I am looking for references regarding the stability / instability of a periodic solution to a partial differential equation / evolution equation in infinite dimensions. Suppose we have a periodic orbit to a partial differential equation and would like to perturb it, is there some Floquet type criterion or other type of criterion one can use to determine questions such as linear instability implying nonlinear instability etc? Are there papers, review papers, text books etc which focus on the stability of periodic orbits to such equations like reaction diffusion, fluids type equations i.e stability / instability criterion around periodic orbits for infinite dimensional systems / partial differential equations, i.e are there some elementary references / review papers etc regarding extending Floquet theory to PDE's and / or studying the stability around periodic orbits by other methods for infinite dimensional systems. Thank you very much.
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There is a wellestablished Floquet theory for PDEs, see e.g. the book by Kuchment (Birkhäuser 1993). Concerning your question about (in)stability, you may want to check the HartmanGrobman theorem. 

