Summary
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 ξ 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 also describe 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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Mayoral, E., Robledo, A. (2006). A Recent Appreciation of the Singular Dynamics at the Edge of Chaos. In: Ausloos, M., Dirickx, M. (eds) The Logistic Map and the Route to Chaos. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-32023-7_19
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DOI: https://doi.org/10.1007/3-540-32023-7_19
Publisher Name: Springer, Berlin, Heidelberg
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