An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps

The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.

An extension of the entropic chaos degree and its positive effect

The Lyapunov exponent is used to quantify the chaos of a dynamical system, by characterizing the exponential sensitivity of an initial point on the dynamical system. However, we cannot directly compute the Lyapunov exponent for a dynamical system without its dynamical equation, although some estimation methods do exist. Information dynamics introduces the entropic chaos degree to measure the strength of chaos of the dynamical system. The entropic chaos degree can be used to compute the strength of chaos with a practical time series. It may seem like a kind of finite space Kolmogorov-Sinai entropy, which then indicates the relation between the entropic chaos degree and the Lyapunov exponent. In this paper, we attempt to extend the definition of the entropic chaos degree on a d-dimensional Euclidean space to improve the ability to measure the stength of chaos of the dynamical system and show several relations between the extended entropic chaos degree and the Lyapunov exponent.

New estimation method of degree of chaos in biosystems

. Potential application of traditional and stochastic methods in assessment of parameters of complex biosystem – complexity is presented. Their amplitude-frequency characteristics, autocorrelation functions A(t), Lyapunov exponents, statistical distribution function f(x) constantly change. In spite of such chaotic dynamics in recorded parameters of tremor, tapping, cardiograms, myograms and other parameters of homeostasis, an order in the dynamics of these processes can be observed. The order is revealed in a change of numbers of sample overlaps that are obtained as a result of some processes. Owing to all the stochastic characteristics constantly change, calculation method for statistical distribution functions with repetitions of identical experimental sets is proposed. In this case, number of sample overlaps (their belonging to the same general population) will numerically present the transition mechanism to order out of chaos or the variation of chaos degree in movement formation and electrobiological muscular activity. The current work shows some typical examples of different physiological human states presented in a form of matrices of paired comparison.

A New Pixels Flipping Method for Huge Watermarking Capacity of the Invoice Font Image

Invoice printing just has two-color printing, so invoice font image can be seen as binary image. To embed watermarks into invoice image, the pixels need to be flipped. The more huge the watermark is, the more the pixels need to be flipped. We proposed a new pixels flipping method in invoice image for huge watermarking capacity. The pixels flipping method includes one novel interpolation method for binary image, one flippable pixels evaluation mechanism, and one denoising method based on gravity center and chaos degree. The proposed interpolation method ensures that the invoice image keeps features well after scaling. The flippable pixels evaluation mechanism ensures that the pixels keep better connectivity and smoothness and the pattern has highest structural similarity after flipping. The proposed denoising method makes invoice font image smoother and fiter for human vision. Experiments show that the proposed flipping method not only keeps the invoice font structure well but also improves watermarking capacity.

How to Classify Influenza A Viruses and Understand Their Severity

As an application of the chaos degree introduced in the framework of information adaptive dynamics, we study the classification of the Influenza A viruses. What evolutional processes determine the severity and the ability for transmission among human of influenza A viruses? We performed phylogenetic classifications of influenza A viruses that were sampled between 1918 and 2009 by using a measure called entropic chaos degree, that was developed through the study of chaos in information dynamics. The phylogenetic analysis of the internal protein (PB2, PB1, PA, NS, M1, M2, NS1, and NS2) indicated that Influenza A viruses adapting to human and transmitting among human were clearly distinguished from swine lineage and avian lineage. Furthermore, the HA, NA, and internal proteins of the influenza strain that caused a pandemic or a severe epidemic with high mortality were phylogenetically different from those from previous pandemic and severe epidemic strains. We have come to the conclusion that the internal protein has a significant impact on the ability for transmission among human. Based of this study, we are convinced that entropic chaos degree is very useful as a measure of understanding the classification and severity of an isolated strain of influenza A virus.

On a Combined Quantum Baker's Map and Its Characterization by Entropic Chaos Degree

Quantum Baker's map is a theoretical model that exhibits chaos in a quantum system. In this paper, we introduce a combined map by combining several quantum Baker's maps. Chaos of such a combined dynamics is studied by the entropic chaos degree.