Dinâmica não-linear e reagente limitante: um exemplo de bifurcação transcrítica

The nonlinear analysis of a general mixed second order reaction was performed, aiming to explore some basic tools concerning the mathematics of nonlinear differential equations. Concepts of stability around fixed points based on linear stability analysis are introduced, together with phase plane and integral curves. The main focus is the chemical relationship between changes of limiting reagent and transcritical bifurcation, and the investigation underlying the conclusion.

Polymeeripinnoitteen aiheuttamat itseherätteiset värähtelyt nipillisessä telasysteemissä

Lukuisissa teollisuussovelluksissa materiaalien, kuten paperin ja teräslevyjen, muokkaamiseen käytettävät pyörivät nippitelat kärsivät aina erilaisten herätteiden synnyttämistä mekaanisista värähtelyistä, jotka voivat aiheuttaa virheitä valmistettaviin tuotteisiin. Tässä työssä tutkittiin viskoelastisia polymeerejä ja polymeeripinnoitteen nipilliseen telasysteemiin synnyttämiä haitallisia itseherätteisiä värähtelyjä. Työn polymeerejä käsittelevässä kirjallisuusosassa luotiin katsaus amorfisten polymeerien fysikaalisiin ominaisuuksiin. Kokeellisessa osuudessa tutkittiin tarkemmin kahden amorfisen telapinnoitepolymeerin termoreologisia ja mekaanisia ominaisuuksia suoritettujen DMTA-mittausten perusteella. Sovittamalla toisen polymeerin master-käyrään yleistetty lineaarisen standardiaineen malli saatiin selville polymeerin mekaaniset parametrit ja approksimaatio sen relaksaatiospektrille. Telapinnoitteen nipilliseen systeemiin synnyttämiä itseherätteisiä värähtelyjä ja niiden seurauksia tarkasteltiin kahdelle telalle ja polymeeripinnoitteelle kehitetyn analyyttisen mallin ja numeeristen laskujen avulla. Pinnoite mallinnettiin lineaarisen standardiaineen mukaisesti. Telasysteemin parametrit määritettiin DMTA-mittaustuloksista ja systeemiä vastaavasta koelaitteesta kokeellisella moodianalyysillä ja elementtimenetelmällä. Numeerisesta stabiilisuusanalyysistä ja liikeyhtälöiden integroinneista saadut tulokset kertovat telapinnoitteen aaltomaisista deformaatiomuodoista ja niiden synnyttämistä taajuusalueittain esiintyvistä epästabiileista värähtelyistä. Telasysteemi on epästabiili pinnoitedeformaatiokuvion systeemiin aiheuttaman herätevoiman taajuuden ollessa lähellä systeemin korkeampaa ominaistaajuutta. Numeerisista tuloksista voitiin ennustaa nopean ja hitaan barringin olemassaolo.

Qualitative Characteristics and Quantitative Measures of Solution's Reliability in Discrete Optimization: Traditional Analytical Approaches, Innovative Computational Methods and Applicability

The purpose of this thesis is twofold. The first and major part is devoted to sensitivity analysis of various discrete optimization problems while the second part addresses methods applied for calculating measures of solution stability and solving multicriteria discrete optimization problems. Despite numerous approaches to stability analysis of discrete optimization problems two major directions can be single out: quantitative and qualitative. Qualitative sensitivity analysis is conducted for multicriteria discrete optimization problems with minisum, minimax and minimin partial criteria. The main results obtained here are necessary and sufficient conditions for different stability types of optimal solutions (or a set of optimal solutions) of the considered problems. Within the framework of quantitative direction various measures of solution stability are investigated. A formula for a quantitative characteristic called stability radius is obtained for the generalized equilibrium situation invariant to changes of game parameters in the case of the H¨older metric. Quality of the problem solution can also be described in terms of robustness analysis. In this work the concepts of accuracy and robustness tolerances are presented for a strategic game with a finite number of players where initial coefficients (costs) of linear payoff functions are subject to perturbations. Investigation of stability radius also aims to devise methods for its calculation. A new metaheuristic approach is derived for calculation of stability radius of an optimal solution to the shortest path problem. The main advantage of the developed method is that it can be potentially applicable for calculating stability radii of NP-hard problems. The last chapter of the thesis focuses on deriving innovative methods based on interactive optimization approach for solving multicriteria combinatorial optimization problems. The key idea of the proposed approach is to utilize a parameterized achievement scalarizing function for solution calculation and to direct interactive procedure by changing weighting coefficients of this function. In order to illustrate the introduced ideas a decision making process is simulated for three objective median location problem. The concepts, models, and ideas collected and analyzed in this thesis create a good and relevant grounds for developing more complicated and integrated models of postoptimal analysis and solving the most computationally challenging problems related to it.

