Attractors play an important role in dynamical systems theory as they capture the large-time behavior of many orbits. I will report on some recent developments in the analysis of a class of attractors. These attractors occur naturally. They are chaotic, or "strange", in the sense that they have complex geometric structures, and their dynamics are unpredictable, generating statistics that resemble those from random stochastic processes.

The first lecture is an overview aimed at a general audience. In the second hour, I will try to paint a geometric picture of these attractors, and in the third, I will demonstrate how to verify the existence of attractors of this type in several situations, including periodically forced limit cycles and Hopf bifurcations for ODEs and evolutionary PDEs.