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Two simple chaotic maps without equilibria are proposed in this paper. All nonlinearities are quadratic and the functions of the right-hand side of the equations are continuous. The procedure of their design is explained and their dynamical properties such as return map, bifurcation diagram, Lyapunov exponents, and basin of attraction are investigated. These maps belong to the hidden attractor category which is a newly introduced category of dynamical system.

Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labeling. Addressing these problems corresponds to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, which cover a large variety of known (and less known) examples of chaos. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map, …).

By treating extensively the Rössler and the Lorenz attractors, we extended the description of branched manifold — the highest known taxonomic rank for classifying chaotic attractor — by a linking matrix (or linker) to multicomponent attractors (bounded by a torus whose genus $g\ge 3$).

The rice (Oryza sativa L.) genome has become the reference genome to which others are compared. Part of the reason for this is that rice has the lowest DNA content of the common cereals and its gene content and gene order are found in other grass species used for food. Having the genome sequence of rice, both japonica and indica, allows comparisons with regard to genomic structure, gene constitution, and gene expression. Map locations for single-copy genes, families of genes, and quantitative trait loci (QTLs) are often compared among species, usually with rice as the reference. Specialized databases have been developed to facilitate cross-species homology relationships relative to genome and EST sequencing, protein structure, gene function, and other useful aspects. The evolutionary relationship of rice and several other cereals such as maize (Zea mays L.) and sorghum is clearly observed when highlighting syntenic regions. The colinearity of rice and American wildrice (Zizania palustris) has been exploited to develop a molecular genetic map and to locate QTLs in wildrice. The goal of this paper is to illustrate the value of rice for comparative genome referencing.

Let T be a competitive map on a rectangular region , and assume T is C¹ in a neighborhood of a fixed point . The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from when both eigenvalues of the Jacobian of T at are nonzero and at least one of them has absolute value less than one, and establish that is an increasing curve that separates into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. These results, known in hyperbolic case, have been used to determine the basins of attraction of hyperbolic equilibrium points and to establish certain global bifurcation results when switching from competitive coexistence to competitive exclusion. The emphasis in applications in this paper is on planar systems of difference equations with nonhyperbolic equilibria, where we establish a precise description of the basins of attraction of finite or infinite number of equilibrium points.

Conventional maps of election results can give a misleading picture of the popular support that candidates have because population is highly non-uniform and equal areas on a map may not correspond to equal numbers of voters. Taking the example of the 2004 United States presidential election, we show how this problem can be corrected using a cartogram — a map in which the sizes of regions such as states are rescaled according to population or some other variable of interest.

All nondegenerate, continuous, piecewise-linear maps on ${ℝ}^2$ with two pieces are equivalent to a member of a four-parameter family of maps known as the two-dimensional border-collision normal form. This paper shows how the powerful technique of renormalization can be applied to this family and reveals previously undescribed bifurcation structure in a succinct way. We partition a parameter region where the family is known to exhibit chaos robustly into infinitely many subregions in an explicit way. We then show the chaotic attractor has different numbers of connected components in different subregions. The results rely on a careful analysis of the global dynamics of the renormalization operator. This is challenging because the operator is essentially a quadratic map on ${ℝ}^4$.

The Cartographical Modeling belongs to the system of common scientific methods we use in search of new knowledge and its proving. The study of spatial relations is based on a map providing the most complete description and comprehension of any territorial problems.

A map gives a new information of more high order on mapping phenomena which is hidden in an initial figures. This new information one have got due to generalization of statistics is of particular value to scientific research and practical needs. The process of generalization results in discovery of the cartographical structures forming a certain system. Analysis of these structures enables the revelation of spatial regularities in disposition, proportion, combination and dynamics of sociodemographic and socioeconomical processes and phenomena.

Besides, the cartographical modeling provides the transition from discrete to continuous knowledge. This is the only method to obtain the continuous picture of spatially unbroken phenomena on the basis of discrete factual information (Aslanicashvili A., 1974). The importance of uninterrupted knowledge contained in the cartographical model is conditioned not only by its possibility to reveal the changes of investigated process or phenomena "from place to place" but also by its potentialities to bring to light a significant spatial relations between them and other social and natural processes and phenomena represented in the given model (map). The new knowledge obtained in the course of modeling serves as a basis for working out of the management decisions.

The comparison of identical models for a few years in succession gives us the notion about the nature and rate of changes and development of spatial structures. The cartographical modeling may be regarded as one of the modification of latent structure analysis which pursues an object to reveal and distinguish the latent groups of population with peculiar social organization, material and cultural consumption, goals, preferences and behaviour.

The permanent observation of current statistical information during a long time creates the necessary grounds for organization of data base. The collection of statistical data, their standardization and compiling of series of relevant maps are integral parts of monitoring as a system of supervision and control after the processes of spatial behaviour of population.

The scientific programme of monitoring includes also the working out of prognoses concerning eventual changes in the course of spatial self-organization of people, providing it with necessary information about possible unfavourable consequences, appraisals of regulation decisions and their efficiency.

Present paper contains the analysis of a spatial behaviour of rural population in Ukraine since the seventies, carried out by means of cartographical modeling of statistical data in the monitoring regime.