A Pluto notebook to do some simple experiments with chaos theory
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### A Pluto.jl notebook ### | |
# v0.19.9 | |
using Markdown | |
using InteractiveUtils | |
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). | |
macro bind(def, element) | |
quote | |
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end | |
local el = $(esc(element)) | |
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) | |
el | |
end | |
end | |
# ╔═╡ b1c4f074-b973-49eb-81cd-7bae1d8e339b | |
begin | |
using PlutoUI | |
using Plots | |
end | |
# ╔═╡ f535d33c-a9b5-4c64-9408-879f2adbfae8 | |
using StatsBase | |
# ╔═╡ 6b7a41a0-2b91-11ed-1db9-099d4ed52a0c | |
md" | |
# _Easy Chaos!_ | |
Pluto is a programming environment for _Julia_, designed to be **interactive** and **helpful**. Changes made anywhere in the notebook affect the entire notebook. Pluto is reactive so you don't need to recalculate cells. | |
This is useful in studying chaos because small changes made to a function can have big effects. You can define a function, run it, and immediately see the effects. | |
" | |
# ╔═╡ fcdfd7bd-47c1-4ab8-a6d6-b3a5cc01b302 | |
md" | |
## _A Linear Equation_ | |
The equation of a line is $y = mx + b$ where $m$ is the slope and $b$ is the $y-$intercept. The slope is how much $y$ changes for each unit change in $x$, so for any starting point $x$, if you move one step to the right to $x + 1$, the new $y-$value becomes $m(x+1) + b = mx + m + b$. Subtract $y = mx + b$ and all that's left is $m$. | |
A step by $1$ in the $x-$direction gives a change by $m$ in the $y-$direction, which is the slope of the line. | |
If $x=0$, then $y = m \times 0 + b = b$ or the $y-$intercept is $b$. Since two points define a line, then start at $b$ on the $y-$axis, move to $x=1$ and add $m$ (or subtract $m$ if $m$ is negative) to get a second point. Draw your line through the two points. | |
The line can also be defined as a function, | |
$$f(x) = mx + b.$$ | |
Let's start by plotting some lines. | |
" | |
# ╔═╡ bde2d54e-d760-4189-ab83-b561f9e14a5b | |
md" | |
We need a way to change the values of $m$ and $b$. First, we'll make a slider for the slope $m$ and set the initial value to $m=1$. | |
" | |
# ╔═╡ a42ec3ae-7ec9-421f-9618-9198c2460807 | |
@bind m Slider(-5:0.1:5, default=1) | |
# ╔═╡ 196417da-9335-40a4-9514-a2da44734275 | |
print("m = ", m) | |
# ╔═╡ 6e5d364b-9786-430f-8322-16eccb8ff39b | |
md" | |
Next, we'll make a slider for $b$ and set it to $b=0$. Both can be varied between $-5$ and $5$. | |
" | |
# ╔═╡ ca11fada-6f31-4a9e-af8b-83237fa766f8 | |
@bind b Slider(-5:0.1:5, default=0) | |
# ╔═╡ 818ad39e-2ea5-4e96-aeaf-34fa53906f73 | |
print("b = ", b) | |
# ╔═╡ 5d5c759a-d405-4342-845f-a6b4eb412ddf | |
begin | |
x = -5:5; | |
y = m .* x .+ b; | |
plot(x,y, label = "f(x) = mx + b", xlims = (-5,5), ylims = (-5,5)) | |
end | |
# ╔═╡ b906dce5-cf29-452d-b286-2fd0dcea1782 | |
md" | |
Notice that the origin of the plot is in the center. Move the sliders for $m$ and $b$ to get a new plot. Interesting, but not chaos. | |
" | |
# ╔═╡ 62349737-9a6d-4e2e-9667-b009ac599c6a | |
md" | |
## _A Little Chaos_ | |
Chaos is doing nearly the same thing over and over and expecting wildly different results. Let's start with the linear equation above, with $m = 1$ and $b = -1$. | |