Development of a Control System for a Tele Operated Mobile Robot

Mobile robots are capable of performing spatial displacement motions in different environments. This motions can be calculated based on sensorial data (autonomous robot) or given by an operator (tele operated robot). This thesis is focused on the latter providing the control architecture which bridges the tele operator and the robot’s locomotion system and end effectors. Such a task might prove overwhelming in cases where the robot comprises a wide variety of sensors and actuators hence a relatively new option was selected: Robot Operating System (ROS). The control system of a new robot will be sketched and tested in a simulation model using ROS together with Gazebo in order to determine the viability of such a system. The simulated model will be based on the projected shape and main features of the real machine. A stability analysis will be performed first theoretically and afterwards using the developed model. This thesis concluded that both the physical properties and the control architecture are feasible and stable settling up the ground for further work with the same robot.

Sur un modèle d'érythropoïèse comportant un taux de mortalité dynamique

Ce mémoire concerne la modélisation mathématique de l’érythropoïèse, à savoir le processus de production des érythrocytes (ou globules rouges) et sa régulation par l’érythropoïétine, une hormone de contrôle. Nous proposons une extension d’un modèle d’érythropoïèse tenant compte du vieillissement des cellules matures. D’abord, nous considérons un modèle structuré en maturité avec condition limite mouvante, dont la dynamique est capturée par des équations d’advection. Biologiquement, la condition limite mouvante signifie que la durée de vie maximale varie afin qu’il y ait toujours un flux constant de cellules éliminées. Par la suite, des hypothèses sur la biologie sont introduites pour simplifier ce modèle et le ramener à un système de trois équations différentielles à retard pour la population totale, la concentration d’hormones ainsi que la durée de vie maximale. Un système alternatif composé de deux équations avec deux retards constants est obtenu en supposant que la durée de vie maximale soit fixe. Enfin, un nouveau modèle est introduit, lequel comporte un taux de mortalité augmentant exponentiellement en fonction du niveau de maturité des érythrocytes. Une analyse de stabilité linéaire permet de détecter des bifurcations de Hopf simple et double émergeant des variations du gain dans la boucle de feedback et de paramètres associés à la fonction de survie. Des simulations numériques suggèrent aussi une perte de stabilité causée par des interactions entre deux modes linéaires et l’existence d’un tore de dimension deux dans l’espace de phase autour de la solution stationnaire.

Analytical Investigations 0n Collapse of Cylindrical Submarine Shells

Submarine hull structure is a watertight envelope, under hydrostatic pressure when in operation. Stiffened cylindrical shells constitute the major portion of these submarine hulls and these thin shells under compression are susceptible to buckling failure. Normally loss of stability occurs at the limit point rather than at the bifurcation point and the stability analysis has to consider the change in geometry at each load step. Hence geometric nonlinear analysis of the shell forms becomes. a necessity. External hydrostatic pressure will follow the deformed configuration of the shell and hence follower force effect has to be accounted for. Computer codes have been developed based on all-cubic axisymmetric cylindrical shell finite element and discrete ring stiffener element for linear elastic, linear buckling and geometric nonIinear analysis of stiffened cylindrical shells. These analysis programs have the capability to treat hydrostatic pressure as a radial load and as a follower force. Analytical investigations are carried out on two attack submarine cylindrical hull models besides standard benchmark problems. In each case, the analysis has been carried out for interstiffener, interdeepframe and interbulkhead configurations. The shell stiffener attachment in each of this configuration has been represented by the simply supported-simply supported, clamped-clamped and fixed-fixed boundary conditions in this study. The results of the analytical investigations have been discussed and the observations and conclusions are described. Rotation restraint at the ends is influential for interstiffener and interbulkhead configurations and the significance of axial restraint becomes predominant in the interbulkhead configuration. The follower force effect of hydrostatic pressure is not significant in interstiffener and interdeepframe configurations where as it has very high detrimental effect on buckling pressure on interbulkhead configuration. The geometric nonlinear interbulkhead analysis incorporating follower force effect gives the critical value of buckling pressure and this analysis is recommended for the determination of collapse pressure of stiffened cylindrical submarine shells.

