(Non)Invariance of dynamical quantities for orbit equivalent flows

We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent non-invariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions

Network synchronization: Optimal and Pessimal Scale-Free Topologies

By employing a recently introduced optimization algorithm we explicitely design optimally synchronizable (unweighted) networks for any given scale-free degree distribution. We explore how the optimization process affects degree-degree correlations and observe a generic tendency towards disassortativity. Still, we show that there is not a one-to-one correspondence between synchronizability and disassortativity. On the other hand, we study the nature of optimally un-synchronizable networks, that is, networks whose topology minimizes the range of stability of the synchronous state. The resulting ``pessimal networks'' turn out to have a highly assortative string-like structure. We also derive a rigorous lower bound for the Laplacian eigenvalue ratio controlling synchronizability, which helps understanding the impact of degree correlations on network synchronizability.Comment: 11 pages, 4 figs, submitted to J. Phys. A (proceedings of Complex Networks 2007

Multiwavelength analysis of brightness variations of 3C~279: Probing the relativistic jet structure and its evolution

We studied the correlation between brightness and polarization variations in 3C~279 at different wavelengths, over time intervals long enough to cover the time lags due to opacity effects. We used these correlations together with VLBI images to constrain the radio and high energy source position.We made 7 mm radio continuum and $R$-band polarimetric observations of 3C~279 between 2009 and 2014. The radio observations were performed at the Itapetinga Radio Observatory, while the polarimetric data were obtained at Pico dos Dias Observatory, both in Brazil. We compared our observations with the $\gamma$-ray Fermi/LAT and $R$-band SMARTS light curves. We found a good correlation between 7~mm and $R$-band light curves, with a delay of $170 \pm 30$ days in radio, but no correlation with the $\gamma$ rays. However, a group of several $\gamma$-ray flares in April 2011 could be associated with the start of the 7 mm strong activity observed at the end of 2011.We also detected an increase in $R$-band polarization degree and rotation of the polarization angle simultaneous with these flares. Contemporaneous VLBI images at the same radio frequency show two new strong components close to the core, ejected in directions very different from that of the jet.The good correlation between radio and $R$-band variability suggests that their origin is synchrotron radiation. The lack of correlation with $\gamma$-rays produced by the Inverse Compton process on some occasions could be due to the lack of low energy photons in the jet direction or to absorption of the high energy photons by the broad line region clouds. The variability of the polarization parameters during flares can be easily explained by the combination of the jet polarization parameters and those of newly formed jet components.Comment: 11 pages, 6 figures, 2 tables. Accepted by A&

Enhancing complex-network synchronization

Heterogeneity in the degree (connectivity) distribution has been shown to suppress synchronization in networks of symmetrically coupled oscillators with uniform coupling strength (unweighted coupling). Here we uncover a condition for enhanced synchronization in directed networks with weighted coupling. We show that, in the optimum regime, synchronizability is solely determined by the average degree and does not depend on the system size and the details of the degree distribution. In scale-free networks, where the average degree may increase with heterogeneity, synchronizability is drastically enhanced and may become positively correlated with heterogeneity, while the overall cost involved in the network coupling is significantly reduced as compared to the case of unweighted coupling.Comment: 4 pages, 3 figure

Network Synchronization, Diffusion, and the Paradox of Heterogeneity

Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states. We show that, for a given degree distribution, the maximum synchronizability is achieved when the network of couplings is weighted and directed, and the overall cost involved in the couplings is minimum. This enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size. Numerical verification of the main results is provided for representative classes of small-world and scale-free networks.Comment: Synchronization in Weighted Network

Signatures of small-world and scale-free properties in large computer programs

A large computer program is typically divided into many hundreds or even thousands of smaller units, whose logical connections define a network in a natural way. This network reflects the internal structure of the program, and defines the ``information flow'' within the program. We show that, (1) due to its growth in time this network displays a scale-free feature in that the probability of the number of links at a node obeys a power-law distribution, and (2) as a result of performance optimization of the program the network has a small-world structure. We believe that these features are generic for large computer programs. Our work extends the previous studies on growing networks, which have mostly been for physical networks, to the domain of computer software.Comment: 4 pages, 1 figure, to appear in Phys. Rev.

Distributed flow optimization and cascading effects in weighted complex networks

We investigate the effect of a specific edge weighting scheme $\sim (k_i k_j)^{\beta}$ on distributed flow efficiency and robustness to cascading failures in scale-free networks. In particular, we analyze a simple, yet fundamental distributed flow model: current flow in random resistor networks. By the tuning of control parameter $\beta$ and by considering two general cases of relative node processing capabilities as well as the effect of bandwidth, we show the dependence of transport efficiency upon the correlations between the topology and weights. By studying the severity of cascades for different control parameter $\beta$, we find that network resilience to cascading overloads and network throughput is optimal for the same value of $\beta$ over the range of node capacities and available bandwidth

On the Klein-Gordon equation and hyperbolic pseudoanalytic function theory

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this paper we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein-Gordon equation with a potential. With the aid of one particular solution we factorize the Klein-Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the considered Klein-Gordon equation. Using hyperbolic pseudoanalytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein-Gordon equation with potential. Finally, we give some examples of application of the proposed procedure