$$y = f(x) = x - 1.$$ | |
Starting at $x = 1$, the output of the function is $y=0$. Using this new value as the input, $f(0) = -1$. If you do this several times you get the uninteresting sequence | |
$$x = \{1,0,-1,-2,-3, \ldots \}.$$ | |
Instead of calculating $x_1 = mx_0 + b$ and then $x_2 = mx_1 + b$ and so on, we can do this in a function *iter_f* with inputs $m,b,x_0,n$ where $x_0$ is the starting point, and $n$ is the number of iterations. | |
" | |
# ╔═╡ 98a59205-bb10-447b-a60f-b1a9494223e3 | |
function iter_f(m,b,x₀,n) | |
y = zeros(1,n) | |
y[1] = x₀ | |
for k = 2:n | |
y[k] = m*y[k-1] + b | |
end | |
return y | |
end | |
# ╔═╡ ff706d3c-89dd-4003-9f03-d3298f6fdbcf | |
md" | |
We can try this function with the equation $y = x - 1$ starting at $x_0 = 0.4961$ and $x_1 = 0.4962$ for $n = 10$ iterations. | |
" | |
# ╔═╡ abe68984-391d-4b7b-bbd6-ebbc1bead377 | |
y₀ = iter_f(1,-1,0.4961,10) | |
# ╔═╡ ccf7564b-4815-4c04-97f4-24a894001f69 | |
y₁ = iter_f(1,-1,0.4962,10) | |
# ╔═╡ af1b038f-284a-41c1-baad-f11d8e002e9b | |
md" | |
Taking the difference between the outputs gives an uninteresting output: | |
" | |
# ╔═╡ 67768d78-689f-46f3-b0d5-7f25cd178249 | |
y₁ - y₀ | |
# ╔═╡ 4148003a-802f-440d-b14f-d982dda368e4 | |
md" | |
Now, let's change the equation slightly to $f(x) = 2x - 1$, and use the same starting points as before. | |
" | |
# ╔═╡ abe779f2-f7ba-4933-9eb8-110a2dfd2624 | |
y₂ = iter_f(2,-1,0.4961,10) | |
# ╔═╡ 9adb8b0f-e80b-408c-b06d-4f53dc5d95d8 | |
y₃ = iter_f(2,-1,0.4962,10) | |
# ╔═╡ 1f48c58d-419a-42d3-b4ee-951161829bab | |
md" | |
The difference between these sequences is: | |
" | |
# ╔═╡ 273bd83b-5d57-40f9-9bc6-dfe124031c36 | |
y₃ - y₂ | |
# ╔═╡ 849730c0-a09f-4280-94c8-430a3f015962 | |
md" | |
Multiply the differences by $10000$ to make the changes more obvious: | |
" | |
# ╔═╡ 6de59735-9dfa-4cb1-81d7-88ec02d57ab6 | |
10000 * (y₃ - y₂) | |
# ╔═╡ 737854c5-5858-4c12-9b74-15f4fa90fc1c | |
md" | |
This is a little bit surprising, but notice that each number is twice the previous number. It looks like the sequence $\{ 2^0, 2^1, 2^2, \dots 2^9 \}$ and could be written as $2^{[0:9]}$: | |
" | |
# ╔═╡ 5b3f0e3d-f1cb-427f-a636-0b58052585af | |
2 .^ (0:9) | |
# ╔═╡ 688be232-c63c-4cc3-9437-fa1b5d0c1af4 | |
md" | |
Let's write out several iterations of $y = mx + b$ in terms of the starting point $x_0$. | |
$\begin{aligned} | |
x_1 &= m x_0 + b \\ | |
x_2 &= m x_1 + b = m (m x_0 + b) + b = m^2 x_0 + m b + b = m^2 x_0 + (m+1) b \\ | |
x_3 &= m x_2 + b = m (m^2 x_0 + (m+1) b) + b = m^3 x_0 + (m^2 + m + 1) b \\ | |
x_4 &= m x_3 + b = m (m^3 x_0 + (m^2 + m +1) b) + b = m^4 x_0 + (m^3 + m^2 + m + 1) b \\ | |
\end{aligned}$ | |
The $n^{th}$ iterate in terms of $x_0$ is | |
$x_n = m^n x_0 + (m^{n-1} + m^{n-2} + \cdots + 1) b.$ | |
What happens if we make a small change in $x_0$? If we change $x_0$ by a small amount, $\epsilon$, then | |
$\tilde{x}_n = m^n (x_0 + \epsilon) + (m^{n-1} + m^{n-2} + \cdots + 1) b$ | |
and the difference between the two results after $n$ iterations is | |
$\tilde{x}_n - x_n = m^n (x_0 + \epsilon) - m^n x_0 = m^n \epsilon.$ | |
So long as $|m| > 1$ then we can make $m^n \epsilon$ as big as we like by iterating enough times, no matter how small $\epsilon$ is. A small change to the initial value of $x_0$ produces as big a change as you like in the final value, $x_n$. | |
