Estabilidade de pontos de equilíbrio e existência de soluções periódicas em alguns modelos bidimensionais

Pós-graduação em Matemática Universitária - IGCE

Characterization of Stability Region for General Autonomous Nonlinear Dynamical Systems

The existing characterization of stability regions was developed under the assumption that limit sets on the stability boundary are exclusively composed of hyperbolic equilibrium points and closed orbits. The characterizations derived in this technical note are a generalization of existing results in the theory of stability regions. A characterization of the stability boundary of general autonomous nonlinear dynamical systems is developed under the assumption that limit sets on the stability boundary are composed of a countable number of disjoint and indecomposable components, which can be equilibrium points, closed orbits, quasi-periodic solutions and even chaotic invariant sets.

TYPE-ZERO SADDLE-NODE BIFURCATIONS AND STABILITY REGION ESTIMATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS

A complete characterization of the stability boundary of a class of nonlinear dynamical systems that admit energy functions is developed in this paper. This characterization generalizes the existing results by allowing the type-zero saddle-node nonhyperbolic equilibrium points on the stability boundary. Conceptual algorithms to obtain optimal estimates of the stability region (basin of attraction) in the form of level sets of a given family of energy functions are derived. The behavior of the stability region and the corresponding estimates are investigated for parameter variation in the neighborhood of a type-zero saddle-node bifurcation value.

Contents of radioactive elements, main chemical composition, and sedimentation rates of bottom sediments from some stations in the Southeast Pacific

Gamma-spectrometric analysis was used for six sediment cores from the area occupied by metalliferous sediments in the Southeast Pacific. In five of these cores vertical distribution curves of 230Th enabled positions of equilibrium points to be determined and sediments to be dated. The ionium curve was normalized for one core. Vertical distribution of 230Th in metalliferous sediments resembles its distribution in normal ocean-floor sediments beyond the area of influence of active ridges. Sedimentation rates lay within the range 0.7-12.3 mm/ky.

Variable threshold algorithm for division of labor analyzed as a dynamical system

Division of labor is a widely studied aspect of colony behavior of social insects. Division of labor models indicate how individuals distribute themselves in order to perform different tasks simultaneously. However, models that study division of labor from a dynamical system point of view cannot be found in the literature. In this paper, we define a division of labor model as a discrete-time dynamical system, in order to study the equilibrium points and their properties related to convergence and stability. By making use of this analytical model, an adaptive algorithm based on division of labor can be designed to satisfy dynamic criteria. In this way, we have designed and tested an algorithm that varies the response thresholds in order to modify the dynamic behavior of the system. This behavior modification allows the system to adapt to specific environmental and collective situations, making the algorithm a good candidate for distributed control applications. The variable threshold algorithm is based on specialization mechanisms. It is able to achieve an asymptotically stable behavior of the system in different environments and independently of the number of individuals. The algorithm has been successfully tested under several initial conditions and number of individuals.

Determinación de la higroscopicidad y comportamiento termodinámico de la madera juvenil y madura a través de sus isotermas de sorción

