Alberto Robledo

Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics.

We uncover the dynamics at the chaos threshold mu infinity of the logistic map and find that it consists of trajectories made of intertwined power laws that reproduce the entire period-doubling cascade that occurs for mu<mu infinity. 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 lambda(q). Our results are an unequivocal validation of the applicability of the nonextensive generalization of Boltzmann-Gibbs statistical mechanics to critical points of nonlinear maps.

Nonextensive Pesin identity: exact renormalization group analytical results for the dynamics at the edge of chaos of the logistic map.

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.

Feigenbaum Graphs: A Complex Network Perspective of Chaos

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.

Tsallis' q index and Mori's q phase transitions at the edge of chaos

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 Moris q -phase transitions of rapidly decreasing strength, each associated with a discontinuity in Feigenbaums 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.

Analytical properties of horizontal visibility graphs in the Feigenbaum scenario.

Time series are proficiently converted into graphs via the horizontal visibility (HV) algorithm, which prompts interest in its capability for capturing the nature of different classes of series in a network context. We have recently shown [B. Luque et al., PLoS ONE 6, 9 (2011)] that dynamical systems can be studied from a novel perspective via the use of this method. Specifically, the period-doubling and band-splitting attractor cascades that characterize unimodal maps transform into families of graphs that turn out to be independent of map nonlinearity or other particulars. Here, we provide an in depth description of the HV treatment of the Feigenbaum scenario, together with analytical derivations that relate to the degree distributions, mean distances, clustering coefficients, etc., associated to the bifurcation cascades and their accumulation points. We describe how the resultant families of graphs can be framed into a renormalization group scheme in which fixed-point graphs reveal their scaling properties. These fixed points are then re-derived from an entropy optimization process defined for the graph sets, confirming a suggested connection between renormalization group and entropy optimization. Finally, we provide analytical and numerical results for the graph entropy and show that it emulates the Lyapunov exponent of the map independently of its sign.

On the relationship between the density functional formalism and the potential distribution theory for nonuniform fluids

It is shown that the variational principle for the grand potential of a nonuniform fluid as a functional of the singlet density yields the potential distribution theory for the equilibrium density. We derive the explicit form that the functional takes for a system of hard rods, and propose an approximate one for hard spheres. Attractive interactions are also considered in mean-field approximation. In all cases the pair direct correlation function of the nonuniform system is obtained and the density gradient expansion of the free energy is investigated.

The liquid–solid transition of the hard sphere system from uniformity of the chemical potential

The search for solidlike singlet distribution functions in a system of hard spheres is undertaken. The analysis is based on the potential‐distribution theory for nonuniform fluids. The ensuing requirement of constancy of the chemical potential throughout the system leads to a nonlinear integro‐difference equation for the singlet density which is exact in one dimension. Bifurcations from its uniform solution, occurring before close packing and of an oscillatory nature, are found for both hard disks and hard spheres, but not for hard rods. The branching solution for hard spheres has the symmetry of a hexagonal close packing lattice, whereas that for hard disks is triangular. An investigation is also made of the stability of the fluid states. It is found that fluid instabilities and bifurcation points appear simultaneously.

Ubiquity of metastable-to-stable crossover in weakly chaotic dynamical systems

We present a comparative study of several dynamical systems of increasing complexity, namely, the logistic map with additive noise, one, two and many globally coupled standard maps, and the Hamiltonian mean field model (i.e., the classical inertial infinitely ranged ferromagnetically coupled XY spin model). We emphasize the appearance, in all of these systems, of metastable states and their ultimate crossover to the equilibrium state. We comment on the underlying mechanisms responsible for these phenomena (weak chaos) and compare common characteristics. We point out that this ubiquitous behavior appears to be associated to the features of the nonextensive generalization of the Boltzmann–Gibbs statistical mechanics.

Horizontal visibility graphs generated by type-I intermittency.

The type-I intermittency route to (or out of) chaos is investigated within the Horizontal Visibility graph theory. For that purpose, we address the trajectories generated by unimodal maps close to an inverse tangent bifurcation and construct, according to the Horizontal Visibility algorithm, their associated graphs. We show how the alternation of laminar episodes and chaotic bursts has a fingerprint in the resulting graph structure. Accordingly, we derive a phenomenological theory that predicts quantitative values of several network parameters. In particular, we predict that the characteristic power law scaling of the mean length of laminar trend sizes is fully inherited in the variance of the graph degree distribution, in good agreement with the numerics. We also report numerical evidence on how the characteristic power-law scaling of the Lyapunov exponent as a function of the distance to the tangent bifurcation is inherited in the graph by an analogous scaling of the block entropy over the degree distribution. Furthermore, we are able to recast the full set of HV graphs generated by intermittent dynamics into a renormalization group framework, where the fixed points of its graph-theoretical RG flow account for the different types of dynamics. We also establish that the nontrivial fixed point of this flow coincides with the tangency condition and that the corresponding invariant graph exhibit extremal entropic properties.

Multifractality and nonextensivity at the edge of chaos of unimodal maps

We examine both the dynamical and the multifractal properties at the chaos threshold of logistic maps with general nonlinearity z>1. First we determine analytically the sensitivity to initial conditions ξt. Then we consider a renormalization group operation on the partition function Z of the multifractal attractor that eliminates one half of the multifractal points each time it is applied. Invariance of Z fixes a length-scale transformation factor 2−η in terms of the generalized dimensions Dβ. There exists a gap Δη in the values of η equal to λq=1/(1−q)=D∞−1−D−∞−1 where λq is the q-generalized Lyapunov exponent and q is the nonextensive entropic index. We provide an interpretation for this relationship—previously derived by Lyra and Tsallis—between dynamical and geometrical properties.