Period doubling and reducibility in the quasi-periodically forced logistic map

We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and its reducibility. This reveals that the parameter values for which the invariant curve doubles its period are contained in regions of the parameter space where the invariant curve is reducible. Then we present two additional studies to explain this fact. In first place we consider the images and the preimages of the critical set (the set where the derivative of the map w.r.t the non-periodic coordinate is equal to zero). Studying these sets we construct constrains in the parameter space for the reducibility of the invariant curve. In second place we consider the reducibility loss of the invariant curve as codimension one bifurcation and we study its interaction with the period doubling bifurcation. This reveals that, if the reducibility loss and the period doubling bifurcation curves meet, they do it in a tangent way

Classical models of urban population density. The case of Barcelona Metropolitan Area

In this paper, we study the classical functional forms density-distance and its relation with the suburbanization for the Barcelona Metropolitan Area. The investigation of population concentration will be based on a mesure of the population density gradients derived from the classical models of urban population density. The monocentric urban density analysis has received considerable attention in urban geography and regional science as an empirical issue as well as from a theorical standpoint. The classic study by Colin Clark (1951), followed by the work of Muth (1969) and Mills (1970, 1972), has led to an extensive body of literature dealing with empirical implementations for a wide range of metropolitan areas and cities, in different countries and moments of time. In this work we analize the classical econometric models of urban density. Some of them have been used in studies about the traffic planning and some others in theoretical models on housing market. Quantitative geography has also attempted to model the urban population density. The functions have been estimated from the data sets of Barcelona Metropolitan Area (for six periods of time, 1975-1998) and eleven subcenters whose population ranges from 50.000 to over 200.000 (for one period, 1991). The data are taken from the municipality and census tract area and population data from the Census of Population

Las "esferas" de los valores

El objetivo de este trabajo radica en mostrar cómo el estudio de los valores debe partir de una clasificación en ámbitos o esferas. La lógica interna de cada una de las esferas es independiente de todas las demás, aunque en los distintos mundos de la vida lo habitual es que una esfera colonice las otras. Finalmente se presenta, a modo de ejemplo, la forma de operar de una de las esferas axiológicas: la política

La oclusiva retrofleja sonora del vasco antiguo: nuevas etimologías

On propose ici l'existence en Basque d'une ancienne consonne (occlusive rétroflexe sonore) qui a affecté l'adaptation phonétique d'emprunts Espagnols et Latins.A Basque phonetic characteristic scantly studied is the alternation between inrtervocalic /r/ and /d/. In this work it will be proposed the origin of such phenomena, giving its origin in the existence in Ancient Basque of a voiced retroflex plosive, a phoneme that also would be used in Ancient Gascon and Ancient Aragonese. Following this supposition, a review is done over those loanwords that present such alternancy /r/ ~ /d/, as to continue then reviewing the primitive Basque lexique, proposing for some words interesting relations between other words that hadn't a clear relation before.Una característica fonética vasca poco analizada es la particular alternancia entre /r/ y /d/ en posición intervocálica. En este trabajo se intenta proponer la causa de este fenómeno, situando su origen en la existencia en el antiguo vasco de una oclusiva retrofleja sonora, fonema que habría existido paralelamente en el gascón medieval y el aragonés medieval. Siguiendo esta premisa, se hace un repaso a los préstamos que presentan la alternancia /r/ ~ /d/, para luego examinar el léxico vasco primitivo, llegando a poderse relacionar de esta manera palabras que antes no tenían una relación evidente