📂Dynamics

Hénon Map

Definition

A Hénon map is defined as a map of a dynamical system as follows. $$\begin{align*} x \mapsto& 1 - a x^{2} + y \\ y \mapsto& b x \end{align*}$$

Explanation

The French astronomer Hénon demonstrated in 1975 that the intriguing phenomena shown by the Poincaré map of a dynamical system expressed as a differential equation can also be studied with a simple simulation using a $2$-dimensional map¹. Below is the phase portrait of the Hénon map when $a = 1.28$ and $b = 0.3$ are true.

The significance of the Hénon map lies in its demonstration that even a non-linear system with just $x^{2}$ input can well represent complex phenomena. In many cases, the Hénon map is the first multi-dimensional map encountered after learning about the logistic map, which is a $1$-dimensional map. Due to its textbook simplicity, or perhaps because of it, it is widely mentioned in numerous studies and literature.

Code

The following is Julia code that reproduces the phase plane of the Hénon map.

using CSV, DataFrames, Plots

a = 1.28
b = 0.3
henon(x,y) = [a - x^2 + b*y, x]

trj_ = [rand(2)]
for t in 1:10000
    push!(trj_, henon(trj_[end]...))
end
data = DataFrame(x = first.(trj_)[100:end], y = last.(trj_)[100:end])
scatter(data.x, data.y, legend = :none, size = [500, 500], msw = 0, ms = 1, xlabel = "x", ylabel = "y")
png("Henon.png")
CSV.write("Henon.csv", data)