using PlutoUI

, Plots

888 ms

Plots.PlotlyBackendPlotlyBackend

plotly()

150 ms

the logistic map

read Sethna sec 4.3

results

$\begin{array}{r} x_{n + 1} = r x_{n} (1 - x_{n}) \end{array}$

r in 1:3 stable
3 < r < 3.45 oscillation between 2 values
r > 3.45 CRAZY

md"

# the logistic map

## read Sethna sec 4.3

## results

```math

\begin{aligned}

x_{n+1} = r \, x_n \, (1 - x_n)

\end{aligned}

```

* r in 1:3 stable

* 3 < r < 3.45 oscillation between 2 values

* r > 3.45 CRAZY

4.5 ms

logistic_map (generic function with 1 method)

function logistic_map(x; r = 0.5)

# iterate once

return r * x * (1 - x)

end

830 μs

slider_N1

slider_N1 = @bind N1 Slider(0:5:80, show_value = true, default = 10)

59.3 ms

slider_rr

2.5

slider_rr = @bind rr Slider(0.0:0.1:4.0, show_value = true, default = 2.5)

49.7 ms

let

# different starting points

x1 = 0.01

x2 = 0.9

x1arr = [x1]

x2arr = [x2]

for i1 = 1:N1

x1 = logistic_map(x1; r = rr)

x2 = logistic_map(x2; r = rr)

push!(x1arr, x1)

push!(x2arr, x2)

end

Plots.scatter(0:N1, x1arr, ylim = (0, 1.1))

Plots.scatter!(0:N1, x2arr, ylim = (0, 1.1), title="the logistic map @ r=$rr", xlabel="no. of iter.")

# analytic limit

Plots.plot!(0:N1, x -> 1 - 1 / rr, label = "1-1/r")

end

1.6 ms

let

# visual aid to understand end points

# r = 3.4 has 2 end points

# solution is at x = f(x) = f(f(x)) = ...

r = 3.4

f(x) = r * x * (1 - x)

Plots.plot(range(0.0, 1.0, length = 20), x -> x, label = "x")

Plots.plot!(range(0.0, 1.0, length = 20), x -> f(x), label = "f(x)")

Plots.plot!(range(0.0, 1.0, length = 20), x -> f(f(x)), label = "f(f(x))")

Plots.plot!(range(0.0, 1.0, length = 20), x -> f(f(f(x))), label = "f(f(f(x)))")

Plots.plot!(range(0.0, 1.0, length = 20), x -> f(f(f(f(x)))), label = "f(f(f(f(x))))")

end

118 ms

invariant density: a histogram of a trajectory at a given r

md"

# invariant density: a histogram of a trajectory at a given r

142 μs

let

# invariant density

# just plot the number of occurances in a histogram

rval = 3.61

x_arr = []

x = 0.28

for i1 = 1:85000

x = logistic_map(x; r = rval)

push!(x_arr, x)

end

# for r = 4

f(x) = 1 / sqrt(x * (1 - x)) / pi

histogram(x_arr, normalize = true, bins = 150, label="numerical")

plot!(0:0.01:1, f,

label="analytic (r=4)",

title="invariant density @ r = $rval"

)

end

148 ms

let

rarr = range(0.5, 4.1, length = 140)

function track(r1)

xarr = []

x = 0.8

for i1 = 1:120

x = logistic_map(x; r = r1)

push!(xarr, x)

end

return xarr

end

function plot_one(i1)

return scatter!(

rarr,

r -> track(r)[end-i1],

markersize = 0.8,

color="black",

ylim = (0, 1),

label = "",

)

end

fig = plot(rarr, r->1-1/r, label="analytic (small r)")

for i1 = 1:20

fig = plot_one(i1)

end

fig

119 ms

other map

$\begin{array}{r} x_{n + 1} = r s i n (π x_{n}) \end{array}$

$\begin{array}{r} x_{n + 1} = x_{n} e^{r (1 - x_{n})} \end{array}$

have similar feature.

md"

# other map

```math

\begin{aligned}

x_{n+1} = r \, sin(\pi \, x_n)

\end{aligned}

```

```math

\begin{aligned}

x_{n+1} = x_n \, e^{\, r \, (1-x_n)}

\end{aligned}

```

have similar feature.

194 μs

let

# there nothing special about the function r*x*(1-x)

# a sine function map will work too

function sin_map(x; r = 0.5)

# iterate once

return r * sin(x * pi)

end

rarr = range(0.0, 1.2, length = 120)

function track(r1)

xarr = []

x = 0.8

for i1 = 1:80

x = sin_map(x; r = r1)

push!(xarr, x)

end

return xarr

end

function plot_one(i1)

return scatter!(

rarr,

r -> track(r)[end-i1],

markersize = 0.8,

ylim = (0, 1.2),

138 ms

let

# yet another map

function exp_map(x; r = 0.5)

# iterate once

return x * exp(r * (1 - x))

end

rarr = range(1.4, 4.0, length = 180)

function track(r1)

xarr = []

x = 0.4

for i1 = 1:80

x = exp_map(x; r = r1)

push!(xarr, x)

end

return xarr

end

function plot_one(i1)

return scatter!(

rarr,

r -> track(r)[end-i1],

markersize = 0.8,

ylim = (0, 4.0),

label = "",

)

end

114 ms

Frontmatter

the logistic map

read Sethna sec 4.3

results

invariant density: a histogram of a trajectory at a given r

other map

further reading