On s'intéresse dans ce travail aux automates cellulaires unidimensionnels qui ont été largement étudiés mais où il reste beaucoup à faire. La théorie spectrale des automates cellulaires a notamment été peu abordée à l'exception de quelques résultats indirects. On cherche a mieux comprendre les cadres topologiques et ergodiques en étudiant l'existence de valeurs propres en particulier celles irrationnelles c'est à dire de la forme e^{2Iπα} où α est un irrationnel et I la racine carrée de l'unité. Cette question ne semble pas avoir été abordée jusqu'à présent. Dans le cadre topologique les résultats sur l'équicontinuité de Kůrka et Blanchard et Tisseur permettent de déduire directement que tout automate cellulaire équicontinu possède des valeurs propres topologiques rationnelles. La densité des points périodiques pour le décalage empêche l'existence de valeurs propres topologiques irrationnelles. La densité des points périodiques pour l'automate cellulaire semble être liée à la question des valeurs propres. Dans le cadre topologique, si l'automate cellulaire possède des points d'équicontinuité sans être équicontinu, la densité des points périodiques a comme conséquence le fait que le spectre représente l'ensemble des racines rationnelles de l'unité c'est à dire tous les nombres de la forme e^{2Iπα} avec α∈Q .Dans le cadre mesuré, la question devient plus difficile, on s'intéresse à la dynamique des automates cellulaires surjectifs pour lesquels la mesure uniforme est invariante en vertu du théorème de Hedlund. La plupart des résultats obtenus demeurent valable dans un cadre plus large. Nous commençons par montrer que les automates cellulaires ayant des points d'équicontinuité ne possèdent pas de valeurs propres mesurables irrationnelles. Ce résultat se généralise aux automates cellulaires possédant des points μ-équicontinu selon la définition de Gilman. Nous démontrons finalement que les automates cellulaires possédant des points μ-équicontinu selon la définition de Gilman possèdent des valeurs propres rationnelles / We investigate properties of one-dimensional cellular automata. This category of cellular automata has been widely studied but many questions are still open. Among them the spectral theory of unidimensional cellular automata is an open field with few indirect results. We want a better understanding of both ergodic and topological aspect by investigating the existence of eigenvalues of cellular automata, in particular irrational ones, i.e., those of the form e^{2Iπα} where α is irrationnal and I the complex root of -1. The last question seems not to have been studied yet.In the topological field the results of Kůrka & Blanchard and Tisseur about equicontinuous cellular automata have as direct consequence that any equicontinuous CA has rational eigenvalues. Density of shift periodic points leads to the impossibility for CA to have topological irrational eigenvalues. The density of periodic points of cellular automata seems to be related with the question of eignevalues. If the CA has equicontinuity points without being equicontinuous, the density of periodic points implies the fact that the spectrum contains all rational roots of the unity, i.e., all numbers of the form e^{2Iπα} with α∈Q .In the measurable field the question becomes harder. We assume that the cellular automaton is surjective, which implies that the uniform measure is invariant. Most results are still available in more general conditions. We first prove that cellular automata with equicontinuity points never have irrational measurable eigenvalues. This result is then generalized to cellular automata with μ-equicontinuous points according to Gilman's classification. We also prove that cellular automata with μ-equicontinuous points have rational eigenvalues

In this thesis I study the effects of gap junctions on pattern formation in a neural field model for working memory. I review known results for the base model (the “Amari model”), then see how the results change for the “gap junction model”. I find steady states of both models analytically and numerically, using lateral inhibition with a step firing rate function, and a decaying oscillatory coupling function with a smooth firing rate function. Steady states are homoclinic orbits to the fixed point at the origin. I also use a method of piecewise construction of solutions by deriving an ordinary differential equation from the partial integro-differential formulation of the model. Solutions are found numerically using AUTO and my own continuation code in MATLAB. Given an appropriate level of threshold, as the firing rate function steepens, the solution curve becomes discontinuous and stable homoclinic orbits no longer exist in a region of parameter space. These results have not been described previously in the literature. Taking a phase space approach, the Amari model is written as a four-dimensional, reversible Hamiltonian system. I develop a numerical technique for finding both symmetric and asymmetric homoclinic orbits. I discover a small separate solution curve that causes the main curve to break as the firing rate function steepens and show there is a global bifurcation. The small curve and the global bifurcation have not been reported previously in the literature. Through the use of travelling fronts and construction of an Evans function, I show the existence of stable heteroclinic orbits. I also find asymmetric steady state solutions using other numerical techniques. Various methods of determining the stability of solutions are presented, including a method of eigenvalue analysis that I develop. I then find both stable and transient Turing structures in one and two spatial dimensions, as well as a Type-I intermittency. To my knowledge, this is the first time transient Turing structures have been found in a neural field model. In the Appendix, I outline numerical integration schemes, the pseudo-arclength continuation method, and introduce the software package AUTO used throughout the thesis.

We investigate chaotic ionization of highly excited hydrogen atom in crossed electric and magnetic fields (Rydberg atom) and intra-molecular relaxation in planar carbonyl sulfide (OCS) molecule. The underlying theoretical framework of our studies is dynamical systems theory and periodic orbit theory. These theories offer formulae to compute expectation values of observables in chaotic systems with best accuracy available in given circumstances, however they require to have a good control and reliable numerical tools to compute unstable periodic orbits. We have developed such methods of computation and partitioning of the phase space of hydrogen atom in crossed at right angles electric and magnetic fields, represented by a two degree of freedom (dof) Hamiltonian system. We discuss extensions to a 3-dof setting by developing the methodology to compute unstable invariant tori, and applying it to the planar OCS, represented by a 3-dof Hamiltonian. We find such tori important in explaining anomalous relaxation rates in chemical reactions. Their potential application in Transition State Theory is discussed.