La respuesta higroscópica de la madera varía a lo largo de la dirección radial del árbol. El aumento de corta de ejemplares jóvenes y el uso de troncos de pequeños diámetros en la industria de los productos forestales, hacen preciso estudiar el comportamiento higroscópico tanto de la madera juvenil como de la madura. Su determinación proporciona información para comprender los mecanismos de sorción. Asimismo, la obtención de las propiedades termodinámicas de ambos tipos de madera facilita la modelización de procesos industriales como el secado o el encolado. En el presente trabajo, se ha comparado el comportamiento higroscópico y las propiedades termodinámicas de la madera juvenil y madura de Abies pinsapo Boissier, Abies alba Mill., Pinus canariensis C. Sm. ex DC., Pinus nigra Arnold, Pinus uncinata Mill. ex Mirb. y Pinus pinea L. Para este propósito se han utilizado las isotermas de sorción obtenidas mediante el método tradicional de sales saturadas descrito por COST Action E8 a 15, 35 y 50ºC en Abies pinsapo, Abies alba, Pinus nigra, Pinus uncinata y Pinus pinea, y a 35 y 50ºC en Pinus canariensis. Igualmente, se ha empleado el equipo dynamic vapor sorption (DVS) en la obtención de las isotermas de Pinus pinea a 35 y 50ºC. El ajuste de las curvas se ha realizado mediante el modelo Guggenheim, Anderson y de Boer-Dent (GAB), cumpliendo todas las muestras los criterios de aceptación establecidos. En el estudio de las isotermas se ha calculado el coeficiente y área de histéresis entre el proceso de adsorción y desorción para cada una de las muestras. Con el fin de comprender el comportamiento higroscópico experimentado por cada madera se ha determinado la composición química, espectros de infrarrojos (FTIR) y difractogramas de rayos X de cada una. Los parámetros termodinámicos - calor isostérico neto y total heat of wetting - se han obtenido a partir de las isotermas de sorción mediante el método de integración de la ecuación de Clausius-Clapeyron. Finalmente, se han comparado los datos obtenidos con el método tradicional de sales saturadas y con dynamic vapor sorption con el propósito de conocer la existencia de similitudes entre ambas metodologías. Los resultados mostraron que los puntos de equilibrio son, en la mayor parte de los casos, superiores en la madera madura frente a la juvenil, y por tanto las isotermas de la madera madura se encuentran siempre por encima de las de la juvenil, debido principalmente a la composición química. Respecto a las propiedades termodinámicas, se ha determinado que la energía involucrada en los procesos de sorción es superior en la madera madura que en la madera juvenil, siendo mayor en el proceso de desorción frente al de adsorción. En la comparación de las metodologías de sales saturadas y dynamic vapor sorption no se han detectado casi diferencias significativas en el proceso de adsorción, mientras que sí se han obtenido en el de desorción. ABSTRACT The hygroscopic response of wood varies throughout the radial direction of the tree. The longer cut of young trees and the use of small-diameter trunks in the forest product industry make it necessary to study the hygroscopic behaviour of both juvenile and mature wood. Determining this behaviour in both types of wood provides information for understanding the sorption mechanisms. Similarly, obtaining the thermodynamic properties of juvenile and mature wood facilitates modelling of industrial processes such as drying and bonding. In this study a comparison was made of the hygroscopic behaviour and thermodynamic properties of juvenile and mature wood of Abies pinsapo Boissier, Abies alba Mill., Pinus canariensis C. Sm. ex DC., Pinus nigra Arnold, Pinus uncinata Mill. ex Mirb. and Pinus pinea L. This was done by obtaining the sorption isotherms using the traditional saturated salt method described by COST Action E8 at 15, 35 and 50ºC in Abies pinsapo, Abies alba, Pinus nigra, Pinus uncinata and Pinus pinea, and at 35 and 50ºC in Pinus canariensis. In addition, dynamic vapour sorption (DVS) was used to obtain the isotherms of Pinus pinea at 35 and 50ºC. The curves were fitted using the Guggenheim, Anderson and de Boer- Dent (GAB) model and all samples met the established acceptance criteria. In the study of the isotherms, the hysteresis coefficient and area of the hysteresis loop between adsorption and desorption were calculated for each sample. To understand the hygroscopic behaviour of juvenile and mature wood, the chemical composition, infrared spectra (FTIR) and X-ray diffractograms of each type of wood were determined. The thermodynamic parameters - net isosteric heat and total heat of wetting - were obtained from the sorption isotherms by applying the integration method of the Clausius-Clapeyron equation. The data obtained using the traditional saturated salt method and with dynamic vapour sorption were compared to determine the similarities between the two methods. The results showed that the equilibrium points are greater in the mature wood than in the juvenile wood in most cases, and therefore the mature wood isotherms are always above the juvenile wood isotherms, mainly because of the chemical composition. As regards the thermodynamic properties, it was determined that the energy involved in the sorption processes is greater in the mature wood than in the juvenile wood, and is greater in the desorption process than in the adsorption process. On comparing the saturated salt and dynamic vapour sorption methods, almost no significant differences were detected in the adsorption process, but significant differences were obtained in the desorption process.

