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.
Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity ζ > 1 at both their pitchfork and tangent bifurcations. These functions have the form of q-exponentials as proposed in Tsallis' generalization of statistical mechanics. We determine the qindices that characterize these universality classes and perform for the first time the calculation of the q-generalized Lyapunov coefficient λq. The pitchfork and the left-hand side of the tangent bifurcations display weak insensitivity to initial conditions, while the right-hand side of the tangent bifurcations presents a 'super-strong' (faster than exponential) sensitivity to initial conditions. We corroborate our analytical results with a priori numerical calculations. *
We show that the dynamical and entropic properties at the chaos threshold of the logistic map are naturally linked through the nonextensive expressions for the sensitivity to initial conditions and for the entropy. We corroborate analytically, with the use of the Feigenbaum renormalization group transformation, the equality between the generalized Lyapunov coefficient lambda(q) and the rate of entropy production, K(q), given by the nonextensive statistical mechanics. Our results advocate the validity of the q -generalized Pesin identity at critical points of one-dimensional nonlinear dissipative maps.
We uncover the dynamics at the chaos threshold µ∞ of the logistic map and find it consists of trajectories made of intertwined power laws that reproduce the entire period-doubling cascade that occurs for µ < µ∞. We corroborate this structure analytically via the Feigenbaum renormalization group (RG) transformation and find that the sensitivity to initial conditions has precisely the form of a q-exponential, of which we determine the q-index and the q-generalized Lyapunov coefficient λq. Our results are an unequivocal validation of the applicability of the non-extensive generalization of Boltzmann-Gibbs (BG) statistical mechanics to critical points of nonlinear maps.Critical points of nonlinear maps offer a suitable playground for testing the validity of the non-extensive generalization of the Botzmann-Gibbs (BG) statistical mechanics proposed by Tsallis over a decade ago [1,2]. Here we describe universal properties related to the dynamics of iterates at the onset of chaos in unimodal maps [3], that provide a literal confirmation of the generalized nonextensive theory. To this end we employ the celebrated one-dimensional logistic map, f µ (x) = 1 − µ |x| 2 , −1 ≤ x ≤ 1, and the properties of its renormalization group (RG) fixed point, to present evidence of previously unexposed scaling properties at the onset of chaos µ = µ ∞ . At this state, the most prominent of the map critical points, the trajectories of the iterates exhibit an intricate structure, that we describe and show is governed by the Feigenbaum's RG transformation [3].The domain of validity of BG statistical mechanics has been implicitly challenged by the proposal of its nonextensive generalization. Subsequent studies have offered experimental and numerical evidence that point out both the inadequacy of the standard BG statistics and the plausible competence of the generalized theory in describing various types of phenomena and systems. This theoretical development represents an exceptional event in the long and trustworthy history of BG statistical mechanics. However, it is still in the process of being converted into a rigorously corroborated and fully understood fact. The suggested circumstances under which the generalized theory is believed to be applicable, at least with regards to non-linear dynamical systems, are those associated to a phase space with power-law sensitivity to initial conditions, to the consequent vanishing of the largest Lyapunov exponent, and to a fractal, or multifractal geometrical structure [2]. Here we show that our results for the dynamics at the onset of chaos in unimodal maps constitute an unequivocal proof of the universal validity of the non-extensive statistics at such critical points.In fact, at the chaos threshold (as well as at other critical points of the map) the Lyapunov exponent λ 1 vanishes, and the sensitivity to initial conditions ξ t , for large iteration time t, ceases to obey exponential behavior, exhibiting instead power-law behavior [4]. In order to describe the dynamics at such critical points, the q...
The recently formulated theory of horizontal visibility graphs transforms time series into graphs and allows the possibility of studying dynamical systems through the characterization of their associated networks. This method leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. We support our claim via the case study of the period-doubling and band-splitting attractor cascades that characterize unimodal maps. We provide a universal analytical description of this classic scenario in terms of the horizontal visibility graphs associated with the dynamics within the attractors, that we call Feigenbaum graphs, independent of map nonlinearity or other particulars. We derive exact results for their degree distribution and related quantities, recast them in the context of the renormalization group and find that its fixed points coincide with those of network entropy optimization. Furthermore, we show that the network entropy mimics the Lyapunov exponent of the map independently of its sign, hinting at a Pesin-like relation equally valid out of chaos.
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