Negative and Nonlinear Response in an Exactly Solved Dynamical Model of Particle Transport

We consider a simple model of particle transport on the line defined by a dynamical map F satisfying F(x+1) = 1 + F(x) for all x in R and F(x) = ax + b for |x| < 0.5. Its two parameters a (`slope') and b (`bias') are respectively symmetric and antisymmetric under reflection x -> R(x) = -x. Restricting ourselves to the chaotic regime |a| > 1 and therein mainly to the part a>1 we study not only the `diffusion coefficient' D(a,b), but also the `current' J(a,b). An important tool for such a study are the exact expressions for J and D as obtained recently by one of the authors. These expressions allow for a quite efficient numerical implementation, which is important, because the functions encountered typically have a fractal character. The main results are presented in several plots of these functions J(a,b) and D(a,b) and in an over-all `chart' displaying, in the parameter plane, all possibly relevant information on the system including, e.g., the dynamical phase diagram as well as invariants such as the values of topological invariants (kneading numbers) which, according to the formulas, determine the singularity structure of J and D. Our most significant findings are: 1) `Nonlinear Response': The parameter dependence of these transport properties is, throughout the `ergodic' part of the parameter plane (i.e. outside the infinitely many Arnol'd tongues) fractally nonlinear. 2) `Negative Response': Inside certain regions with an apparently fractal boundary the current J and the bias b have opposite signs.Comment: corrected typos and minor reformulations; 28 pages (revtex) with 7 figures (postscript); accepted for publication in JS

Approximate entropy of network parameters

We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We firstly define a purely structural entropy obtained by computing the approximate entropy of the so called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erd\H{o}s-R\'enyi networks. By using classical results of Pincus, we show that our entropy measure converges with network size to a certain binary Shannon entropy. On a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs permit to naturally associate to a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.Comment: 11 pages, 5 EPS figure

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.Comment: 8 figure

Quantum Graphs: A simple model for Chaotic Scattering

We connect quantum graphs with infinite leads, and turn them to scattering systems. We show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics: typical poles, delay time and conductance distributions, Ericson fluctuations, and when considered statistically, the ensemble of scattering matrices reproduce quite well the predictions of appropriately defined Random Matrix ensembles. The underlying classical dynamics can be defined, and it provides important parameters which are needed for the quantum theory. In particular, we derive exact expressions for the scattering matrix, and an exact trace formula for the density of resonances, in terms of classical orbits, analogous to the semiclassical theory of chaotic scattering. We use this in order to investigate the origin of the connection between Random Matrix Theory and the underlying classical chaotic dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the fore-front of the research in chaotic scattering and related fields.Comment: 28 pages, 13 figures, submitted to J. Phys. A Special Issue -- Random Matrix Theor

Cross-section Fluctuations in Open Microwave Billiards and Quantum Graphs: The Counting-of-Maxima Method Revisited

The fluctuations exhibited by the cross-sections generated in a compound-nucleus reaction or, more generally, in a quantum-chaotic scattering process, when varying the excitation energy or another external parameter, are characterized by the width Gamma_corr of the cross-section correlation function. In 1963 Brink and Stephen [Phys. Lett. 5, 77 (1963)] proposed a method for its determination by simply counting the number of maxima featured by the cross sections as function of the parameter under consideration. They, actually, stated that the product of the average number of maxima per unit energy range and Gamma_corr is constant in the Ercison region of strongly overlapping resonances. We use the analogy between the scattering formalism for compound-nucleus reactions and for microwave resonators to test this method experimentally with unprecedented accuracy using large data sets and propose an analytical description for the regions of isolated and overlapping resonances