Necessary Condition of Self-Organisation in Nonextensive Open Systems.

In this paper, we focus on evolution from an equilibrium state in a power law form by means of q-exponentials to an arbitrary one. Introducing new q-Gibbsian equalities as the necessary condition of self-organization in nonextensive open systems, we theoretically show how to derive the connections between q-renormalized entropies (ΔS˜q) and q-relative entropies (KLq) in both Bregman and Csiszar forms after we clearly explain the connection between renormalized entropy by Klimantovich and relative entropy by Kullback-Leibler without using any predefined effective Hamiltonian. This function, in our treatment, spontaneously comes directly from the calculations. We also explain the difference between using ordinary and normalized q-expectations in mean energy calculations of the states. To verify the results numerically, we use a toy model of complexity, namely the logistic map defined as Xt+1=1-aXt2, where a∈[0,2] is the map parameter. We measure the level of self-organization using two distinct forms of the q-renormalized entropy through period doublings and chaotic band mergings of the map as the number of periods/chaotic-bands increase/decrease. We associate the behaviour of the q-renormalized entropies with the emergence/disappearance of complex structures in the phase space as the control parameter of the map changes. Similar to Shiner-Davison-Landsberg (SDL) complexity, we categorize the tendencies of the q-renormalized entropies for the evaluation of the map for the whole control parameter space. Moreover, we show that any evolution between two states possesses a unique q=q* value (not a range for q values) for which the q-Gibbsian equalities hold and the values are the same for the Bregmann and Csiszar forms. Interestingly, if the evolution is from a=0 to a=ac≃1.4011, this unique q* value is found to be q*≃0.2445, which is the same value of qsensitivity given in the literature.

Recurrence Quantification Analysis at work: Quasi-periodicity based interpretation of gait force profiles for patients with Parkinson disease.

In this letter, making use of real gait force profiles of healthy and patient groups with Parkinson disease which have different disease severity in terms of Hoehn-Yahr stage, we calculate various heuristic complexity measures of the recurrence quantification analysis (RQA). Using this technique, we are able to evince that entropy, determinism and average diagonal line length (divergence) measures decrease (increases) with increasing disease severity. We also explain these tendencies using a theoretical model (based on the sine-circle map), so that we clearly relate them to decreasing degree of irrationality of the system as a course of gait's nature. This enables us to interpret the dynamics of normal/pathological gait and is expected to increase further applications of this technique on gait timings, gait force profiles and combinations of them with various physiological signals.

Generalized Pesin-Like Identity and Scaling Relations at the Chaos Threshold of the Rössler System.

In this paper, using the Poincaré section of the flow we numerically verify a generalization of a Pesin-like identity at the chaos threshold of the Rössler system, which is one of the most popular three-dimensional continuous systems. As Poincaré section points of the flow show similar behavior to that of the logistic map, for the Rössler system we also investigate the relationships with respect to important properties of nonlinear dynamics, such as correlation length, fractal dimension, and the Lyapunov exponent in the vicinity of the chaos threshold.

Entropy-based complexity measures for gait data of patients with Parkinson's disease.

Shannon, Kullback-Leibler, and Klimontovich's renormalized entropies are applied as three different complexity measures on gait data of patients with Parkinson's disease (PD) and healthy control group. We show that the renormalized entropy of variability of total reaction force of gait is a very efficient tool to compare patients with respect to disease severity. Moreover, it is a good risk predictor such that the sensitivity, i.e., the percentage of patients with PD who are correctly identified as having PD, increases from 25% to 67% while the Hoehn-Yahr stage increases from 2.5 to 3.0 (this stage goes from 0 to 5 as the disease severity increases). The renormalized entropy method for stride time variability of gait is found to correctly identify patients with a sensitivity of 80%, while the Shannon entropy and the Kullback-Leibler relative entropy can do this with a sensitivity of only 26.7% and 13.3%, respectively.

Probability densities for the sums of iterates of the sine-circle map in the vicinity of the quasiperiodic edge of chaos.

We investigate the probability density of rescaled sum of iterates of sine-circle map within quasiperiodic route to chaos. When the dynamical system is strongly mixing (i.e., ergodic), standard central limit theorem (CLT) is expected to be valid, but at the edge of chaos where iterates have strong correlations, the standard CLT is not necessarily valid anymore. We discuss here the main characteristics of the probability densities for the sums of iterates of deterministic dynamical systems which exhibit quasiperiodic route to chaos. At the golden-mean onset of chaos for the sine-circle map, we numerically verify that the probability density appears to converge to a q-Gaussian with q<1 as the golden mean value is approached.