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Register a new userStatistical characterization of discrete conservative systems: The web map
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We numerically study the two-dimensional, area preserving, web map. When the map is governed by ergodic behavior, it is, as expected, correctly described by Boltzmann-Gibbs statistics, based on the additive entropic functional $S_{BG}[p(x)] = -kint dx,p(x) ln p(x)$. In contrast, possible ergodicity breakdown and transitory sticky dynamical behavior drag the map into the realm of generalized $q$-statistics, based on the nonadditive entropic functional $S_q[p(x)]=kfrac{1-int dx,[p(x)]^q}{q-1}$ ($q in {cal R}; S_1=S_{BG}$). We statistically describe the system (probability distribution of the sum of successive iterates, sensitivity to the initial condition, and entropy production per unit time) for typical values of the parameter that controls the ergodicity of the map. For small (large) values of the external parameter $K$, we observe $q$-Gaussian distributions with $q=1.935dots$ (Gaussian distributions), like for the standard map. In contrast, for intermediate values of $K$, we observe a different scenario, due to the fractal structure of the trajectories embedded in the chaotic sea. Long-standing non-Gaussian distributions are characterized in terms of the kurtosis and the box-counting dimension of chaotic sea.
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The standard map, paradigmatic conservative system in the $(x,p)$ phase space, has been recently shown to exhibit interesting statistical behaviors directly related to the value of the standard map parameter $K$. A detailed numerical description is achieved in the present paper. More precisely, for large values of $K$, the Lyapunov exponents are neatly positive over virtually the entire phase space, and, consistently with Boltzmann-Gibbs (BG) statistics, we verify $q_{text{ent}}=q_{text{sen}}=q_{text{stat}}=q_{text{rel}}=1$, where $q_{text{ent}}$ is the $q$-index for which the nonadditive entropy $S_q equiv k frac{1-sum_{i=1}^W p_i^q}{q-1}$ (with $S_1=S_{BG} equiv -ksum_{i=1}^W p_i ln p_i$) grows linearly with time before achieving its $W$-dependent saturation value; $q_{text{sen}}$ characterizes the time increase of the sensitivity $xi$ to the initial conditions, i.e., $xi sim e_{q_{text{sen}}}^{lambda_{q_{text{sen}}} ,t};(lambda_{q_{text{sen}}}>0)$, where $e_q^z equiv[1+(1-q)z]^{1/(1-q)}$; $q_{text{stat}}$ is the index associated with the $q_{text{stat}}$-Gaussian distribution of the time average of successive iterations of the $x$-coordinate; finally, $q_{text{rel}}$ characterizes the $q_{text{rel}}$-exponential relaxation with time of the entropy $S_{q_{text{ent}}}$ towards its saturation value. In remarkable contrast, for small values of $K$, the Lyapunov exponents are virtually zero over the entire phase space, and, consistently with $q$-statistics, we verify $q_{text{ent}}=q_{text{sen}}=0$, $q_{text{stat}} simeq 1.935$, and $q_{text{rel}} simeq1.4$. The situation corresponding to intermediate values of $K$, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG or $q$-statistical behavior are observed.
In recent years, statistical characterization of the discrete conservative dynamical systems (more precisely, paradigmatic examples of area-preserving maps such as the standard and the web maps) has been analyzed extensively and shown that, for larger parameter values for which the Lyapunov exponents are largely positive over the entire phase space, the probability distribution is a Gaussian, consistent with Boltzmann-Gibbs (BG) statistics. On the other hand, for smaller parameter values for which the Lyapunov exponents are virtually zero over the entire phase space, we verify this distribution appears to approach a $q$-Gaussian (with $q simeq 1.935$), consistent with $q$-statistics. Interestingly, if the parameter values are in between these two extremes, then the probability distributions happen to exhibit a linear combination of these two behaviours. Here, we numerically show that the Harper map is also in the same universality class of the maps discussed so far. This constitutes one more evidence on the robustness of this behavior whenever the phase space consists of stable orbits. Then, we propose a generalization of the standard map for which the phase space includes many sticky regions, changing the previously observed simple linear combination behavior to a more complex combination.
To characterize local finite-time properties associated with transient chaos in open dynamical systems, we introduce an escape rate and fractal dimensions suitable for this purpose in a coarse-grained description. We numerically illustrate that these quantifiers have a considerable spread across the domain of the dynamics, but their spatial variation, especially on long but non-asymptotic integration times, is approximately consistent with the relationship that was recognized by Kantz and Grassberger for temporally asymptotic quantifiers. In particular, deviations from this relationship are smaller than differences between various locations, which confirms the existence of such a dynamical law and the suitability of our quantifiers to represent underlying dynamical properties in the non-asymptotic regime.
We use direct statistical simulation (DSS) to find the low-order statistics of the well-known dynamical system, the Lorenz63 model. Instead of accumulating statistics from numerical simulation of the dynamical systems, we solve the equations of motion for the statistics themselves after closing them by making several different choices for the truncation. Fixed points of the statistics are obtained either by time evolving, or by iterative methods. Statistics so obtained are compared to those found by the traditional approach.
In contrary to other 1D momentum-conserving lattices such as the Fermi-Pasta-Ulam $beta$ (FPU-$beta$) lattice, the 1D coupled rotator lattice is a notable exception which conserves total momentum while exhibits normal heat conduction behavior. The temperature behavior of the thermal conductivities of 1D coupled rotator lattice had been studied in previous works trying to reveal the underlying physical mechanism for normal heat conduction. However, two different temperature behaviors of thermal conductivities have been claimed for the same coupled rotator lattice. These different temperature behaviors also intrigue the debate whether there is a phase transition of thermal conductivities as the function of temperature. In this work, we will revisit the temperature dependent thermal conductivities for the 1D coupled rotator lattice. We find that the temperature dependence follows a power law behavior which is different with the previously found temperature behaviors. Our results also support the claim that there is no phase transition for 1D coupled rotator lattice. We also give some discussion about the similarity of diffusion behaviors between the 1D coupled rotator lattice and the single kicked rotator also called the Chirikov standard map.
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