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Register a new userConvergence of the critical attractor of dissipative maps: Log-periodic oscillations, fractality and nonextensivity
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For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase space volume occupied by the ensemble W(t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractors fractal dimension governed by the inflexion of the map near its extremal point. Further, we investigate the temporal evolution of W(t) for the circle map whose critical attractor is dense. In this case, we found W(t) to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of nonextensive Tsallis entropies.
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For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase space volume occupied by the ensemble $W(t)$ depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractors fractal dimension governed by the inflexion of the map near its extremal point. Further, we investigate the temporal evolution of $W(t)$ for the circle map whose critical attractor is dense. In this case, we found $W(t)$ to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of non-extensive Tsallis entropies.
We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the ($z_1,z_2$)-{it logarithmic map}, corresponds to a generalization of the $z$-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the $z$-logistic map is numerically consistent with a $q$-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy $S_q$. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to $q$-Gaussian attractor distributions. We also study the generalized $q$-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The $q$-sensitivity indices are obtained as well. Our results are, like those for the $z$-logistic maps, numerically compatible with the $q$-generalization of a Pesin-like identity for ensemble averages.
Using numerical simulations of a simple sea-coast mechanical erosion model, we investigate the effect of spatial long-range correlations in the lithology of coastal landscapes on the fractal behavior of the corresponding coastlines. In the model, the resistance of a coast section to erosion depends on the local lithology configuration as well as on the number of neighboring sea sides. For weak sea forces, the sea is trapped by the coastline and the eroding process stops after some time. For strong sea forces erosion is perpetual. The transition between these two regimes takes place at a critical sea force, characterized by a fractal coastline front. For uncorrelated landscapes, we obtain, at the critical value, a fractal dimension D=1.33, which is consistent with the dimension of the accessible external perimeter of the spanning cluster in two-dimensional percolation. For sea forces above the critical value, our results indicate that the coastline is self-affine and belongs to the Kardar-Parisi-Zhang universality class. In the case of landscapes generated with power-law spatial long-range correlations, the coastline fractal dimension changes continuously with the Hurst exponent H, decreasing from D=1.34 to 1.04, for H=0 and 1, respectively. This nonuniversal behavior is compatible with the multitude of fractal dimensions found for real coastlines.
We study the effects of dissipative boundaries in many-body systems at continuous quantum transitions, when the parameters of the Hamiltonian driving the unitary dynamics are close to their critical values. As paradigmatic models, we consider fermionic wires subject to dissipative interactions at the boundaries, associated with pumping or loss of particles. They are induced by couplings with a Markovian baths, so that the evolution of the system density matrix can be described by a Lindblad master equation. We study the quantum evolution arising from variations of the Hamiltonian and dissipation parameters, starting at t=0 from the ground state of the critical Hamiltonian. Two different dynamic regimes emerge: (i) an early-time regime for times t ~ L, where the competition between coherent and incoherent drivings develops a dynamic finite-size scaling, obtained by extending the scaling framework describing the coherent critical dynamics of the closed system, to allow for the boundary dissipation; (ii) a large-time regime for t ~ L^3 whose dynamic scaling describes the late quantum evolution leading to the t->infty stationary states.
An updated review [1] of nonextensive statistical mechanics and thermodynamics is colloquially presented. Quite naturally the possibility emerges for using the value of q-1 (entropic nonextensivity) as a simple and efficient manner to provide, at least for some classes of systems, some characterization of the degree of what is currently referred to as complexity [2]. A few historical digressions are included as well.
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