Power-law sensitivity to initial conditions, characterizing the behaviour of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear 1D logistic-like maps xt+1 = 1 − a |xt| z , (z > 1; 0 < a ≤ 2; t = 0, 1, 2, . . .).The main ingredient of our approach is the generalized deviation law lim ∆x(0)→01−q (equal to e λ 1 t for q = 1, and proportional, for large t, to t 1 1−q for q = 1; q ∈ R is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter q and the fractal dimension d f of the onset-to-chaos attractor is revealed: q appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to −∞ when d f varies from 1 (nonfractal, ergodic-like, limit) to zero. 05.45.+b; 05.20.-y; 05.90.+m

We investigate the origin of the deviations from the classical Darcy law by numerical simulation of the Navier-Stokes equations in two-dimensional disordered porous media. We apply the Forchheimer equation as a phenomenological model to correlate the variations of the friction factor for different porosities and flow conditions. At sufficiently high Reynolds numbers, when inertia becomes relevant, we observe a transition from linear to nonlinear behavior which is typical of experiments. We find that such a transition can be understood and statistically characterized in terms of the spatial distribution of kinetic energy in the system. [S0031-9007(99)09541-1] PACS numbers: 47.55.Mh, 47.11. + j A standard approach in the investigation of singlephase fluid flow in microscopically disordered and macroscopically homogeneous porous media is to characterize the system in terms of Darcy's law [1][2][3], which assumes that a global index, the permeability k, relates the average fluid velocity V through the pores with the pressure drop DP measured across the system,Here L is the length of the sample in the flow direction and m is the viscosity of the fluid. However, in order to understand the interplay between porous structure and fluid flow, it is necessary to examine local aspects of the pore space morphology and relate them to the relevant mechanisms of momentum transfer (viscous and inertial forces). This has been accomplished in previous studies [4][5][6][7][8][9][10] where computational simulations based on a detailed description of the pore space have been quite successful in predicting permeability coefficients and validating well-known relations on real porous materials. In spite of its great applicability, the concept of permeability as a global index for flow should be restricted to viscous flow conditions or, more precisely, to small values of the Reynolds number. Unlike the sudden transition from laminar to turbulent flow in pipes and channels where there is a critical Reynolds number value separating the two regimes, experimental studies on flow through porous media have shown that the passage from linear (Darcy's law) to nonlinear behavior is more likely to be gradual (see Dullien [1], and references therein). It has then been argued [1] and confirmed by numerical simulations [11,12] that the contribution of inertia to the flow in the pore space should also be examined in the framework of the laminar flow regime before assuming that fully developed turbulence effects are present and relevant to momentum transport. Here we show by direct simulation of the Navier-Stokes equations that the departure from Darcy's law in flow through high porosity percolation structures (e . e c , when e c is the critical percolation porosity) and at sufficiently high Reynolds numbers can also be explained in terms of the inertial contribution to the laminar fluid flow through the void space. The calculations we perform do not apply for unstable or turbulent Reynolds conditions. We then demonstrate that it is possible to statisti...

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In the Sznajd model of sociophysics on the square lattice, neighbors having the same opinion convince their neighbors of this opinion. We study scaling of the cluster growth. The spreading-of-damage technique is applied for the spread of opinions. We study the time evolution of the damage and compare it with the magnetization evolution. We also compare this model with the Ising model at low temperatures. It was recently shown that the distribution of votes in Brazilian elections follows a power law behavior with exponent ≃ -1.0. A model for elections based on the Sznajd model is proposed. The exponent obtained for the distribution of votes during the transient agrees with that obtained for elections.

Voting data from city-councillors, state and federal deputies elections are analyzed and considered as a response function of a social system with underlying dynamics leading to complex behavior. The voting results from the last two general Brazilian elections held in 1998 and 2000 are then used as representative data sets. We show that the voting distributions follow a generalized Zipf's Law which has been recently proposed within a nonextensive statistics framework. Moreover, the voting distribution for city-councillors is clearly distinct from those of state and federal deputies in the sense that the latter depicts a higher degree of nonextensivity. We relate this finding with the different degrees of complexity corresponding to local and non-local voting processes. 89.75.Da, 05.65.+b,