📂Dynamics

Ikeda Map

Definition

A map that defines a dynamical system in the complex plane $\mathbb{C}$ is known as the Ikeda map. $$z \mapsto \mu z \exp \left( i \left[ a + {\frac{ b }{ |z|^{2} + 1 }} \right] \right) + c$$ When a complex number is set as $z = x + iy$, the Ikeda map is generally expressed in terms of the real number $x, y$ and parameter $t$, as $a = 0.4$, $b = -6$, $c = 1$ as follows. $$\begin{align*} x \mapsto& 1 + \mu \left( x \cos t - y \sin t \right) \\ y \mapsto& \mu \left( x \sin t + y \cos t \right) \\ t = & 0.4 - {\frac{ 6 }{ 1 + x^{2} + y^{2} }} \end{align*}$$

Description

The Ikeda map was proposed by the Japanese physicist Kensuke Ikeda in a 1979 paper, simplifying the phenomenon of light passing through an optical resonator¹. Typically, the bifurcation parameter is $\mu$, and it becomes chaotic when $\mu \ge 0.6$.

From a research perspective, understanding such a system can add value to a paper. Chaotic systems among $2$-dimensional maps can be found abundantly, including the famous Hénon map, but the Ikeda map is mathematically more complex².

Code

Here is the Julia code that calculates the trajectory of the Ikeda map.

using CSV, DataFrames, Plots

μ = .9
function ikeda(x,y)
    φ = .4 - 6/(1 + x^2 + y^2)
    cosφ = cos(φ)
    sinφ = sin(φ)
    return [1 + μ*(x*cosφ - y*sinφ), μ*(x*sinφ + y*cosφ)]
end

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