A Pluto notebook to do some simple experiments with chaos theory

EasyChaos.jl

### 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
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StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"

[compat]
Plots = "~1.31.7"
PlutoUI = "~0.7.40"
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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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"
"""

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