C. Hinsley
16 May 2025
The 20th-century Nizhny Novgorod school of dynamical systems did a lot of work to lay out the basic mechanisms of chaotic evolution in diffeomorphisms and systems of differential equations. One of the successes of this line of research was the study of Shilnikov's chaotic saddle-focus homoclinics.
Figure 1. A Shilnikov saddle-focus homoclinic in $\mathbb{R}^3$, shown in red. A neighborhood of the saddle-focus equilibrium shown in purple, with the domain of the Shilnikov map in green. Nearby are countably many saddle periodic orbits as illustrated in blue and yellow.
Gonchenko, Kazakov, and Turaev conjecture (the “P or Q conjecture”, cf. [Gon1]) that the sorts of chaotic attractors that appear in finite-dimensional diffeomorphisms or smooth flows fall into one or the other class, the pseudohyperbolic attractors, or the quasiattractors described by Afraimovich and Shilnikov. The primary distinction between these two classes is that quasiattractors generally exhibit stable periodic orbits, which for example can often be seen in parameter sweeps for measures of complexity as stability windows, while there cannot be stable non-chaotic sets in pseudohyperbolic attractors.