Interactive Poincaré Surface of Section

InteractiveDynamics.interactive_poincaresos — Function

interactive_poincaresos(cds, plane, idxs, complete; kwargs...)

Launch an interactive application for exploring a Poincaré surface of section (PSOS) of the continuous dynamical system cds. Requires DynamicalSystems.

The plane can only be the Tuple type accepted by DynamicalSystems.poincaresos, i.e. (i, r) for the ith variable crossing the value r. idxs gives the two indices of the variables to be displayed, since the PSOS plot is always a 2D scatterplot. I.e. idxs = (1, 2) will plot the 1st versus 2nd variable of the PSOS. It follows that plane[1] ∉ idxs must be true.

complete is a three-argument function that completes the new initial state during interactive use, see below.

The function returns: figure, laststate with the latter being an observable containing the latest initial state.

Keyword Arguments

direction, rootkw : Same use as in DynamicalSystems.poincaresos.
tfinal = (1000.0, 10.0^4) : A 2-element tuple for the range of values for the total integration time (chosen interactively).
color : A function of the system's initial condition, that returns a color to plot the new points with. The color must be RGBf0/RGBAf0. A random color is chosen by default.
labels = ("u₁" , "u₂") : Scatter plot labels.
scatterkwargs = (): Named tuple of keywords passed to scatter.
diffeq... : Any extra keyword arguments are passed into init of DiffEq.

Interaction

The application is a standard scatterplot, which shows the PSOS of the system, initially using the system's u0. Two sliders control the total evolution time and the size of the marker points (which is always in pixels).

Upon clicking within the bounds of the scatter plot your click is transformed into a new initial condition, which is further evolved and its PSOS is computed and then plotted into the scatter plot.

Your click is transformed into a full D-dimensional initial condition through the function complete. The first two arguments of the function are the positions of the click on the PSOS. The third argument is the value of the variable the PSOS is defined on. To be more exact, this is how the function is called:

x, y = mouseclick; z = plane[2]
newstate = complete(x, y, z)

The complete function can throw an error for ill-conditioned x, y, z. This will be properly handled instead of breaking the application. This newstate is also given to the function color that gets a new color for the new points.

source

To generate the video at the start of this page you can run

using InteractiveDynamics, GLMakie, OrdinaryDiffEq, DynamicalSystems
diffeq = (alg = Vern9(), abstol = 1e-9, reltol = 1e-9)

hh = Systems.henonheiles()

potential(x, y) = 0.5(x^2 + y^2) + (x^2*y - (y^3)/3)
energy(x,y,px,py) = 0.5(px^2 + py^2) + potential(x,y)
const E = energy(get_state(hh)...)

function complete(y, py, x)
    V = potential(x, y)
    Ky = 0.5*(py^2)
    Ky + V ≥ E && error("Point has more energy!")
    px = sqrt(2(E - V - Ky))
    ic = [x, y, px, py]
    return ic
end

plane = (1, 0.0) # first variable crossing 0

# %% Coloring points using the Lyapunov exponent
function λcolor(u)
    λ = lyapunovs(hh, 4000; u0 = u)[1]
    λmax = 0.1
    return RGBf0(0, 0, clamp(λ/λmax, 0, 1))
end

state, scene = interactive_poincaresos(hh, plane, (2, 4), complete;
labels = ("q₂" , "p₂"),  color = λcolor, diffeq...)