There you have it. Chaos! | |
" | |
# ╔═╡ b336d8f7-12bd-4535-b030-2e04dc470a1b | |
md" | |
## _Moar Chaos_ | |
With a slightly more complicated equation, chaos becomes even more interesting. Instead of iterating a linear equation, let's use the logistic function (see [The Growing Gap](https://wildpeaches.xyz/blog/the-growing-gap/) for an application of logistic functions) | |
$F(x) = cx(1-x).$ | |
For $c = 4$, this is an inverted parabola that has a maximum at $(0.5,1.0)$. If you draw a the line $y=x$ it intersects the parabola at $(0,0)$ and $(0.75,0.75)$. This line is useful for following the iterations or *orbits* of the function. | |
Start with a point on the $x-$axis, say $x_0 = 0.2$ (point *A*). Find the value of $F(0.2) = 4 \times 0.2 \times (1 - 0.2) = 0.64$. This point *B* is on the parabola above point *A*. | |
We want to iterate $F(x)$ by using the $y-$coordinate of *B* as the next input. We could calculate $F(0.64) = 4 \times 0.64 \times (1 - 0.64) = 0.9216$ and we'll want to write a Julia function to do that, but it's also useful to see how the orbit evolves. | |
Draw a horizontal line until you get to the line $y=x$ at point *C*. Because *C* is on the $45 \degree$ line, then $x_C = y_C$ so the coordinates are $C = (0.64,0.64)$. The $x-$coordinate of *C* is the $y-$coordinate of *B*, so *C* becomes the next iterate. From *C*, draw a vertical line until you intersect the parabola at *D* which has coordinates $(0.64,0.9216)$. | |
This process can be repeated as often as you like and will show the trajectory of the function $F(x)$. | |
" | |
# ╔═╡ 8a0869f7-e36d-4596-8af2-b7c683841413 | |
md" | |
## _An Iterator for the Logistic Function_ | |
Like the *iter_f* function above, we can write a Julia function to iterate the logistic function. The inputs will be the starting point, $x_0$, the constant $c$, and the number of iterations $n$ with a default value of $n=100$. | |
This new function, *iter_logistic* will return the orbit, $y$. | |
" | |
# ╔═╡ 4f65c087-ee86-4b75-9898-5168c4d65be7 | |
function iter_logistic(x₀,c,n=100) | |
# Initialize the trajectory, starting at x₀ | |
y = zeros(n) | |
y[1] = x₀ | |
# Loop over the remaining iterates | |
for k = 2:n | |
y[k] = c*y[k-1]*(1-y[k-1]) | |
end | |
# Return the trajectory | |
return y | |
end | |
# ╔═╡ 24888486-b966-4ba3-aba8-98e610db998d | |
md" | |
Let's try an example with $x_0 = 0.2$ | |
" | |
# ╔═╡ e385cba0-fe32-4d63-a76a-a313d52d9745 | |
@bind x₀ Slider(0:0.01:1, default=0.2) | |
# ╔═╡ 7d82e877-8038-429a-8d6d-2975086c21b1 | |
print("x₀ = ", x₀) | |
# ╔═╡ c0bf7079-cf86-491b-94c4-4ca23dd96a11 | |
md" | |
and $c = 4$. | |
" | |
# ╔═╡ a3d611c4-0baf-4f4c-9613-9715437933d4 | |
@bind c Slider(0:0.1:5, default=4) | |
# ╔═╡ 7b70ab97-e529-45cc-9587-c3f92ccffccf | |
print("c = ", c) | |
# ╔═╡ 3f777025-6a7c-4c34-a5be-fbd00b868e6f | |
y_logistic = iter_logistic(x₀,c) | |
# ╔═╡ 278bd787-0918-4d42-9dcc-8421586fbddf | |
md" | |
Now we can plot the trajectory (and try changing the value of $x_0$): | |
" | |
# ╔═╡ d224e1a1-6f6d-45e6-b188-6af9476e70d2 | |
plot(y_logistic) | |
# ╔═╡ dce69a86-6e27-4711-ad94-a94f57df3404 | |
md" | |
