We study the dynamics of iterates at the transition to chaos in the logistic
map and find that it is constituted by an infinite family of Mori's q-phase
transitions. Starting from Feigenbaum's σ function for the diameters
ratio, we determine the atypical weak sensitivity to initial conditions ξt​ associated to each q-phase transition and find that it obeys the form
suggested by the Tsallis statistics. The specific values of the variable q at
which the q-phase transitions take place are identified with the specific
values for the Tsallis entropic index q in the corresponding ξt​. We
describe too the bifurcation gap induced by external noise and show that its
properties exhibit the characteristic elements of glassy dynamics close to
vitrification in supercooled liquids, e.g. two-step relaxation, aging and a
relationship between relaxation time and entropy.Comment: Proceedings of: Verhulst 200 on Chaos, Brussels 16-18 September 2004,
Springer Verlag, in pres