Entropies for severely contracted configuration space: dynamics at or near the onset of chaos

”We consider a largely unexplored research area of statistical physics and nonlinear dynamics, the limit of validity of ordinary Boltzmann-Gibbs statistical mechanics. We present our arguments in terms of the celebrated transitions to chaos while applications span interdisciplinary complex systems of current interest. More specifically, the circumstances under which this largely unchallenged branch of physics gives way to a generalized form are clarified, and correspond to extreme contraction of configuration space, such that the measure of the space of obtainable configurations vanishes with respect to that of the original. We illustrate this reduction by considering prototypical low-dimensional nonlinear maps where the attractors at the transitions to chaos drive the confinement of trajectory positions. We show that generalized entropy expressions describe the process for the three routes to chaos: intermittency, period doublings and quasi periodicity. We refer to natural phenomena in complex systems where these conditions manifest.”