB
Markdown
82 lines
3.3 KiB
Markdown
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# Testing Project for ASPP 2023 Mexico
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## Exercise 1 -- @parametrize and the logistic map
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Make a file `logistic.py` and `test_logistic.py` in the same folder as this
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readme and the `plot_logfun.py` file. Implement the code for the logistic map
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in the `logistic.py` file:
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a) Implement the logistic map f(𝑥)=𝑟∗𝑥∗(1−𝑥) . Use `@parametrize`
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to test the function for the following cases:
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```
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x=0.1, r=2.2 => f(x, r)=0.198
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x=0.2, r=3.4 => f(x, r)=0.544
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x=0.75, r=1.7 => f(x, r)=0.31875
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```
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b) Implement the function `iterate_f` that runs `f` for `it`
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iterations, each time passing the result back into f.
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Use `@parametrize` to test the function for the following cases:
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```
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x=0.1, r=2.2, it=1 => iterate_f(it, x, r)=[0.198]
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x=0.2, r=3.4, it=4 => f(x, r)=[0.544, 0.843418, 0.449019, 0.841163]
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x=0.75, r=1.7, it=2 => f(x, r)=[0.31875, 0.369152]
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```
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c) Import and call the `plot_trajectory` function from the `plot_logfun`
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module to look at the trajectories generated by your code. The `plot_logfun`
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imports and uses your `logistic.py` code. Import the module
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and call the function in a new `plot_script.py` file.
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Try with values `r<3`, `r>4` and `3<r<4` to get a feeling for how the function
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behaves differently with different parameters. Note that your input x0 should
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be between 0 and 1.
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## Exercise 2 -- Check the convergence of an attractor using fuzzing
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a) Write a numerical fuzzing test that checks that, for `r=1.5`, all
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starting points converge to the attractor `f(x, r) = 1/3`.
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b) Use `pytest.mark` to mark the tests from the previous exercise with one mark
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(they relate to the correct implementation of the logistic map) and the
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test from this exercise with another (relates to the behavior of the logistic
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map). Try executing first the first set of tests and then the second set of
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tests separately.
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## Exercise 3 -- Chaotic behavior
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Some r values for `3<r<4` have some interesting properties. A chaotic
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trajectory doesn't diverge but also doesn't converge.
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## Visualize the bifurcation diagram
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a) Use the `plot_trajectory` function from the `plot_logfun` module using your
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implementation of `f` and `iterate_f` to look at the bifurcation diagram.
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The script generates an output image, `bifurcation_diagram.png`.
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b) Write a test that checks for chaotic behavior when r=3.8. Run the
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logistic map for 100000 iterations and verify the conditions for
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chaotic behavior:
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1) The function is deterministic: this does not need to be tested in
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this case
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2) Orbits must be bounded: check that all values are between 0 and 1
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3) Orbits must be aperiodic: check that the last 1000 values are all
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different
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4) Sensitive dependence on initial conditions: this is the bonus
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exercise below
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The test should check conditions 2) and 3)!
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## Bonus Exercise 4 -- The Butterfly Effect
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For the same value of `r`, test the sensitive dependence on initial
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conditions, a.k.a. the butterfly effect. Use the following definition of SDIC.
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>`f` is a function and `x0` and `y0` are two possible seeds.
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>If `f` has SDIC then:
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>there is a number `delta` such that for any `x0` there is a `y0` that is not
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>more than `init_error` away from `x0`, where the initial condition `y0` has
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>the property that there is some integer n such that after n iterations, the
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>orbit is more than `delta` away from the orbit of `x0`. That is
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>|xn-yn| > delta
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