DYNA 2008, in Honor of Maurício Peixoto and David Rand, University of Minho, Braga, Portugal, September 8-12, 2008
We present an approach to network control analysis that applies to some important time-dependent dynamical states for both autonomous and non-autonomous dynamical systems. In particular, the theory applies to periodic solutions of autonomous and periodically forced differential equations. The key results are summation theorems that substantially generalise previous results. These results can be interpreted...
We introduce a renormalization scheme to study the asymptotic dynamical behaviour of a family of mechanical systems with non-isochronous potentials with an elliptic equilibrium. This renormalization scheme acts on a family of orbits of these mechanical systems, all of which contained on neighbourhoods of the elliptic equilibrium, by rescaling space and shifting time in an appropriate way. We will...
The aim of this work is to show the relationship between the fundamentals of the economy and social changes in a framework of the General Equilibrium Theory. To analyze this relationship we introduce the Negishi map. This map makes evident the social impact of the efficient reassignments of the economical resources. The social and economic changes occur along the graph of this map. A deeper analysis...
We consider pure exchange economies whose consumption spaces are Banach lattices. The utility functions are strictly concave, Gateaux differentiable, and not necessarily separable. Following the Negishi approach and using the excess utility function, we introduce a notion of social equilibria. We show that there exists a bijective correspondence between this set and the set of Walrasian equilibria...
New populational growth models, proportional to beta densities, with shape parameters p and 2, where p > 1, and Malthusian parameter r, are developed. For p > 2, these models exhibit natural Allee effect. However, in the case of 1 < p ≤ 2, the proposed models do not include this effect. In order to inforce it, we deduce alternative models and investigate their dynamical behaviour. The Verhulst...
In this paper, we apply the following four power indices to the Portuguese Parliament: Shapley–Shubik index, Banzhaf index, Deegan–Packel index and Public Good Index. We also present the main notions related with simple games and discuss the features of each power index by means of their axiomatic characterizations.
We show a simple methodology (or scheme to work) to study comparative-static effects in some models of the theory of the firm under uncertainty. We present this methodology in detail for a basic production model with only one decision variable (SANDMO’s model). Then we sketch it for a model with two decision variables (HOLTHAUSEN’s model with a forward market), and for a model of optimal allocation...
Let f and g be C r unimodal maps, with r ≥ 3, topologically conjugated by h and without periodic attractors. If h is differentiable at a point p in the expanding set E(f), with h′(p)≠0, then, there is an open renormalization interval J such that h is a C r diffeomorphism in the basin B(J) of J, and h is not differentiable at any...
In a joint work with A. Castro, V. Pinheiro (both from Federal Univ. of Bahia) and M. J. Pacifico (Federal University of Rio de Janeiro), we construct a multidimensional flow exhibiting a Rovella-like attractor: a compact transitive invariant set with an equilibrium accumulated by regular orbits and a partially hyperbolic splitting of the tangent bundle with a multidimensional non-uniformly expanding...
This review examines some recent work on robust heteroclinic networks that can appear as attractors for coupled dynamical systems. We focus on coupled phase oscillators and discuss a number of nonlinear dynamical phenomena that are atypical in systems without some coupling structure. The phenomena we discuss include heteroclinic cycles and networks between partially synchronized states. These networks...
We consider an international trade economical model where two firms of different countries compete in quantities and can use three different strategies: (i) repeated collusion, (ii) deviation from the foreigner firm followed by punishment by the home country and then followed by repeated Cournot, or (iii) repeated deviation followed by punishment. In some cases (ii) and (iii) can be interpreted as...
The concepts involved with fractional calculus (FC) theory are applied in almost all areas of science and engineering. Its ability to yield superior modeling and control in many dynamical systems is well recognized. In this article, we will introduce the fundamental aspects associated with the application of FC to the control of dynamic systems.
We describe some well known properties of Wigner measures and then analyze some connections with Quantum Iterated Function Systems.
Burgers equation $$\frac{\partial u} {\partial t} + u\frac{\partial u} {\partial x} = \delta \frac{{\partial }^{2}u} {\partial {x}^{2}} + f\left (x\right )$$ is one of the simplest partial nonlinear differential equation which can develop discontinuities, being the driven equation used to explore unidimensional “turbulence”. For low values of the viscosity coefficient δ, by discretization through...
More than thirty years have passed since Newhouse (Am. J. Math. 99:1061–1087, 1977) published a dichotomy on C 1 area-preserving diffeomorphisms. Here we revisit some central results on surface conservative C 1-diffeomorphisms by presenting, in particular, a new proof of Newhouse’s theorem and also by proving some, although folklore, not yet proved results on this setting. We intend...
Dynamic regime theory is used in a growing number of disciplines to understand, manage, and predict system behavior. A variety of mathematical models have been developed for seemingly disparate systems, however the similarity of these models suggests that the systems could be approached as a collection of samples. A multidisciplinary meta-analysis of dynamic regime models could yield several benefits...
An introduction is provided to the theory of elasticity in general relativity. Important tensors appearing in this context are presented. In particular, attention is focussed on the elasticity difference tensor, for which an algebraic analysis is performed. Applications are given to static and non-static spherically symmetric configurations. For the latter, dynamical equations are obtained characterizing...
We present a brief summary of the results of Charters et al. [1] where a simple model of a massive inflation field ϕ coupled to another scalar filed χ with interaction term g 2ϕ2χ2 for the first stage of preheating, and we give a full description of the dynamics of the χ field modes, including the behaviour of the phase, in terms of the iteration of a simple family of circle maps.
We apply the SIR model to study the evolution of Measles and Hepatitis C in Portugal using data from 1996 until 2007. We use our results to forecast the evolution of those viruses in subsequent years.