(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

Hausdorff dimension of repellors in low sensitive systems

Methods to estimate the Hausdorff dimension of invariant sets of scattering systems are presented. Based on the levels' hierarchical structure of the time delay function, these techniques can be used in systems whose future-invariant-set codimensions are approximately equal to or greater than one. The discussion is illustrated by a numerical example of a scatterer built with four hard spheres located at the vertices of a regular tetrahedron.Comment: 9 pages, 5 figures, accepted in Physics Letters

FRW cosmologies between chaos and integrability

A recent paper by Castagnino, Giacomini and Lara concludes that there is no chaos in a conformally coupled closed Friedmann-Robertson-Walker universe, which is in apparent contradiction with previous works. We point out that although nonchaotic the quoted system is nonintegrable.Comment: Revtex, 2 pages, no figure

How big is too big? Critical Shocks for Systemic Failure Cascades

External or internal shocks may lead to the collapse of a system consisting of many agents. If the shock hits only one agent initially and causes it to fail, this can induce a cascade of failures among neighoring agents. Several critical constellations determine whether this cascade remains finite or reaches the size of the system, i.e. leads to systemic risk. We investigate the critical parameters for such cascades in a simple model, where agents are characterized by an individual threshold \theta_i determining their capacity to handle a load \alpha\theta_i with 1-\alpha being their safety margin. If agents fail, they redistribute their load equally to K neighboring agents in a regular network. For three different threshold distributions P(\theta), we derive analytical results for the size of the cascade, X(t), which is regarded as a measure of systemic risk, and the time when it stops. We focus on two different regimes, (i) EEE, an external extreme event where the size of the shock is of the order of the total capacity of the network, and (ii) RIE, a random internal event where the size of the shock is of the order of the capacity of an agent. We find that even for large extreme events that exceed the capacity of the network finite cascades are still possible, if a power-law threshold distribution is assumed. On the other hand, even small random fluctuations may lead to full cascades if critical conditions are met. Most importantly, we demonstrate that the size of the "big" shock is not the problem, as the systemic risk only varies slightly for changes of 10 to 50 percent of the external shock. Systemic risk depends much more on ingredients such as the network topology, the safety margin and the threshold distribution, which gives hints on how to reduce systemic risk.Comment: 23 pages, 7 Figure

Mixmaster chaos

The significant discussion about the possible chaotic behavior of the mixmaster cosmological model due to Cornish and Levin [J.N. Cornish and J.J. Levin, Phys. Rev. Lett. 78 (1997) 998; Phys. Rev. D 55 (1997) 7489] is revisited. We improve their method by correcting nontrivial oversights that make their work inconclusive to precisely confirm their result: ``The mixmaster universe is indeed chaotic''.Comment: 9 pages, 4 figure

A model for cascading failures in complex networks

Large but rare cascades triggered by small initial shocks are present in most of the infrastructure networks. Here we present a simple model for cascading failures based on the dynamical redistribution of the flow on the network. We show that the breakdown of a single node is sufficient to collapse the efficiency of the entire system if the node is among the ones with largest load. This is particularly important for real-world networks with an highly hetereogeneous distribution of loads as the Internet and electrical power grids.Comment: 4 pages, 4 figure

Reactive dynamics of inertial particles in nonhyperbolic chaotic flows

Anomalous kinetics of infective (e.g., autocatalytic) reactions in open, nonhyperbolic chaotic flows are important for many applications in biological, chemical, and environmental sciences. We present a scaling theory for the singular enhancement of the production caused by the universal, underlying fractal patterns. The key dynamical invariant quantities are the effective fractal dimension and effective escape rate, which are primarily determined by the hyperbolic components of the underlying dynamical invariant sets. The theory is general as it includes all previously studied hyperbolic reactive dynamics as a special case. We introduce a class of dissipative embedding maps for numerical verification.Comment: Revtex, 5 pages, 2 gif figure

Large-scale structural organization of social networks

The characterization of large-scale structural organization of social networks is an important interdisciplinary problem. We show, by using scaling analysis and numerical computation, that the following factors are relevant for models of social networks: the correlation between friendship ties among people and the position of their social groups, as well as the correlation between the positions of different social groups to which a person belongs.Comment: 5 pages, 3 figures, Revte

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

Integrating fluctuations into distribution of resources in transportation networks

We propose a resource distribution strategy to reduce the average travel time in a transportation network given a fixed generation rate. Suppose that there are essential resources to avoid congestion in the network as well as some extra resources. The strategy distributes the essential resources by the average loads on the vertices and integrates the fluctuations of the instantaneous loads into the distribution of the extra resources. The fluctuations are calculated with the assumption of unlimited resources, where the calculation is incorporated into the calculation of the average loads without adding to the time complexity. Simulation results show that the fluctuation-integrated strategy provides shorter average travel time than a previous distribution strategy while keeping similar robustness. The strategy is especially beneficial when the extra resources are scarce and the network is heterogeneous and lowly loaded.Comment: 14 pages, 4 figure