Photo by Angel Luciano on Unsplash

From Logistic Function to Logistic Map, to Chaos

The animated route to chaos…

From the famous logistic function comes a very simple (yet very interesting), discrete version: x[n+1]=γ x[n](1−x[n]). Let’s plot it out using some γ values.

logimap(x, γ) = γ*x*(1-x)
x = 0.001
γ = 2.0
ts = 1:100
logi = Float64[]
for t in ts
    x = logimap(x, γ)
    push!(logi, x)
end
lb = @sprintf "γ=%2.1f" γ
plot(ts, logi, xlabel="\$n\$",ylabel="\$x[n]\$",
     xlims=(0,101),ylims=(0.0,1.0),label=lb)

Logistic map quickly converges within a few tens of steps.

As seen from the plot above where two cases are shown, the logistic map quickly “converges”:

• With γ=2.0, the map iterations quickly lead to a fixed point at 0.5;
• With γ=3.5, a period-4 cycle is the converged result.

The convergence seems boring: after a few iterations the whole sequence would simply repeat itself. What if we have other γ values?

Bifurcation and the phase space

Since x(1−x)≤(1/4)[x+(1−x)]²=1/4 (equality stands with x=1/2), to concentrate on x values between 0 and 1, we consider only γ≤4. Also, with 0<x<1, for γ≤1, obviously γx(1−x)<γx≤x, so the…