Crossover from critical to chaotic attractor dynamics in logistic and circle maps

A key feature of the Tsallis statistics is its crossover to the Boltzmann-Gibbs (BG) statistics. We illustrate the mechanisms for this crossover at or near critical attractors in one-dimensional maps. First, we consider the intermittency route to chaos and explain how the crossover is due to either the feedback feature from chaotic regions into the neighborhood of the tangency, to a shift from tangency, or to perturbation by noise. We describe the role of the crossover in the related dynamics of critical clusters at thermal critical states. Secondly, we consider the onset of chaos via period doubling bifurcations and discuss two different ways to bring about the crossover. One corresponds to a shift of the map into a2 n -band chaotic attractor and the other to a perturbation of the Feigenbaum attractor with additive noise. We recall that in the latter case we obtain the properties of glassy dynamics close to vitrification in structural glass formers. Finally, we comment briefly on the equivalent properties for the quasiperiodic onset of chaos and its application to the localization transition in incommensurate systems. In all cases we indicate that the critical dynamics is based on a set of Mori’s q-phase transitions and find that the value of q at each transition corresponds to the same special value for the entropic index q.