Escher style

of a rational map with a neutral fixed point

Two different immersions of the projective plane, both with a self-intersection set consisting of an immersed circle with a single triple point. The left is the well-known Boy’s surface;  the right is a lesser-known model studied by S. Goodman and M. Kossowski (http://dx.doi.org/10.1016/j.difgeo.2009.01.011 ). Computer models by A. Mellnik  (http://surfaces.gotfork.net/ )

of a doubly periodic meromorphic function

of an immersion of the projective plane with a single triple point that is not Boy's surface  (image by Alex Mellnik)

The "lumpy torus" is a surface of revolution with one fat part and one skinny part.  In the study of linear partial differential equations on the lumpy torus, one can separate variables in the rotation direction, resulting in a two dimensional effective phase space.  The resulting dynamical system has a stable critical point from the fat part of the torus and an unstable critical point from the skinny part.

circles approaching the limit set (image by Alex French)

This represents projection onto a pendulum-like dynamical system adhered to closely by light pulses oscillating rapidly between wave-guides in a nonlinear optic but damping its way past the separatrix and towards an equilibrium point in the phase plane for the pendulum due to coupling to radiation.