Try adjusting the value for $c$. When $c=4$ the plot seems pretty chaotic. What happens if you change $x_0$ from $0.2$ to $0.19$? This should show that the trajectory is sensitive to the initial value of $x_0$. | |
Next, try $c = 0.5$ and set $x_0 = 0.1$. Is the plot chaotic? Is it sensitive to initial conditions? | |
" | |
# ╔═╡ b24ff792-40fb-41a8-bb19-c5bdcc6407eb | |
md" | |
## _Autocorrelation and Orbits_ | |
While the plot of the logistic function trajectory for $c=4$ appears chaotic, you might be able to convince yourself that there are repeating patterns. The repetitions are far from identical, but if you imagine making a copy of the plot and shifting it over a bit, it might line up. | |
This is what autocorrelation does. The amount of the shift is controlled by a lag $\tau$. For each $y[t]$ autocorrelation compares it to $y[t - \tau]$. The output of the autocorrelation function is a number $r_k$ between $-1$ and $1$, where | |
$r_k = \frac{\sum_{t = \tau + 1}^N (y[t] - \mu)(y[t - \tau] - \mu)}{\sum_{t=1}^N (y[t] - \mu)^2}$ | |
and $\mu$ is the average of all the $y$'s. If $r_k = 1$ then $y[t] = y[t + \tau]$ for every $t$, $r_k = 0$ means there is no correlation, and $r_k = -1$ says that $y[k] = -y[k + \tau]$. | |
An orbit is the plot of $y[k]$ on the $x-$axis and $y[k+\tau]$ on the $y-$axis, with the dots connected. | |
Let's start by plotting the autocorrelation of the iterated logistic function for different values of $\tau$. | |
Rather than scrolling back up to set $x_0$ and $c$, let's just create two new variables and call them $xP$ and $cP$. | |
" | |
# ╔═╡ 2f693c11-24e0-40be-8ab1-87025b3f9fa5 | |
@bind xP Slider(0:0.01:1, default=0.2) | |
# ╔═╡ 02824a9f-4a3c-48fe-b2cf-78765dfc2687 | |
print("xP = ", xP) | |
# ╔═╡ 1668e558-f6e9-494f-b2aa-988d3e628b44 | |
@bind cP Slider(0:0.01:5, default=4.0) | |
# ╔═╡ a021a79e-19f9-4e95-be3d-d38abc10e167 | |
print("cP = ", cP) | |
# ╔═╡ c2f15ca4-9fc3-48c9-8dce-1a0f242dd6ba | |
begin | |
τ = 1:50 | |
yC = iter_logistic(xP,cP,100) | |
rk = StatsBase.autocor(yC,τ) | |
plot(rk) | |
end | |
# ╔═╡ 43820ce5-7c0b-4cf2-ac97-951f4b178a2c | |
md" | |
Choose the autocorrelation step $\tau$: | |
" | |
# ╔═╡ cf8a8df9-36a8-4eee-8315-22adc7891c01 | |
@bind τ₀ Slider(0:10, default=1) | |
# ╔═╡ 8c176990-c4fd-4838-947c-91889e2b7ebf | |
print("τ₀ = ", τ₀) | |
# ╔═╡ 5b1d1aea-675d-4451-a846-5cf509864c49 | |
begin | |
plot(yC[1:end-τ₀],yC[τ₀+1:end],seriestype = :scatter, title = "Logistic Map Orbit", xlims = (0,1), ylims = (0,1.1)) | |
plot!(yC[1:end-τ₀],yC[τ₀+1:end]) | |
xlabel!("y[t]") | |
ylabel!("y[t + τ]") | |
end | |
# ╔═╡ 2ec58a85-d22f-4e25-a28d-173660f74181 | |
md" | |
The autocorrelation plot seems about as chaotic as the logistic function, but when $\tau_0 = 1$, the phase portrait is very smooth. In fact, the outline of the blue dots (y1) looks just like the logistic function! | |
This should be expected since $y_{n+1} = f(y_n)$ for every $n$. | |
Why choose $\tau_0 = 1$? That choice of $\tau_0$ plots the orbit of the function. The orange lines (y2) show which point follows the current location. On the left side, the path seems to be mostly upwards towards a nearby point. On the right, points are sent back to the left half except those near $x = 1$. | |
