• Plasma currents in space -  very simple models (unpublished yet) which are trending towards an interest I've had since ~2011 in "Birkeland current end effects", and possible near-zero-order-with-axial-distance relations (minus dissipative processes including properties analogous to [resistance, capacitance, inductance, memristance]).  But "violent" dynamics have yet to be addressed by the group in question, which are possibly key (their work does relate back to authors who have addressed those issues).
  
• "Fractional calculus" -  Have any of you been using this?!?   Concepts around "fractional order calculus" (differentiation, integration) have been around for at least 30 years, but I only just ran across them earlier this month in an unpublished paper (but there are many published papers out there).  Various terms are used instead of "fractional" - eg "fractonial" (?), but strangely, I have yet to see the term "fractal" - perhaps for a reason as I am just starting the reading. 
  At present, I don't seethe key difference in the formulations compared to standard differentiation (at least in the paper I am looking at) other than a gamma term.
  

An enticing comment by an author :  "...    The strength of derivatives of fractional order is their ability to describe real situations more adequately than integer-order derivatives, especially when the  problem has memory or hereditary properties    ...".

I have no idea of how this concept overlaps or relates to the "chaotic transition" analysis that Cornelius and Sylvia have been using, and at this early stage of my reading, the way that "fractional calculus" has been approached seems strange to me (as if it is missing key concepts and opportunities), but time will tell...  It does strike me as also being strange that Mandelbrot's fractals don't appear more commonly in science, but perhaps that is because classical models don't generate chaotic-type behaviours, and therefore "can't see a need for them", even when model descriptions fall far short of reality?  Or maybe their productive application is far more restrictive than I imagine?

My current "fractional calculus" reading is focused on a particular neural network implementation for dynamical systems, so it is a bit skewed.  However, one of the first thoughts that comes to mind (after the perennial issues of solution [existence, uniqueness, optimality, stability, local versus global]) is the implementation of "fractional calculus" within the tremendously successful NN framework of Approximate Dynamic Programming (ultimately for control theory and application (current fad "optimal and adaptive" (not just parameter tuning, but automated model development where algorithmic solutions aren't possible and numerical methods are inadequate, albeit "perhaps" useful sometimes.