(almost) Kam tori
Summary:
A failed attempt to find KAM tori, that still yeilded a neat result. Click the title for a brief description of what I was looking for (KAM tori), and what I got.
Link to sketch: ⭐"꙰"꙰"꙰"꙰"꙰"꙰"꙰"꙰⭐
A failed attempt to find some KAM tori. When a system has lots of conserved quantities, its dynamics becomes very regular. The level sets of the conserved quantities are tori, and everything circles around the tori at a constant rate. Perturbing the system breaks these symmetries, so we lose our pretty tori. KAM theorem tells us how: A torus with a rational ratio of frequencies collapses into several smaller tori with regular dynamics, interspersed with areas of hyperbolic dynamics. This process repeats and repeats, forming a fractal of tori surrounded by chaos. The picture below shows a cross-section of some KAM tori, taken from Abraham-Marsden Foundation of Mechanics:
Incidentally, the tori with sufficiently irrational frequency ratios remain, separating the chaotic regions.
This sketch is my attempt to see the invariant torus decomposition in the flesh. I start with the simplest integrable system, rotation by a radius-dependent angle, and perturb it. Unfortunately, I couldn’t see much recognizable in the orbits of this system. But, at some point I goofed up and forgot to update the rotation angle for each point, so every point was running under its own dynamical system. This made some pretty mesmerizing dances when perturbed, as each particle spirals in or blossoms out.