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We study the dynamics of iterates at the transition to chaos in the logisticmap and find that it is constituted by an infinite family of Mori's $q$-phasetransitions. Starting from Feigenbaum's $\sigma $ function for the diametersratio, we determine the atypical weak sensitivity to initial conditions $\xi_{t}$ associated to each $q$-phase transition and find that it obeys the formsuggested by the Tsallis statistics. The specific values of the variable $q$ atwhich the $q$-phase transitions take place are identified with the specificvalues for the Tsallis entropic index $q$ in the corresponding $\xi_{t}$. Wedescribe too the bifurcation gap induced by external noise and show that itsproperties exhibit the characteristic elements of glassy dynamics close tovitrification in supercooled liquids, e.g. two-step relaxation, aging and arelationship between relaxation time and entropy.Comment: Proceedings of: Verhulst 200 on Chaos, Brussels 16-18 September 2004,  Springer Verlag, in pres

A Recent Appreciation of the Singular Dynamics at the Edge of Chaos