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Orbit diagrams

An orbit diagram is a way to visualize the asymptotic behaviour of a map, when a parameter of the system is changed. In practice an orbit diagram is a simple plot that plots the last n states of a dynamical system at a given parameter, repeated for all parameters in a range of interest. While this concept can apply to any kind of system, it makes most sense in discrete time dynamical systems. See Chapter 4 of Nonlinear Dynamics, Datseris & Parlitz, Springer 2022, for a more involved discussion on orbit diagrams for both discrete and continuous time systems.

orbitdiagram

Deterministic iterated map

For example, let's compute the famous orbit diagram of the logistic map:

using ChaosTools, CairoMakie

logistic_rule(x, p, n) = @inbounds SVector(p[1]*x[1]*(1-x[1]))
logistic = DeterministicIteratedMap(logistic_rule, [0.4], [4.0])

i = 1
parameter = 1
pvalues = 2.5:0.004:4
n = 2000
Ttr = 2000
output = orbitdiagram(logistic, i, parameter, pvalues; n, Ttr)

L = length(pvalues)
x = Vector{Float64}(undef, n*L)
y = copy(x)
for j in 1:L
    x[(1 + (j-1)*n):j*n] .= pvalues[j]
    y[(1 + (j-1)*n):j*n] .= output[j]
end

fig, ax = scatter(x, y; axis = (xlabel = L"r", ylabel = L"x"),
    markersize = 0.8, color = ("black", 0.05),
)
ax.title = "Logistic map orbit diagram"
xlims!(ax, pvalues[1], pvalues[end]); ylims!(ax,0,1)
fig

Stroboscopic map

The beauty of orbitdiagram is that it can be directly applied to any kind of DynamicalSystem. The most useful cases are the already seen DeterministicIteratedMap, but also PoincareMap and StroboscopicMap. Here is an example of the orbit diagram for the Duffing oscillator (making the same as Figure 9.2 of Nonlinear Dynamics, Datseris & Parlitz, Springer 2022).

using ChaosTools, CairoMakie

function duffing_rule(u,p,t)
    d, a, ω = p
    du1 =  u[2]
    du2 =  -u[1] - u[1]*u[1]*u[1] - d*u[2] + a*sin(ω*t)
    return SVector(du1, du2)
end
T0 = 25.0
p0 = [0.1, 7, 2π/T0]
u0 = [1.1, 1.1]
ds = CoupledODEs(duffing_rule, u0, p0)
duffing = StroboscopicMap(ds, T0)

# We want to change both the parameter `ω`, but also the
# period of the stroboscopic map. `orbitdiagram` allows this!
Trange = range(8, 26; length = 201)
ωrange = @. 2π / Trange
n = 200
output = orbitdiagram(duffing, 1, 3, ωrange; n, u0, Ttr = 100, periods = Trange)

L = length(Trange)
x = Vector{Float64}(undef, n*L)
y = copy(x)
for j in 1:L
    x[(1 + (j-1)*n):j*n] .= Trange[j]
    y[(1 + (j-1)*n):j*n] .= output[j]
end

fig, ax = scatter(x, y; axis = (xlabel = L"T", ylabel = L"u_1"),
    markersize = 8, color = ("blue", 0.25),
)
ylims!(ax, -1, 1)
fig

Pro tip: to actually make Fig. 9.2 you'd have to do two modifications: first, pass periods = Trange ./ 2, so that points are recorded every half period. Then, at the very end, do y[2:2:end] .= -y[2:2:end] so that the symmetric orbits are recorded as well