arXiv subjects

M. L. Lyra

Publications and source records attributed to M. L. Lyra.

41 records · Page 3Linked to original sources

arXiv2001

Asymmetric unimodal maps at the edge of chaos

We numerically investigate the sensitivity to initial conditions of asymmetric unimodal maps $x_{t+1} = 1-a|x_t|^{z_i}$ ($i=1,2$ correspond to $x_t>0$ and $x_t<0$ respectively, $z_i >1$, $0<a\leq 2$, $t=0,1,2,...$) at the edge of chaos. We employ three distinct algorithms to characterize the power-law sensitivity to initial conditions at the edge of chaos, namely: direct measure of the divergence of initially nearby trajectories, the computation of the rate of increase of generalized nonextensive entropies $S_q$ and multifractal analysis. The first two methods provide consistent estimates for the exponent governing the power-law sensitivity. In addition to this, we verify that the multifractal analysis does not provide precise estimates of the singularity spectrum $f(α)$, specially near its extremal points. Such feature prevents to perform a fine check of the accuracy of the scaling relation between $f(α)$ and the entropic index $q$, thus restricting the applicability of the multifractal analysis for studing the sensitivity to initial conditions in this class of asymmetric maps.

cond-mat.stat-mech↗

arXiv2000

Critical behavior of the S=1/2 Heisenberg ferromagnet: A Handscomb quantum Monte Carlo study

We investigate the critical relaxational dynamics of the S=1/2 Heisenberg ferromagnet on a simple cubic lattice within the Handscomb prescription on which it is a diagrammatic series expansion of the partition function that is computed by means of a Monte Carlo procedure. Using a phenomenological renormalization group analysis of graph quantities related to the spin susceptibility and order parameter, we obtain precise estimates for the critical exponents relations $γ/ ν= 1.98\pm 0.01 $ and $β/ν= 0.512 \pm 0.002$ and for the Curie temperature $k_BT_c/J = 1.6778 \pm 0.0002$. The critical correlation time of both energy and susceptibility is also computed. We found that the number of Monte Carlo steps needed to generate uncorrelated diagram configurations scales with the system's volume. We estimate the efficiency of the Handscomb method comparing its ability in dealing with the critical slowing down with that of other quantum and classical Monte Carlo prescriptions.

cond-mat.stat-mech↗

arXiv2000

Convergence of the critical attractor of dissipative maps: Log-periodic oscillations, fractality and nonextensivity

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 attractor's 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.

cond-mat.stat-mech↗

arXiv1997

Nonextensivity and multifractality in low-dimensional dissipative systems

Power-law sensitivity to initial conditions at the edge of chaos provides a natural relation between the scaling properties of the dynamics attractor and its degree of nonextensivity as prescribed in the generalized statistics recently introduced by one of us (C.T.) and characterized by the entropic index $q$. We show that general scaling arguments imply that $1/(1-q) = 1/α_{min}-1/α_{max}$, where $α_{min}$ and $α_{max}$ are the extremes of the multifractal singularity spectrum $f(α)$ of the attractor. This relation is numerically checked to hold in standard one-dimensional dissipative maps. The above result sheds light on a long-standing puzzle concerning the relation between the entropic index $q$ and the underlying microscopic dynamics.

cond-mat.stat-mech↗

arXiv1997

Power-Law Sensitivity to Initial Conditions within a Logistic-like Family of Maps: Fractality and Nonextensivity

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 $x_{t+1} = 1 - a | x_t |^z, (z > 1; 0 < a \le 2; t=0,1,2,...)$ The main ingredient of our approach is the generalized deviation law $\lim_{Δx(0) -> 0} Δx(t) / Δx(0)} = [1+(1-q)λ_q t]^{1/(1-q)}$ (equal to $e^{λ_1 t}$ for q=1, and proportional, for large t, to $t^{1/(1-q)}$ for $q \ne 1; q \in R$ 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 -infinity when d_f varies from 1 (nonfractal, ergodic-like, limit) to zero.

cond-mat.stat-mech↗