Nonautonomous difference equations with applications

This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation. In the second part we present some concrete models that are used in ecology/biology and economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models. One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter space of the autonomous Ricker competition model. Since the dynamics of the Ricker competition model is similar to the logistic competition model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results. Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model.

Renormalization in few-body nuclear physics

Renormalized fixed-point Hamiltonians are formulated for systems described by interactions that originally contain point-like singularities (as the Dirac-delta and/or its derivatives). They express the renormalization group invariance of quantum mechanics. The present approach for the renormalization scheme relies on a subtracted T-matrix equation.

Fixed points on torus fiber bundles over the circle

The main purpose of this work is to study fixed points of fiber-preserving maps over the circle S-1 for spaces which axe fibrations over S-1 and the fiber is the torus T. For the case where the fiber is a surface with nonpositive Euler characteristic, we establish general algebraic conditions, in terms of the fundamental group and the induced homomorphism, for the existence of a deformation of a map over S-1 to a fixed point, free map. For the case where the fiber is a torus, we classify all maps over S-1 which can be deformed fiberwise to a fixed point free map.

Coincidence points of fiber maps on S-n-bundles

In this note we study coincidence of pairs of fiber-preserving maps f, g : E-1 -> E-2 where E-1, E-2 are S-n-bundles over a space B. We will show that for each homotopy class vertical bar f vertical bar of fiber-preserving maps over B, there is only one homotopy class vertical bar g vertical bar such that the pair (f, g), where vertical bar g vertical bar = vertical bar tau circle f vertical bar can be deformed to a coincidence free pair. Here tau : E-2 -> E-2 is a fiber-preserving map which is fixed point free. In the case where the base is S-1 we classify the bundles, the homotopy classes of maps over S-1 and the pairs which can be deformed to coincidence free. At the end we discuss the self-coincidence problem. (C) 2010 Elsevier B.V. All rights reserved.

Abelianized obstruction for fixed points of fiber-preserving maps of surface bundles

Let f: M -> M be a fiber-preserving map where S -> M -> B is a bundle and S is a closed surface. We study the abelianized obstruction, which is a cohomology class in dimension 2, to deform f to a fixed point free map by a fiber-preserving homotopy. The vanishing of this obstruction is only a necessary condition in order to have such deformation, but in some cases it is sufficient. We describe this obstruction and we prove that the vanishing of this class is equivalent to the existence of solution of a system of equations over a certain group ring with coefficients given by Fox derivatives.

Fixed points on Klein bottle fiber bundles over the circle

The main purpose of this work is to study fixed points of fiber-preserving maps over the circle S(1) for spaces which are fiber bundles over S(1) and the fiber is the Klein bottle K. We classify all such maps which can be deformed fiberwise to a fixed point free map. The similar problem for torus fiber bundles over S(1) has been solved recently.

Absence of Cooper-type bound states in three- and few-electron systems

It is shown that the appearance of a fixed-point singularity in the kernel of the two-electron Cooper problem is responsible for the formation of the Cooper pair for an arbitrarily weak attractive interaction between two electrons. This singularity is absent in the problem of three and few superconducting electrons at zero temperature on the full Fermi sea. Consequently, such three- and few-electron systems on the full Fermi sea do not form Cooper-type bound states for an arbitrarily weak attractive pair interaction.

Some dynamical properties of a classical dissipative bouncing ball model with two nonlinearities

Some dynamical properties for a bouncing ball model are studied. We show that when dissipation is introduced the structure of the phase space is changed and attractors appear. Increasing the amount of dissipation, the edges of the basins of attraction of an attracting fixed point touch the chaotic attractor. Consequently the chaotic attractor and its basin of attraction are destroyed given place to a transient described by a power law with exponent -2. The parameter-space is also studied and we show that it presents a rich structure with infinite self-similar structures of shrimp-shape. © 2013 Elsevier B.V. All rights reserved.

