Pseudohyperbolicity is a weakened form of hyperbolicity in which the tangent space near an attractor splits into a strongly contracting subspace and a transversal subspace on which volumes expand exponentially. Attractors satisfying it are genuine strange attractors — every orbit has a positive maximal Lyapunov exponent and no stable periodic orbits appear under small smooth perturbations — yet, unlike hyperbolic or Lorenz attractors, they tolerate simple homoclinic tangencies. Students learn the definition for multidimensional diffeomorphisms, the necessary Lyapunov-exponent conditions in three dimensions, how simple versus non-simple tangencies preserve or destroy the property, and how to tell a genuine attractor from a quasiattractor.
This tutorial paper by Gonchenko et al. provides a comprehensive introduction to pseudohyperbolic attractors and chaotic dynamics in finite-dimensional smooth systems. It covers how dynamical chaos m…