Studies on Some Aspects of Light Beam Propagation Through Certain Nonlinear Media

This thesis deals with the study of light beam propagation through different nonlinear media. Analytical and numerical methods are used to show the formation of solitonS in these media. Basic experiments have also been performed to show the formation of a self-written waveguide in a photopolymer. The variational method is used for the analytical analysis throughout the thesis. Numerical method based on the finite-difference forms of the original partial differential equation is used for the numerical analysis.In Chapter 2, we have studied two kinds of solitons, the (2 + 1) D spatial solitons and the (3 + l)D spatio-temporal solitons in a cubic-quintic medium in the presence of multiphoton ionization.In Chapter 3, we have studied the evolution of light beam through a different kind of nonlinear media, the photorcfractive polymer. We study modulational instability and beam propagation through a photorefractive polymer in the presence of absorption losses. The one dimensional beam propagation through the nonlinear medium is studied using variational and numerical methods. Stable soliton propagation is observed both analytically and numerically.Chapter 4 deals with the study of modulational instability in a photorefractive crystal in the presence of wave mixing effects. Modulational instability in a photorefractive medium is studied in the presence of two wave mixing. We then propose and derive a model for forward four wave mixing in the photorefractive medium and investigate the modulational instability induced by four wave mixing effects. By using the standard linear stability analysis the instability gain is obtained.Chapter 5 deals with the study of self-written waveguides. Besides the usual analytical analysis, basic experiments were done showing the formation of self-written waveguide in a photopolymer system. The formation of a directional coupler in a photopolymer system is studied theoretically in Chapter 6. We propose and study, using the variational approximation as well as numerical simulation, the evolution of a probe beam through a directional coupler formed in a photopolymer system.

Strait dynamics of tropical tidal inlets

The dynamics and associated stability analysis of tidal inlets situated on the southwest coast of India, namely Andhakaranazhi (90 45 J OO JJN and 760 17 J 29 JJ E) and the other at Cochin harbour inlet (90 58 1 04 J1N and 760 14 1 50 1J E) have beenconducted. A detailed study on the inlet regime of Cochin barmouth (permanent in nature) was attempted so as to elucidate information on: (a) channel characteristics (b) tidal hydraulics and (c) stability of the inlet. In this connection, a naturally occurring seasonal sandbar formation at Andhakaranazhi, near Sherthallay, about 20 km south of Cochin inlet, was also chosen as a site ofstudy brought out conclusively the dynamical study. The aspects of ( 1) tidal influx/out flux (2) channel morphology (3) sedimentation regime and (4) stability and factors related to stability of these locations. The above aspects are supported by suitable mathematical formulations to describe the associated coastal processes, wherever applicable

Onset And Characterisation Of Chaos In A Two Dimensional Nonlinear Deterministic Model

Nature is full of phenomena which we call "chaotic", the weather being a prime example. What we mean by this is that we cannot predict it to any significant accuracy, either because the system is inherently complex, or because some of the governing factors are not deterministic. However, during recent years it has become clear that random behaviour can occur even in very simple systems with very few number of degrees of freedom, without any need for complexity or indeterminacy. The discovery that chaos can be generated even with the help of systems having completely deterministic rules - often models of natural phenomena - has stimulated a lo; of research interest recently. Not that this chaos has no underlying order, but it is of a subtle kind, that has taken a great deal of ingenuity to unravel. In the present thesis, the author introduce a new nonlinear model, a ‘modulated’ logistic map, and analyse it from the view point of ‘deterministic chaos‘.

" Nonlinear Signal Processing of Electroencephalogram .' Application in the Study of Neurodynamics. "

Interfacings of various subjects generate new field ofstudy and research that help in advancing human knowledge. One of the latest of such fields is Neurotechnology, which is an effective amalgamation of neuroscience, physics, biomedical engineering and computational methods. Neurotechnology provides a platform to interact physicist; neurologist and engineers to break methodology and terminology related barriers. Advancements in Computational capability, wider scope of applications in nonlinear dynamics and chaos in complex systems enhanced study of neurodynamics. However there is a need for an effective dialogue among physicists, neurologists and engineers. Application of computer based technology in the field of medicine through signal and image processing, creation of clinical databases for helping clinicians etc are widely acknowledged. Such synergic effects between widely separated disciplines may help in enhancing the effectiveness of existing diagnostic methods. One of the recent methods in this direction is analysis of electroencephalogram with the help of methods in nonlinear dynamics. This thesis is an effort to understand the functional aspects of human brain by studying electroencephalogram. The algorithms and other related methods developed in the present work can be interfaced with a digital EEG machine to unfold the information hidden in the signal. Ultimately this can be used as a diagnostic tool.

Interface dynamics in planar neural field models

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.

