Is there an extension of the ideas of Nonlinear Dynamics beyond Ordinary Differential Equations (ODEs), specifically a Partial Differential Equations (PDEs) "version" or more generalized, yet similar, subject area?
In particular, I'm wondering if there are extensions which use some of the same ideas and techniques, such as perturbations, multiple scales, linearization, bifurcations, chaos, and control, to name a few. One of my dynamics professors has alluded to this, mentioning that Complex Ginzburg-Landau and Nonlinear Schrodinger equations are a typical result of the perturbation analysis we do with ODEs, but applied to PDEs. However, I can't seem to find any literature about this subject.
I have done some reading on Evolution Equations, which seem to be somewhat similar to what I'm looking for, but many of the techniques for general Evolution Equations and PDEs in these texts are very different from what we use with ODEs.
I would appreciate any references to literature, either in the form of textbooks or papers, where I can learn more about these things, if they exist.