How to build a Poincaré section for a 6th order system?

General Usage
1

Hello everyone, I want to plot the Poincaré section for the system:

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Following the example of this tutorial, I wrote the following code:

plane = (3, 0.0)


fig = Figure(resolution = (600,300))
axx = Axis(fig[1,1]; xlabel = L"x_1", ylabel = L"x_2")
axy = Axis(fig[1,2]; xlabel = L"y_1", ylabel = L"y_2")
for i in 1:100
    rng = MersenneTwister(i);
    u0 = randn!(rng, zeros(6))
    psos = poincaresos(hr, plane, 20000.0; u0 = u0)
    scatter!(axx, psos[:, 1], psos[:, 4]; marketsize = 2.0)
    scatter!(axy, psos[:, 2], psos[:, 5]; marketsize = 2.0)
end

fig

And the same for plane = (6, 0.0). But I want to get a cross section when both the third variable (h_1) and the sixth variable (h_2) are zero at the same time.

Can I do it somehow? I tried to draw a plane through 2 six-dimensional points as in the example, but in this case the cross() function does not work.

2

Share your whole code.

4

Here is my code

5

An easy way to do this is with polynomial interpolation. If your Poincare section is defined as h(x) = 0, integrate until h(x) changes sign. Then, for an nth-order integration scheme, make an nth-order polynomial interpolant of each the component of x as a function of h values, and interpolate to h=0.

If you use the Polynomial.jl package you can do this in a few lines of code. It’ll be allocating, and maybe there’ll be some cost to tracking h(x). Probably DifferentialEquations.jl has a slcik and efficient method.

2 Likes
6

You just make it a ContinuousCallback with the h(x) as the condition function.

7

Something I didn’t understand from your first post was the goal of finding a Poincare section satisfying two conditions, h1(x) = 0 and h2(x) = 0. In general there’s no reason to expect that with only one independent variable (time t) you’ll be able to find points x(t) that satisfy two conditions simultaneously.

8

By definition, a Poincare section for the flow associated to a system of n first order differential equations is a (n-1)-dimensional surface (usually a hyperplane in R^n), that is transversal to the flow.
To choose a right Poincare section, not just a hyperplane of equation x_i =0 (or y_i=0 or zi_0) you should associate the 6d vector field,
V(x_1, x_2, y_1, y_2, z_1, z2), having as coordinates the right-hand sides of your system.

The general equation of a hyperplane in R^6 is Ax_1+B_x_2+Cy_1+Dy_2+Ez_1+Fz_2+G=0, and its normal is the vector, N, of coordinates [A, B, C, D, E, F]. If the dot product
of the vector field V and the normal, N, is nonzero
at any point of the hyperplane, <V(x_1, x_2, y_1, y_2, z_1, z_2), N> \neq 0, then that hyperplane is a right choice as a Poincare section.
Hence assigning particular values to A,B, C, D, E, F, you should find a Poincare section, if any.