Local Bifurcations in a Nonlinear Model of a Bioreactor

This paper is partially supported by the Bulgarian Science Fund under grant Nr. DO 02– 359/2008.

Simultaneous Control of Chaotic Systems Using RBF Networks

Chaos control is a concept that recently acquiring more attention among the research community, concerning the fields of engineering, physics, chemistry, biology and mathematic. This paper presents a method to simultaneous control of deterministic chaos in several nonlinear dynamical systems. A radial basis function networks (RBFNs) has been used to control chaotic trajectories in the equilibrium points. Such neural network improves results, avoiding those problems that appear in other control methods, being also efficient dealing with a relatively small random dynamical noise.

One-Parameter Bifurcation Analysis of Dynamical Systems using Maple

This paper presents two algorithms for one-parameter local bifurcations of equilibrium points of dynamical systems. The algorithms are implemented in the computer algebra system Maple 13 © and designed as a package. Some examples are reported to demonstrate the package’s facilities.

Resource allocation in multi-user telecommunication networks

This dissertation discussed resource allocation mechanisms in several network topologies including infrastructure wireless network, non-infrastructure wireless network and wire-cum-wireless network. Different networks may have different resource constrains. Based on actual technologies and implementation models, utility function, game theory and a modern control algorithm have been introduced to balance power, bandwidth and customers' satisfaction in the system. ^ In infrastructure wireless networks, utility function was used in the Third Generation (3G) cellular network and the network was trying to maximize the total utility. In this dissertation, revenue maximization was set as an objective. Compared with the previous work on utility maximization, it is more practical to implement revenue maximization by the cellular network operators. The pricing strategies were studied and the algorithms were given to find the optimal price combination of power and rate to maximize the profit without degrading the Quality of Service (QoS) performance. ^ In non-infrastructure wireless networks, power capacity is limited by the small size of the nodes. In such a network, nodes need to transmit traffic not only for themselves but also for their neighbors, so power management become the most important issue for the network overall performance. Our innovative routing algorithm based on utility function, sets up a flexible framework for different users with different concerns in the same network. This algorithm allows users to make trade offs between multiple resource parameters. Its flexibility makes it a suitable solution for the large scale non-infrastructure network. This dissertation also covers non-cooperation problems. Through combining game theory and utility function, equilibrium points could be found among rational users which can enhance the cooperation in the network. ^ Finally, a wire-cum-wireless network architecture was introduced. This network architecture can support multiple services over multiple networks with smart resource allocation methods. Although a SONET-to-WiMAX case was used for the analysis, the mathematic procedure and resource allocation scheme could be universal solutions for all infrastructure, non-infrastructure and combined networks. ^

Solitons in Exciton-Polariton System: Reduced Modelling, Analysis, and Numerical Studies

In this dissertation, we study the behavior of exciton-polariton quasiparticles in semiconductor microcavities, under the sourceless and lossless conditions.

First, we simplify the original model by removing the photon dispersion term, thus effectively turn the PDEs system to an ODEs system,

and investigate the behavior of the resulting system, including the equilibrium points and the wave functions of the excitons and the photons.

Second, we add the dispersion term for the excitons to the original model and prove that the band of the discontinuous solitons now become dark solitons.

Third, we employ the Strang-splitting method to our sytem of PDEs and prove the first-order and second-order error bounds in the $H^1$ norm and the $L_2$ norm, respectively.

Using this numerical result, we analyze the stability of the steady state bright soliton solution. This solution revolves around the $x$-axis as time progresses

and the perturbed soliton also rotates around the $x$-axis and tracks closely in terms of amplitude but lags behind the exact one. Our numerical result shows orbital

stability but no $L_2$ stability.

