When the Cauchy problem to a PDE blows up, it can often be analyzed using self-similar variables. In the reference:
Eggers, J., & Fontelos, M. A. (2008). The role of self-similarity in singularities of partial differential equations. Nonlinearity, 22(1).
they mention that one of the possible blowup profiles is that of a chaotic attractor. They concoct an example using the Lorenz equation and make some references to papers which hypothesize that the self-similar dynamics is chaotic.
I am wondering what other examples are there of chaotic self-similar dynamics in PDEs? Preferably in somewhat less contrived examples.