Try setting $\tau_0 = 0$ and you'll see that the orbit becomes the line $y=x$, because there's perfect correlation between $y[t]$ and $y[t + \tau_0]$. The plot doesn't provide any insight. Try some other values of $\tau_0$ to see what happens. | |
With Pluto, you can change the input values to a function to immediately see the effects. | |
" | |
# ╔═╡ 00000000-0000-0000-0000-000000000001 | |
PLUTO_PROJECT_TOML_CONTENTS = """ | |
[deps] | |
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" | |
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" | |
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" | |
[compat] | |
Plots = "~1.31.7" | |
PlutoUI = "~0.7.40" | |
StatsBase = "~0.33.21" | |
""" | |
# ╔═╡ 00000000-0000-0000-0000-000000000002 | |
PLUTO_MANIFEST_TOML_CONTENTS = """ | |
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version = "0.4.0+1" | |
[[deps.Xorg_xcb_util_renderutil_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e" | |
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e" | |
version = "0.3.9+1" | |
[[deps.Xorg_xcb_util_wm_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] | |
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67" | |
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361" | |
version = "0.4.1+1" | |
[[deps.Xorg_xkbcomp_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"] | |
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b" | |
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4" | |
version = "1.4.2+4" | |
[[deps.Xorg_xkeyboard_config_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"] | |
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d" | |
uuid = "33bec58e-1273-512f-9401-5d533626f822" | |
version = "2.27.0+4" | |
[[deps.Xorg_xtrans_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845" | |
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" | |
version = "1.4.0+3" | |
[[deps.Zlib_jll]] | |
deps = ["Libdl"] | |
uuid = "83775a58-1f1d-513f-b197-d71354ab007a" | |
[[deps.Zstd_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927" | |
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" | |
version = "1.5.2+0" | |
[[deps.libaom_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4" | |
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b" | |
version = "3.4.0+0" | |
[[deps.libass_jll]] | |
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] | |
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47" | |
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" | |
version = "0.15.1+0" | |
[[deps.libblastrampoline_jll]] | |
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"] | |
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" | |
[[deps.libfdk_aac_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" | |
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" | |
version = "2.0.2+0" | |
[[deps.libpng_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] | |
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" | |
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" | |
version = "1.6.38+0" | |
[[deps.libvorbis_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] | |
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c" | |
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" | |
version = "1.3.7+1" | |
[[deps.nghttp2_jll]] | |
deps = ["Artifacts", "Libdl"] | |
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" | |
