Poincaré Sections by Inverse Cubic Hermite Interpolation

A Poincaré section is a trace in a plane, say q=0, of an orbiting trajectory coming up out of the plane. Periodic orbits show up as a single point in such a plane, which the orbit passes through repeatedly. Quasiperiodic orbits show up as closed curves, indicating that the orbits take place in a torus. Chaotic orbits have more complicated behaviour. If the trajectory is computed by a discrete numerical method such as the second-order, symplectic, Størmer–Verlet method (also called the leapfrog method), the points on the orbit will not usually hit the transecting plane exactly. Instead, there will be a point with q < 0 and then the next point will have q > 0. One wants to interpolate between those two points to find just where the continuous trajectory cuts the section. For a second-order method like the leapfrog method, cubic Hermite interpolation is accurate enough and frequently used. But a simpler method exists, namely inverse cubic Hermite interpolation, which will be explained in this paper. We demonstrate on the Henon–Heiles model. We also demonstrate use of the Maple Compiler to make the simulations faster, and plots[pointplot] to make the plotting faster.