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Register a new userDynamical robustness of discrete conservative systems: Harper and generalized standard maps
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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.
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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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