Affinity of the Arov Entropy

According to the van der Waals picture, attractive and repulsive forces play distinct roles in the structure of simple fluids. Here, we examine their roles in dynamics; specifically, in the degree of deterministic chaos using the Kolmogorov-Sinai (KS) entropy rate and the spectra of Lyapunov exponents. With computer simulations of three-dimensional Lennard-Jones and Weeks-Chandler-Andersen fluids, we find repulsive forces dictate these dynamical properties, with attractive forces reducing the KS entropy at a given thermodynamic state. Regardless of interparticle forces, the maximal Lyapunov exponent is intensive for systems ranging from 200 to 2000 particles. Our finite-size scaling analysis also shows that the KS entropy is both extensive (a linear function of system-size) and additive. Both temperature and density control the "dynamical chemical potential," the rate of linear growth of the KS entropy with system size. At fixed system-size, both the KS entropy and the largest exponent exhibit a maximum as a function of density. We attribute the maxima to the competition between two effects: as particles are forced to be in closer proximity, there is an enhancement from the sharp curvature of the repulsive potential and a suppression from the diminishing free volume and particle mobility. The extensivity and additivity of the KS entropy and the intensivity of the largest Lyapunov exponent, however, hold over a range of temperatures and densities across the liquid and liquid-vapor coexistence regimes.

Now we want to discuss a number of topics that are essential for an understanding of dynamical systems theory and which also play a role in a more detailed discussion of the relation between transport theory and dynamical systems theory. We begin with a discussion of the Kolmogorov-Sinai (KS) entropy, which is a characteristic property of those deterministic dynamical systems with ‘randomness’ properties similar to Bernoulli shifts discussed earlier. The KS entropy is essential for formulating the escape-rate expressions for transport coefficients, to be discussed in Chapters 11 and 12.

The problem of simultaneously approximating a vector of irrational numbers with rationals is analyzed in a geometrical setting using notions of dynamical systems theory. We discuss here a (vectorial) multidimensional continued-fraction algorithm (MCFA) of additive type, the generalized mediant algorithm (GMA), and give a geometrical interpretation to it. We calculate the invariant measure of the GMA shift as well as its Kolmogorov-Sinai (KS) entropy for arbitrary number of irrationals. The KS entropy is related to the growth rate of denominators of the Euclidean algorithm. This is the first analytical calculation of the growth rate of denominators for any MCFA.

Kolmogorov–Sinai entropy from the ordinal viewpoint

The protocol followed was analogous to that by Garner and Ling [2014]. High spectral Entropy and high spectral Detrended Fluctuation Analysis have been established as more sensitive to the chaotic responses in forward problems in subjects with type 1 diabetes mellitus (T1DM), chronic obstructive pulmonary disease (COPD), youth obesity and child malnutrition. From 50 linearly increasing starting points, we applied the objective functions (hsCTF1 to hsCTF7) to attain an optimized value, [Formula: see text] which should increase the chaotic response when substituted back into the appropriate model. The mean percentage increase in Kolmogorov–Sinai Entropy and [Formula: see text]-values when the models were compared statistically by Kruskal–Wallis and One-way analysis of variance (ANOVA1) tests were monitored. Normal distributions were considered by Anderson–Darling, Ryan–Joiner and Lilliefors tests. Effect sizes by Cohen’ [Formula: see text] and Hedges [Formula: see text] were computed as sometimes the ANOVA1 and Kruskal–Wallis tests were not discriminative. The optimal statistical results were Duffing (hsCTF1: 29.4%); Brusselator (hsCTF6: 24.7%); Lorenz (hsCTF6: 28.8%). Multivariate analysis was required to determine which of the seven combinations of chaotic globals were most influential. Following Principal Component Analysis (PCA) hsCTF1 and hsCTF3 were the preferred objective functions delivering the highest increases in mean KS-Entropy with most influential PCA. These methods could be advantageous to increase yields and consequently profits in industrial processes such as yeast glycolysis for alcoholic beverages, Systems Biology, inverse and optimization problems, generally.

We report a study of the classical scattering of a point particle from three hard circular discs in a plane, which we propose as a model of an idealized unimolecular fragmentation. The system possesses a fractal and chaotic metastable classical state. On the basis of a coding of the system dynamics, we develop a method to construct the invariant probability measure and to calculate the particle escape rate, the Hausdorff dimension, the Kolmogorov–Sinai entropy per unit time and the mean largest Lyapunov exponent of the repellor. The relations between these characteristics of the system dynamics are discussed. In particular, we show that, in general, chaos inhibits escaping from the metastable state. The theory is compared with numerical simulations. We also introduce the classical tools necessary for the semiclassical quantization of the dynamics; the latter is discussed in the following paper.