Multiple stationary solutions of an irradiated slab

A mathematical model describing the heat budget of an irradiated medium is introduced. The one-dimensional form of the equations and boundary conditions are presented and analysed. Heat transport at one face of the slab occurs by absorption (and reflection) of an incoming beam of short-wave radiation with a fraction of this radiation penetrating into the body of the slab, a diffusive heat flux in the slab and a prescribed incoming heat flux term. The other face of the slab is immersed in its own melt and is considered to be a free surface. Here, temperature continuity is prescribed and evolution of the surface is determined by a Stefan condition. These boundary conditions are flexible enough to describe a range of situations such as a laser shining on an opaque medium, or the natural environment of polar sea ice or lake ice. A two-stream radiation model is used which replaces the simple Beer’s law of radiation attenuation frequently used for semi-infinite domains. The stationary solutions of the governing equations are sought and it is found that there exists two possible stationary solutions for a given set of boundary conditions and a range of parameter choices. It is found that the existence of two stationary solutions is a direct result of the model of radiation absorption, due to its effect on the albedo of the medium. A linear stability analysis and numerical calculations indicate that where two stationary solutions exist, the solution corresponding to a larger thickness is always stable and the solution corresponding to a smaller thickness is unstable. Numerical simulations reveal that when there are two solutions, if the slab is thinner than the smaller stationary thickness it will melt completely, whereas if the slab is thicker than the smaller stationary thickness it will evolve toward the larger stationary thickness. These results indicate that other mechanisms (e.g. wave-induced agglomeration of crystals) are necessary to grow a slab from zero initial thickness in the parameter regime that yields two stationary solutions.

A mathematical model of crystallization in an emulsion

A mathematical model incorporating many of the important processes at work in the crystallization of emulsions is presented. The model describes nucleation within the discontinuous domain of an emulsion, precipitation in the continuous domain, transport of monomers between the two domains, and formation and subsequent growth of crystals in both domains. The model is formulated as an autonomous system of nonlinear, coupled ordinary differential equations. The description of nucleation and precipitation is based upon the Becker–Döring equations of classical nucleation theory. A particular feature of the model is that the number of particles of all species present is explicitly conserved; this differs from work that employs Arrhenius descriptions of nucleation rate. Since the model includes many physical effects, it is analyzed in stages so that the role of each process may be understood. When precipitation occurs in the continuous domain, the concentration of monomers falls below the equilibrium concentration at the surface of the drops of the discontinuous domain. This leads to a transport of monomers from the drops into the continuous domain that are then incorporated into crystals and nuclei. Since the formation of crystals is irreversible and their subsequent growth inevitable, crystals forming in the continuous domain effectively act as a sink for monomers “sucking” monomers from the drops. In this case, numerical calculations are presented which are consistent with experimental observations. In the case in which critical crystal formation does not occur, the stationary solution is found and a linear stability analysis is performed. Bifurcation diagrams describing the loci of stationary solutions, which may be multiple, are numerically calculated.

A mathematical model of the sterol regulatory element binding protein 2 cholesterol biosynthesis pathway

Cholesterol is one of the key constituents for maintaining the cellular membrane and thus the integrity of the cell itself. In contrast high levels of cholesterol in the blood are known to be a major risk factor in the development of cardiovascular disease. We formulate a deterministic nonlinear ordinary differential equation model of the sterol regulatory element binding protein 2 (SREBP-2) cholesterol genetic regulatory pathway in an hepatocyte. The mathematical model includes a description of genetic transcription by SREBP-2 which is subsequently translated to mRNA leading to the formation of 3-hydroxy-3-methylglutaryl coenzyme A reductase (HMGCR), a main precursor of cholesterol synthesis. Cholesterol synthesis subsequently leads to the regulation of SREBP-2 via a negative feedback formulation. Parameterised with data from the literature, the model is used to understand how SREBP-2 transcription and regulation affects cellular cholesterol concentration. Model stability analysis shows that the only positive steady-state of the system exhibits purely oscillatory, damped oscillatory or monotic behaviour under certain parameter conditions. In light of our findings we postulate how cholesterol homestasis is maintained within the cell and the advantages of our model formulation are discussed with respect to other models of genetic regulation within the literature.

Phytoplankton–zooplankton dynamics in the ‘presence’ or ‘absence’ of toxic phytoplankton

The inhibitory effects of toxin-producing phytoplankton (TPP) on zooplankton modulate the dynamics of marine plankton. In this article, we employ simple mathematical models to compare theoretically the dynamics of phytoplankton–zooplankton interaction in situations where the TPP are present with those where TPP are absent. We consider two sets of three-component interaction models: one that does not include the effect of TPP and the other that does. The negative effects of TPP on zooplankton is described by a non-linear interaction term. Extensive theoretical analyses of the models have been performed to understand the qualitative behaviour of the model systems around every possible equilibria. The results of local-stability analysis and numerical simulations demonstrate that the two model-systems differ qualitatively with regard to oscillations and stability. The model system that does not include TPP is asymptotically stable around the coexisting equilibria, whereas, the system that includes TPP oscillates for a range of parametric values associated with toxin-inhibition rate and competition coefficients. Our analysis suggests that the qualitative dynamics of the plankton–zooplankton interactions are very likely to alter due to the presence of TPP species, and therefore the effects of TPP should be considered carefully while modelling plankton dynamics.