A recent appreciation of the singular dynamics at the edge of chaos

Details A recent appreciation of the singular dynamics at the edge of chaos A recent appreciation of the singular dynamics at the edge of chaos

2005-01-17

arxiv.org/abs/cond-mat/0501398v1

Physics

Condensed Matter

Statistical Mechanics

Proceedings of: Verhulst 200 on Chaos, Brussels 16-18 September 2004, Springer Verlag, in press

Scientific paper

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 $\sigma $ function for the diameters ratio, we determine the atypical weak sensitivity to initial conditions $\xi _{t}$ 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 $\xi_{t}$. 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.

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