Tsallis’qindex and Mori’sqphase transitions at the edge of chaos

Abstract

We uncover the basis for the validity of the Tsallis statistics at the onset of chaos in logistic maps. The dynamics within the critical attractor is found to consist of an infinite family of Mori's q -phase transitions of rapidly decreasing strength, each associated with a discontinuity in Feigenbaum's trajectory scaling function sigma. The value of q at each transition corresponds to the same special value for the entropic index q, such that the resultant sets of q-Lyapunov coefficients are equal to the Tsallis rates of entropy evolution.

Equivalence among different formalisms in the Tsallis entropy framework

“…Remark that this relation embodies both dualities q → 2−q and q → 1/q. These two transformations appear recurrently, alone or combined, in the framework of the generalized statistical mechanics based on the Tsallis entropy [33,34,35,36,37,38].…”

Quasiperiodic graphs at the onset of chaos

“…Studies of entropy growth associated with an initial distribution of positions with iteration time t of several chaotic maps [34] have established that a linear growth occurs during an intermediate stage in the evolution of the entropy, after an initial transient dependent on the initial distribution and before an asymptotic approach to a constant equilibrium value. In relation to this it was found, both at the period doubling [11,12] and at the quasiperiodic golden ratio [22] transitions to chaos, that (i) there is no initial transient if the initial distribution is uniform and defined around a small interval of an attractor position, and (ii) the distribution remains uniform for an extended period of time due to the subexponential dynamics. In Fig.…”

“…We refer to Pesin-like identities as those that were first found to occur at the period-doubling transitions to chaos that link generalized Lyapunov exponents to entropy growth rates at finite, but all, iteration times [11,12]. Recently [7] these identities were retrieved in a network context via the HV method.…”

Some Open Points in Nonextensive Statistical Mechanics

“…(20) and (21). These equations only reflect the maximal values of an entire family, fully (and not only asymptotically) described in [Robledo, 2006;Mayoral & Robledo , 2005]. The rigorous necessary and sufficient conditions for behaviors such as those indicated in Eqs.…”