I can’t understand this sentence unfortunately. In any case you indeed need D+1 real numbers to uniquely define a hyperplane in D dimensions. See the wikipedia page on hyperplanes.
Your dynamical system is 4D. So the state space is 4D. You can’t define a hyper plane in 4 dimensions using only three points, you need at least four. See geometry - Is a hyperplane defined by four points? - Mathematics Stack Exchange
It isn’t the poincare map that is causing the problems. It is your lack of knowledge and/or understanding of hyperplanes. Have a look at the wikipedia page on hyper planes. There you will find the equation that I quote in the documentation string of poincaresos , that says you need to provide a vector of D+1 real numbers to define a hyperplane in D-dimensional space. Here it is: Hyperplane - Wikipedia
In fact, in the very first published tutorial of DynamicalSystems.jl , published 4 years ago, I show exactly what you want to do here, making a Poincare Section on a hyperplane connecting the (unstable) fixed ponts of a dynamical system. Here you are: https://www.youtube.com/live/13hqE_1a158?feature=share&t=2277
and here is the code that does the same thing from the test suite. The comment in the code block literally says # create hyperplane spanned by these two points. DynamicalSystemsBase.jl/poincaremap_tests.jl at main · JuliaDynamics/DynamicalSystemsBase.jl (github.com)
Do you see how gis_plane(μ) = [cross(Np(μ), Nm(μ))..., 0] is a vector of four numbers even though the Gissinger system I use it for is 3-dimensional?