The Control of a Mathematical Analog of a Tentacle

A 2-dimensional dynamic analog of squid tentacles was presented. The tentacle analog consists of a multi-cell structure, which can be easily replicated to a large scale. Each cell of the model is a quadrilateral with unit masses at the corners. Each side of the quadrilateral is a spring-damper system in parallel. The spring constants are the controls for the system. The dynamics are subject to the constraint that the area of each quadrilateral must remain constant. The system dynamics was analyzed, and various equilibrium points were found with different controls. Then these equilibrium points were further determined experimentally, demonstrated to be asymptotically stable. A simulation built in MATLAB was used to find the convergence rates under different controls and damping coefficients. Finally, a control scheme was developed and used to drive the system to several configurations observed in real tentacle.

Turbulence measurements in an equilibrium axisymmetric wall jet

This paper reports measurements of turbulent quantities in an axisymmetric wall jet subjected to an adverse pressure gradient in a conical diffuser, in such a way that a suitably defined pressure-gradient parameter is everywhere small. Self-similarity is observed in the mean velocity profile, as well as the profiles of many turbulent quantities at sufficiently large distances from the injection slot. Autocorrelation measurements indicate that, in the region of turbulent production, the time scale of ν fluctuations is very much smaller than the time scale of u fluctuations. Based on the data on these time scales, a possible model is proposed for the Reynolds stress. One-dimensional energy spectra are obtained for the u, v and w components at several points in the wall jet. It is found that self-similarity is exhibited by the one-dimensional wavenumber spectrum of $\overline{q^2}(=\overline{u^2}+\overline{v^2}+\overline{w^2})$, if the half-width of the wall jet and the local mean velocity are used for forming the non-dimensional wavenumber. Both the autocorrelation curves and the spectra indicate the existence of periodicity in the flow. The rate of dissipation of turbulent energy is estimated from the $\overline{q^2}$ spectra, using a slightly modified version of a previously suggested method.

Turbulence measurements in an ‘equilibrium’ axisymmetric wall jet Journal

This paper reports measurements of turbulent quantities in an axisymmetric wall jet subjected to an adverse pressure gradient in a conical diffuser, in such a way that a suitably defined pressure-gradient parameter is everywhere small. Self-similarity is observed in the mean velocity profile, as well as the profiles of many turbulent quantities at sufficiently large distances from the injection slot. Autocorrelation measurements indicate that, in the region of turbulent production, the time scale of ν fluctuations is very much smaller than the time scale of u fluctuations. Based on the data on these time scales, a possible model is proposed for the Reynolds stress. One-dimensional energy spectra are obtained for the u, v and w components at several points in the wall jet. It is found that self-similarity is exhibited by the one-dimensional wavenumber spectrum of $\overline{q^2}(=\overline{u^2}+\overline{v^2}+\overline{w^2})$, if the half-width of the wall jet and the local mean velocity are used for forming the non-dimensional wavenumber. Both the autocorrelation curves and the spectra indicate the existence of periodicity in the flow. The rate of dissipation of turbulent energy is estimated from the $\overline{q^2}$ spectra, using a slightly modified version of a previously suggested method.

Fixed-orientation equilateral triangle matching of point sets

Given a point set P and a class C of geometric objects, G(C)(P) is a geometric graph with vertex set P such that any two vertices p and q are adjacent if and only if there is some C is an element of C containing both p and q but no other points from P. We study G(del)(P) graphs where del is the class of downward equilateral triangles (i.e., equilateral triangles with one of their sides parallel to the x-axis and the corner opposite to this side below that side). For point sets in general position, these graphs have been shown to be equivalent to half-Theta(6) graphs and TD-Delaunay graphs. The main result in our paper is that for point sets P in general position, G(del)(P) always contains a matching of size at least vertical bar P vertical bar-1/3] and this bound is tight. We also give some structural properties of G(star)(P) graphs, where is the class which contains both upward and downward equilateral triangles. We show that for point sets in general position, the block cut point graph of G(star)(P) is simply a path. Through the equivalence of G(star)(P) graphs with Theta(6) graphs, we also derive that any Theta(6) graph can have at most 5n-11 edges, for point sets in general position. (C) 2013 Elsevier B.V. All rights reserved.