We consider the Kolmogorov–Sinai entropy for dilute gases ofN hard disks or spheres. This can be expanded in density as , with a the diameterof the sphere or disk, n the density, and d the dimensionality of the system. We estimate the constantB by solving a linear differential equation for the approximate distribution of eigenvaluesof the inverse radius of curvature tensor. We compare the resulting values ofB both to previous estimates and to existing simulation results, finding very good agreementwith the latter. Also, we compare the distribution of eigenvalues of the inverse radius ofcurvature tensor resulting from our calculations to new simulation results. For most of thespectrum the agreement between our calculations and the simulations again is verygood.

Conditional entropy of ordinal patterns

The coupled map lattices (CML) is a spatiotemporal chaotic system with complex dynamic behavior. In this paper, we propose a spatiotemporal chaotic system with a novel pseudo-random coupling method based on elementary cellular automata (ECA), and introduce different perturbations into lattices in each iteration according to ECA. We investigate the spatiotemporal dynamic properties and chaotic behaviors of the proposed system such as bifurcation diagrams, Kolmogorov Sinai entropy, and uniformity. Moreover, the randomness of sequences generated by the proposed system and the correlation between any two lattices are discussed. Theory analyses and simulations indicate that the new system has better performance in complexity, ergodic and unpredictability than other CML systems such as adjacent CML and nonlinear CML based on fractional order logistic equation, etc. Furthermore, the correlation coefficient between any two lattices in the proposed system is significantly lower than other systems, and another advantage of the proposed system is utilizing the output of ECA to perturb the chaotic system which can effectively alleviate the dynamical degradation in digital system. The excellent performance of the proposed system demonstrates that it has great potential for crypto-system.

In n-scroll attractors design using active devices, the behavior of the nonlinear part is, often, modeled by using a piecewise linear (PWL) approach. Although little effort is required to build a PWL model, the main performance parameters associated with the active device to be used in the physical design are not included. Moreover, the main metrics used for quantifying chaos are: Lyapunov exponent (LE), Kolmogorov–Sinai entropy, Kaplan–Yorke dimension, fractal power spectra and fractal dimension. Among them, the former not only is used as indicator of chaotic motion when a positive LE is found, but if its value positively increases, then the unpredictability grade of the chaotic system also increases. All reported works until today about the computed LEs on chaotic systems are derived using a PWL approach. This is a serious drawback since PWL models cannot model a dynamic behavior, and hence, introducing a level of error during the numerical analysis becomes evident when experimental and numerical results are examined at high frequency. Under this scheme, new results on the accurate computing of LEs for a chaotic oscillator based on a nonlinear function called saturated nonlinear function series (SNFS) are introduced. A mathematical model is used for characterizing the dynamical behavior of the SNFS that includes the main performance parameters of Op Amps such as slew rate, DC gain, gain bandwidth product and dynamic range, showing a much better accuracy than the traditional PWL approach. This behavioral modeling demonstrates that the positive LE increases in relation to the number of scrolls and even is also depending on the frequency of the system, overthrowing the results that have been handled in previous works, where a constant value of the LE was obtained when the number of scrolls increased.

Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov–Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov–Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.

Information theory, synchronization and topological order in complete dynamical networks of discontinuous maps

The Arnold cat map, the Ulam method and time reversal

Kolmogorov–Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

Our thermodynamic approach to the study and design of robust optimal control processes in nonlinear (in general global unstable) dynamic systems used soft computing based on genetic algorithms with a fitness function as minimum entropy production. Control objects were nonlinear dynamic systems involving essentially nonlinear stochastic differential equations. An algorithm was developed for calculating entropy production rate in control object motion and in control systems. Part 1 discusses relation of the Lyapunov function (measure of stability) and the entropy production rate (physical measure of controllability). This relation was used to describe the following qualitative properties and important relations: dynamic stability motion (Lyapunov function), Lyapunov exponent and Kolmogorov-Sinai entropy, physical entropy production rates, and symmetries group representation in essentially nonlinear systems as coupled oscillator models. Results of computer simulation are presented for entropy-like dynamic behavior for typical benchmarks of dynamic systems such as Van der Pol, Duffing, and Holmes-Rand, and coupled oscillators. Parts 2 and 3 discuss the application of this approach to simulation of dynamic entropy-like behavior and optimal benchmark control as a 2-link manipulator in a robot for service use and nonlinear systems under stochastic excitation.