Spiralling dynamics near heteroclinic networks

There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. We construct explicitly a {two parameter family of vector fields} on the three-dimensional sphere \EU^3, whose flow has a spiralling attractor containing the following: two hyperbolic equilibria, heteroclinic trajectories connecting them {transversely} and a non-trivial hyperbolic, invariant and transitive set. The spiralling set unfolds a heteroclinic network between two symmetric saddle-foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols, {coexisting with Newhouse phenonema}. The vector field is the restriction to \EU^3 of a polynomial vector field in \RR^4. In this article, we also identify global bifurcations that induce chaotic dynamics of different types.Comment: change in one figur

Monetary Poverty, Material Deprivation and Consistent Poverty in Portugal

In this paper we use the Portuguese component of the European Union Statistics on Income and Living Conditions (EU-SILC) to develop a measure of consistent poverty in Portugal. It is widely agreed that being poor does not simply mean not having enough monetary resources. It also reflects a lack of access to the resources required to enjoy a minimum standard of living and participation in the society one belongs to. The coexistence of material deprivation and monetary poverty leads to the concept of consistent poverty. The assessment of material deprivation and the identification of the households and individuals living in consistent poverty could become essential parts of the national anti-poverty strategy and crucial instruments in the definition of the target groups in social policy.Multidimensional poverty measurement, material deprivation, consistent poverty, Portugal.

Dense heteroclinic tangencies near a Bykov cycle

This article presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where trajectories turn in opposite directions near the two nodes --- we say that the nodes have different chirality. We show that in the set of vector fields defined on a three-dimensional manifold, there is a class where tangencies of the invariant manifolds of two hyperbolic saddle-foci occur densely. The class is defined by the presence of the Bykov cycle, and by a condition on the parameters that determine the linear part of the vector field at the equilibria. This has important consequences: the global dynamics is persistently dominated by heteroclinic tangencies and by Newhouse phenomena, coexisting with hyperbolic dynamics arising from transversality. The coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic and non-hyperbolic dynamics, in contrast with the case where the nodes have the same chirality. We illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three-dimensional sphere

On Takens' Last Problem: tangencies and time averages near heteroclinic networks

We obtain a structurally stable family of smooth ordinary differential equations exhibiting heteroclinic tangencies for a dense subset of parameters. We use this to find vector fields $C^2$-close to an element of the family exhibiting a tangency, for which the set of solutions with historic behaviour contains an open set. This provides an affirmative answer to Taken's Last Problem (F. Takens (2008) Nonlinearity, 21(3) T33--T36). A limited solution with historic behaviour is one for which the time averages do not converge as time goes to infinity. Takens' problem asks for dynamical systems where historic behaviour occurs persistently for initial conditions in a set with positive Lebesgue measure. The family appears in the unfolding of a degenerate differential equation whose flow has an asymptotically stable heteroclinic cycle involving two-dimensional connections of non-trivial periodic solutions. We show that the degenerate problem also has historic behaviour, since for an open set of initial conditions starting near the cycle, the time averages approach the boundary of a polygon whose vertices depend on the centres of gravity of the periodic solutions and their Floquet multipliers. We illustrate our results with an explicit example where historic behaviour arises $C^2$-close of a $\textbf{SO(2)}$-equivariant vector field

Global bifurcations close to symmetry

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection - the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies - trajectories that follow the original cycle n times around before they arrive at the other node. Each n-pulse heteroclinic tangency is accumulated by a sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions

How to comply with environmental regulations? The role of information

The effectiveness of environmental regulation can be viewed as conditioned by the action of at least two main agents: the regulated firms and the public agency (the regulator). The agency’s role is, on one hand, to enact environmental regulations and, on the other, to monitor firms’ environmental behavior and enforce environmental regulations. The regulated firms, on the other hand, must be informed about the legal limits imposed on them and subsequently they must be able to comply with those limits. Using a questionnaire on the pulp and paper industry in Portugal we found that firms decision to comply with environmental regulations is strongly influenced by firms’ information on its legal obligations and that this effect is stronger for smaller firms. Moreover larger and younger firms are less likely to comply with environmental regulations than smaller and older firms. With respect to the public agency’s behavior, we found that greater monitoring efforts are directed towards larger and younger firms, as well as towards those firms most likely to cause higher pollution levels.

Conditional tests for elliptical symmetry using robust estimators

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behaviour under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behaviour of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. This comparison was done looking not only at the observed level and power but we rather use the size-corrected relative exact power which provides a tool to assess the test statistic skill to detect alternatives. We also provide a numerical comparison with other competing tests.Comment: In press in Communications in Statistics: Theory and Methods, 201

Conceptualizando… De anormal a pessoa em situação de deficiência: revisitando e contextualizando conceitos

EntretextosSabendo que os conceitos são portadores de significado e indutores de comportamentos, importa enquadrá-los nos contextos sócio-educativos em que se originam e evoluem. Como mostra a reflexão que se segue, as diferentes denominações das situações de deficiência reflectem ou tentam reflectir os diferentes “olhares” que a sociedade em geral e a escola em particular veiculam e operacionalizam. Partindo do conceito de anormal, de utilização ainda bem recente (anos setenta, século passado), hoje considerado um insulto, até ao conceito de pessoa em situação de deficiência, hoje o mais adequado, mas muito pouco utilizado, fez-se um longo percurso conceptual, na forma e no conteúdo (deficiente e pessoa com deficiência), que é também um bocadinho da história das mentalidades e dos comportamentos, no que à deficiência diz respeito.Bearing in mind that the concepts convey meaning and induce behaviours, it is important to make them fit into the socio-educational contexts from where they originate and evolve. As shown in the discussion bellow, the different denominations of disabilities reflect or attempt to reflect the different "looks" that society in general and the school in particular convey and make operational. Beginning at the concept of abnormal that has been used until very recently (the seventies, last century) and is now considered as an insult, to reach the concept of a person in the context of disability, now considered the most appropriate but very little used, there was a long conceptual way, in form and content (abnormal, person with disability), which is also in part the history of mentalities and behaviour, with regard to disability