Existence of Nowhere Differentiable Boundaries in a Realistic Map

If the world is a differential equation, as Boltzmann thought, the problem of the origin of the "nowhere differentiable nature of reality" (cf. Mandelbrot [1]) poses itself [2]. To our knowledge, only two differentiable maps exist in the literature which produce nowhere differentiable behavior, the 3-D diffeomorphisms indicated by Grebogi et al. [3] and by Kaneko [4], The first map [3] which contains both a nowherin differentiable basin boundary and a nowhere differentiable attractor is, however, nongeneric. That is, it contains a 2-variable subystem which is Hamiltonian as a forcing subsystem, namely, Arnold's cat map. The second map [4] possesses a nowhere differentiable torus at exactly one point in parameter space. We consider the following generic 3-D map: