Chimera state in coupled map lattice of matrices

In recent years, a lot of research has focused on understanding the behavior of when synchronous and asynchronous phases occur, that is, the existence of chimera states in various networks. Chimera states have wide-range applications in many disciplines including biology, chemistry, physics, or engineering. The object of research in this paper is a coupled map lattice of matrices when each node is described by an iterative map of matrices of order two. A regular topology network of iterative maps of matrices was formed by replacing the scalar iterative map with the iterative map of matrices in each node. The coupled map of matrices is special in a way where we can observe the effect of divergence. This effect can be observed when the matrix of initial conditions is a nilpotent matrix. Also, the evolution of the derived network is investigated. It is found that the network of the supplementary variable $\mu$ can evolve into three different modes: the quiet state, the state of divergence, and the formation of divergence chimeras. The space of parameters of node coupling including coupling strength $\varepsilon$ and coupling range $r$ is also analyzed in this study. Image entropy is applied in order to identify chimera state parameter zones.

Evolution of Electoral Preferences for a Regime of Three Political Parties

In this article, we use a discrete system to study the opinion dynamics regarding the electoral preferences of a nontendentious group of agents. To measure the level of preference, a continuous opinion space is used, in which the preference (opinion) can evolve from any political option, to any other; for a regime of three parties, a circle is the convenient space. To model a nonbiased society, new agents are considered. Besides their opinion, they have a new attribute: an individual iterative monoparametric map that imitates a process of internal reflection, allowing them to update their opinion in their own way. These iterative maps introduce six fixed points on the opinion space; the points’ stability depends on the sign of the parameter. When the latter is positive, three attractors are identified with political options, while the repulsors are identified with the antioptions (preferences diametrically opposed to each political choice). In this new model, pairs of agents interact only if their respective opinions are alike; a positive number called confidence bound is introduced with this purpose; if opinions are similar, they update their opinion considering each other’s opinion, while if they are not alike, each agent updates her opinion considering only her individual map. In addition, agents give a certain level of trust (weight) to other agent’s opinions; this results in a positive stochastic matrix of weights which models the social network. The model can be reduced to a pair of coupled nonlinear difference equations, making extracting analytical results possible: a theorem on the conditions governing the existence of consensus in this new artificial society. Some numerical simulations are provided, exemplifying the analytical results.

Nonlinear dynamics and chaos

Simple maps and dynamical systems are used to explore chaos in nature. The discussion starts with a review of the properties of nonlinear ordinary differential equations, including the useful concepts of phase portraits, fixed points, and limit cycles. These notions are developed further in an examination of iterative maps that reveal chaotic behavior. Next, the damped driven oscillator is used to illustrate the Lyapunov exponent that can be used to quantify chaos. The famous KAM theorem on the conditions under which chaotic behavior occurs in physical systems is also presented. The principle is illustrated with the Hénon-Heiles model of a star in a galactic environment and billiard models that describe the motion of balls in closed two-dimensional regions.

On High Order Barycentric Root-Finding Methods

To approximate a simple root of a real function f we construct a family of iterative maps, which we call Newton-barycentric functions, and analyse their convergence order. The performance of the resulting methods is illustrated by means of numerical examples.

CHAOS AND MUSIC: FROM TIME SERIES ANALYSIS TO EVOLUTIONARY COMPOSITION

Music is an amalgam of logic and emotion, order and dissonance, along with many combinations of contradicting notions which allude to deterministic chaos. Therefore, it comes as no surprise that several research works have examined the utilization of dynamical systems for symbolic music composition. The main motivation of the paper at hand is the analysis of the tonal composition potentialities of several discrete dynamical systems, in comparison to genuine human compositions. Therefore, a set of human musical compositions is utilized to provide "compositional guidelines" to several dynamical systems, the parameters of which are properly adjusted through evolutionary computation. This procedure exposes the extent to which a system is capable of composing tonal sequences that resemble human composition. In parallel, a time series analysis on the genuine compositions is performed, which firstly provides an overview of their dynamical characteristics and secondly, allows a comparative analysis with the dynamics of the artificial compositions. The results expose the tonal composition capabilities of the examined iterative maps, providing specific references to the tonal characteristics that they can capture.