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

E. Mayoral·A. Robledo
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Abstract

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 $ξ_{t}$ 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 $ξ_{t}$. 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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E. Mayoral, A. Robledo. 2005-01-17. A recent appreciation of the singular dynamics at the edge of chaos. https://doi.org/10.1007/3-540-32023-7_19

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