Análise da dinâmica de um sistema vibrante não ideal de dois graus de liberdade

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

As funções do grupo de renormalização no modelo com auto-interação quártica fermiônica

Modelos com interações quárticas fermiônicas tem sido estudadas para clarificar aspectos conceituais e possíveis aplicações em teoria quântica de campos. Neste trabalho apresentamos a estrutura do grupo de renormalização no modelo de Nambu-Jona-Lasinio até a ordem de 1-loop. O modelo é não renormalizável perturbativamente, no sentido usual de contagem de potência, mas é tratado como uma teoria efetiva, válida numa escala de energia onde p << ^, sendo p o momento externo do loop e ^ um parâmetro de escala de massa que caracteriza o acoplamento do vértice não renormalizável. Esclarecemos a estrutura tensorial dos vértices de interação e calculamos as funções do grupo de renormalização. A análise dos pontos fixos da teoria também é apresentada e discutida usando o formalismo de redução das constantes de acoplamento proposto por Zimmermann. Encontramos a baixas eneergias a origem como ponto fixo infravermelho estável e um ponto fixo não trivial ultravioleta estável, indicando a consistência perturbativa se o momento é pequeno.

Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation

Convergence to a period one fixed point is investigated for both logistic and cubic maps. For the logistic map the relaxation to the fixed point is considered near a transcritical bifurcation while for the cubic map it is near a pitchfork bifurcation. We confirmed that the convergence to the fixed point in both logistic and cubic maps for a region close to the fixed point goes exponentially fast to the fixed point and with a relaxation time described by a power law of exponent -1. At the bifurcation point, the exponent is not universal and depends on the type of the bifurcation as well as on the nonlinearity of the map.

Non-perturbative fixed points and renormalization group improved effective potential

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Involutions whose fixed set has three or four components: a small codimension phenomenon

Let T : M → M be a smooth involution on a closed smooth manifold and F = n j=0 F j the fixed point set of T, where F j denotes the union of those components of F having dimension j and thus n is the dimension of the component of F of largest dimension. In this paper we prove the following result, which characterizes a small codimension phenomenon: suppose that n ≥ 4 is even and F has one of the following forms: 1) F = F n ∪ F 3 ∪ F 2 ∪ {point}; 2) F = F n ∪ F 3 ∪ F 2 ; 3) F = F n ∪ F 3 ∪ {point}; or 4) F = F n ∪ F 3 . Also, suppose that the normal bundles of F n, F 3 and F 2 in M do not bound. If k denote the codimension of F n, then k ≤ 4. Further, we construct involutions showing that this bound is best possible in the cases 2) and 4), and in the cases 1) and 3) when n is of the form n = 4t, with t ≥ 1.

Métodos numéricos para encontrar zeros de funções: aplicações para o Ensino Médio

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Untersuchungen zum nichtstörungstheoretischen Renormierungsverhalten der Quanten-Einstein-Gravitation

Deutsche Version: Zunächst wird eine verallgemeinerte Renormierungsgruppengleichung für die effektiveMittelwertwirkung der EuklidischenQuanten-Einstein-Gravitation konstruiert und dann auf zwei unterschiedliche Trunkierungen, dieEinstein-Hilbert-Trunkierung und die$R^2$-Trunkierung, angewendet. Aus den resultierendenDifferentialgleichungen wird jeweils die Fixpunktstrukturbestimmt. Die Einstein-Hilbert-Trunkierung liefert nebeneinem Gaußschen auch einen nicht-Gaußschen Fixpunkt. Diesernicht-Gaußsche Fixpunkt und auch der Fluß in seinemEinzugsbereich werden mit hoher Genauigkeit durch die$R^2$-Trunkierung reproduziert. Weiterhin erweist sichdie Cutoffschema-Abhängigkeit der analysierten universellenGrößen als äußerst schwach. Diese Ergebnisse deuten daraufhin, daß dieser Fixpunkt wahrscheinlich auch in der exaktenTheorie existiert und die vierdimensionaleQuanten-Einstein-Gravitation somit nichtperturbativ renormierbar sein könnte. Anschließend wird gezeigt, daß der ultraviolette Bereich der$R^2$-Trunkierung und somit auch die Analyse des zugehörigenFixpunkts nicht von den Stabilitätsproblemen betroffen sind,die normalerweise durch den konformen Faktor der Metrikverursacht werden. Dadurch motiviert, wird daraufhin einskalares Spielzeugmodell, das den konformen Sektor einer``$-R+R^2$''-Theorie simuliert, hinsichtlich seinerStabilitätseigenschaften im infraroten (IR) Bereichstudiert. Dabei stellt sich heraus, daß sich die Theorieunter Ausbildung einer nichttrivialen Vakuumstruktur auf dynamische Weise stabilisiert. In der Gravitation könnteneventuell nichtlokale Invarianten des Typs $intd^dx,sqrt{g}R (D^2)^{-1} R$ dafür sorgen, daß der konformeSektor auf ähnliche Weise IR-stabil wird.