Assimilating any nanostructure with a fractal, in the most general Mandelbrot's sense, non - differentiable behaviors in their dynamics on the heat transfer phenomena are analyzed. As such, nanostructure dynamics on the heat transfer in the form of Schrodinger - type various regimes imply "holographic implementation" of the thermal fields through groupal invariance of SL(2R) type. Then, by means of previous groupal invariance as synchronization group between any nanostructure entities, both the phases and the amplitudes of the entities are affected from a homographic perspective. In a special case of synchronization of nanostructure entities, given by Riccati type gauge, period doubling, damping oscillations, self - modulation and chaotic regimes emerge as natural behaviors in the nanostructure dynamics of the heat transfer processes. The present model can also be applied to a large class of nanostructures (i.e. polymeric biocomposites, "liquid wood", temperature - depending drug release systems etc.)