[[deps.p7zip_jll]] | |
deps = ["Artifacts", "Libdl"] | |
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" | |
[[deps.x264_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" | |
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" | |
version = "2021.5.5+0" | |
[[deps.x265_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] | |
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" | |
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" | |
version = "3.5.0+0" | |
[[deps.xkbcommon_jll]] | |
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] | |
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a" | |
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" | |
version = "1.4.1+0" | |
""" | |
# ╔═╡ Cell order: | |
# ╟─6b7a41a0-2b91-11ed-1db9-099d4ed52a0c | |
# ╟─fcdfd7bd-47c1-4ab8-a6d6-b3a5cc01b302 | |
# ╠═b1c4f074-b973-49eb-81cd-7bae1d8e339b | |
# ╟─bde2d54e-d760-4189-ab83-b561f9e14a5b | |
# ╟─a42ec3ae-7ec9-421f-9618-9198c2460807 | |
# ╟─196417da-9335-40a4-9514-a2da44734275 | |
# ╟─6e5d364b-9786-430f-8322-16eccb8ff39b | |
# ╟─ca11fada-6f31-4a9e-af8b-83237fa766f8 | |
# ╟─818ad39e-2ea5-4e96-aeaf-34fa53906f73 | |
# ╟─5d5c759a-d405-4342-845f-a6b4eb412ddf | |
# ╟─b906dce5-cf29-452d-b286-2fd0dcea1782 | |
# ╟─62349737-9a6d-4e2e-9667-b009ac599c6a | |
# ╠═98a59205-bb10-447b-a60f-b1a9494223e3 | |
# ╟─ff706d3c-89dd-4003-9f03-d3298f6fdbcf | |
# ╠═abe68984-391d-4b7b-bbd6-ebbc1bead377 | |
# ╠═ccf7564b-4815-4c04-97f4-24a894001f69 | |
# ╟─af1b038f-284a-41c1-baad-f11d8e002e9b | |
# ╠═67768d78-689f-46f3-b0d5-7f25cd178249 | |
# ╟─4148003a-802f-440d-b14f-d982dda368e4 | |
# ╠═abe779f2-f7ba-4933-9eb8-110a2dfd2624 | |
# ╠═9adb8b0f-e80b-408c-b06d-4f53dc5d95d8 | |
# ╟─1f48c58d-419a-42d3-b4ee-951161829bab | |
# ╠═273bd83b-5d57-40f9-9bc6-dfe124031c36 | |
# ╟─849730c0-a09f-4280-94c8-430a3f015962 | |
# ╠═6de59735-9dfa-4cb1-81d7-88ec02d57ab6 | |
# ╟─737854c5-5858-4c12-9b74-15f4fa90fc1c | |
# ╠═5b3f0e3d-f1cb-427f-a636-0b58052585af | |
# ╟─688be232-c63c-4cc3-9437-fa1b5d0c1af4 | |
# ╟─b336d8f7-12bd-4535-b030-2e04dc470a1b | |
# ╟─8a0869f7-e36d-4596-8af2-b7c683841413 | |
# ╠═4f65c087-ee86-4b75-9898-5168c4d65be7 | |
# ╟─24888486-b966-4ba3-aba8-98e610db998d | |
# ╟─e385cba0-fe32-4d63-a76a-a313d52d9745 | |
# ╟─7d82e877-8038-429a-8d6d-2975086c21b1 | |
# ╟─c0bf7079-cf86-491b-94c4-4ca23dd96a11 | |
# ╟─a3d611c4-0baf-4f4c-9613-9715437933d4 | |
# ╟─7b70ab97-e529-45cc-9587-c3f92ccffccf | |
# ╠═3f777025-6a7c-4c34-a5be-fbd00b868e6f | |
# ╟─278bd787-0918-4d42-9dcc-8421586fbddf | |
# ╠═d224e1a1-6f6d-45e6-b188-6af9476e70d2 | |
# ╟─dce69a86-6e27-4711-ad94-a94f57df3404 | |
# ╟─b24ff792-40fb-41a8-bb19-c5bdcc6407eb | |
# ╟─2f693c11-24e0-40be-8ab1-87025b3f9fa5 | |
# ╟─02824a9f-4a3c-48fe-b2cf-78765dfc2687 | |
# ╟─1668e558-f6e9-494f-b2aa-988d3e628b44 | |
# ╟─a021a79e-19f9-4e95-be3d-d38abc10e167 | |
# ╠═f535d33c-a9b5-4c64-9408-879f2adbfae8 | |
# ╠═c2f15ca4-9fc3-48c9-8dce-1a0f242dd6ba | |
# ╟─43820ce5-7c0b-4cf2-ac97-951f4b178a2c | |
# ╟─cf8a8df9-36a8-4eee-8315-22adc7891c01 | |
# ╟─8c176990-c4fd-4838-947c-91889e2b7ebf | |
# ╠═5b1d1aea-675d-4451-a846-5cf509864c49 | |
# ╟─2ec58a85-d22f-4e25-a28d-173660f74181 | |
# ╟─00000000-0000-0000-0000-000000000001 | |
# ╟─00000000-0000-0000-0000-000000000002 |
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