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.
So, ultimately, the P or Q conjecture states that, if a system (under a diffeomorphism or a smooth flow) has a chaotic attractor which isn’t pseudohyperbolic, then any $\mathcal{C}^1$-small perturbation of the system results in the appearance of stable orbits. A counterexample to the conjecture would come in the form of a robust non-pseudohyperbolic chaotic global CRH attractor.
Kneading diagrams for ODE flows were introduced in [Shi2] to assist with parameter space exploration for systems reducible via Poincare return maps to unimodal maps of an interval. The motivation was that a single trajectory per parameter value could be integrated to give global topological information about the state space, permitting an inexpensive scan over the parameter space to look for bifurcation sets before coming back with a numerical bifurcation continuation toolkit like AUTO or MATCONT to study the bifurcations at the contours of the kneading diagrams in more detail. One could think of bifurcation continuation as searching for objects in a dark room, where kneading diagrams amount to flipping on a light switch first. Beforehand, Lyapunov exponent scans were popular, but they take much longer integration times to converge than kneading scans and do not necessarily catch all bifurcations, particularly if they do not lie at the boundary of a parameter set exhibiting a chaotic dynamical regime. Additionally, Lyapunov exponents cannot generally provide global information about the attractor from a single trajectory, although in the case of such “unimodal” attractors (in the sense of reducing to a unimodal interval map) any lost global information is more subtle.
[Bon1]
[Gon1]
[Kra1]
[Shi1]