📂Dynamics

로치 맵

Definitions ¹

A map defined as follows is referred to as the Lozi map.
$$\begin{align*} x \mapsto& 1 - a \left| x \right| + y \\ y \mapsto& b x \end{align*}$$

Description

The Lozi map is a map where only $x^{2}$ is replaced by $\left| x \right|$ from the Hénon map. It is continuous but has points $x = 0$ that are non-differentiable. Its bifurcation diagram and Lyapunov spectrum are as follows.

Code

The following is the code to reproduce the bifurcation diagram of the Lozi map using Julia.

b = 0.3
hrzn = []; vrtc = []
for a = 0.5:0.001:1.8
    Lozi(x) = [1 + x[2] - a*abs(x[1]), b*x[1]]
    v_ = [[.0, .0]]
    for k in 1:1000
        push!(v_, Lozi(v_[end]))
    end
    push!(hrzn, fill(a, 500))
    push!(vrtc, first.(v_[(end-500):end]))
end
p1 = plot(legend = :none, xlabel = "a", ylabel = "x", title = "Lozi map")
for ak in eachindex(hrzn)
    scatter!(p1, hrzn[ak], vrtc[ak], color = :black, ms = 0.5, ma = 0.5)
end
plot!(p1, xlims = [0.5, 1.5]);
png("Lozi_map.png")