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<p dir="ltr">The increasing availability of spatio-temporal information in the form of detailed time-resolved images sampled at very high framing rates has resulted in a search for mathematical techniques capable of extracting and relaying the pertinent underlying physics governing complex flows. Analysis relying on the usage of a solitary spectral, correlation, or modal decomposition techniques to identify dynamically significant information from large datasets may give an incomplete description of these phenomena. Moreover, fully resolved spatio-temporal measurements of these complex flow fields are needed for a complete and accurate description across a wide spectrum of length and time scales. The primary goals of this dissertation are address these challenges in two key aspects: (1) to improve and demonstrate the novel application of complementary data analysis and modal decomposition techniques to quantify the dynamics of flow systems that exhibit intricate patterns and behaviors in both space and time, and (2) to make advancements in achieving and characterizing high-resolution and high-speed quantitative measurements of turbulent mixing fields.</p><p dir="ltr">In the first goal, two canonical flow fields are considered, including an acoustically excited co-axial jet and a bluff-body stabilized flame. The local susceptibility of a nonreacting, cryogenic, coaxial-jet, rocket injector to transverse acoustics is characterized by applying dynamical systems theory in conjunction with complementary wavelet-based spectral decomposition to high-speed backlit images of flow field. The local coupling of the jet with external acoustics is studied as a function of the relative momentum flux ratio between the outer and inner jets, giving a quantitative description of the dynamical response of each jet to external acoustics as a function of the downstream distance from the nozzle.</p><p dir="ltr">Bluff bodies are a common feature in the design of propulsion systems owing to their ability to act as flame holders. The reacting wake behind the bluff body consists of a recirculation bubble laden with hot-products and wrapped between separated shear layers. The wake region of a bluff body is systematically investigated utilizing a technique known as robust dynamic mode decomposition (DMD) to discern the onset of the thermoacoustic instability mode, which is highly detrimental to aerospace propulsion systems. The approach enables quantification of the spatial distribution and behavior of coherent structures observed from different flows as a function of the equivalence ratio.</p><p dir="ltr">As modal decomposition techniques employ a certain degree of averaging in time, a novel space-and-time local filtering technique utilizing the well-defined characteristics of wavelets is introduced with a goal of temporally resolving the spatial evolution of irregular flow instabilities associated with specific frequencies. This provides insight into the existence of transient sub-modal characteristics representing intermittencies within seemingly stable modes. The flow fields obtained from the same two canonical flows are interrogated to demonstrate the utility of the technique. It has been shown that temporally resolved flow features obtained from wavelet filtering satisfactorily track the same modal featured derived from DMD, but reveal sub-modal spatial distortions or local non-stationarity of specific modal frequencies on a frame-by-frame basis.</p><p dir="ltr">Finally, to improve the ability to study the dynamical behavior of complex flows across the full range of spatio-temporal scales present, advancements are reported in the spatial and temporal quantitative measurement of the scalar quantities in turbulent mixing fields utilizing 100 kHz planar laser-induced fluorescence (PLIF) and Rayleigh scattering imaging of acetone. The imaging system provided a resolution of 55 µm with a field-of-view mapping of 18.5 µm/pixel on the camera sensor, which is three times better spatial resolution than the previous reported work to-date for similar flow fields that were investigated at 1/10^th the current measurement rate. The power spectra of instantaneous mixture fraction fluctuations adhere to Kolmogorov's well-established -5/3 law, showing that the technique captures a significant range of dissipation scales. This observation underscores the ability to study mixing dynamics throughout the turbulent by fully resolving scalar fluctuations up to 30 kHz. This enhanced spatio-temporal resolution allows for a more detailed investigation of the dynamical behavior of turbulent flows with complex modal interactions down to the smallest diffusion limited mixing scales.</p>

Systems biology is an emergent field focused on developing a system-level understanding of biological systems. In the last decade advances in genomics, transcriptomics and proteomics have gathered a remarkable amount data enabling the possibility of a system-level analysis to be grounded at a molecular level. The reverse-engineering of biochemical networks from experimental data has become a central focus in systems biology. A variety of methods have been proposed for the study and identification of the system's structure and/or dynamics. The objective of this dissertation is to introduce and propose solutions to some of the challenges inherent in reverse-engineering of biological systems. First, previously developed reverse engineering algorithms are studied and compared using data from a simulated network. This study draws attention to the necessity for a uniform benchmark that enables an ob jective comparison and performance evaluation of reverse engineering methods. Since several reverse-engineering algorithms require discrete data as input (e.g. dynamic Bayesian network methods, Boolean networks), discretization methods are being used for this purpose. Through a comparison of the performance of two network inference algorithms that use discrete data (from several different discretization methods) in this work, it has been shown that data discretization is an important step in applying network inference methods to experimental data. Next, a reverse-engineering algorithm is proposed within the framework of polynomial dynamical systems over finite fields. This algorithm is built for the identification of the underlying network structure and dynamics; it uses as input gene expression data and, when available, a priori knowledge of the system. An evolutionary algorithm is used as the heuristic search method for an exploration of the solution space. Computational algebra tools delimit the search space, enabling also a description of model complexity. The performance and robustness of the algorithm are explored via an artificial network of the segment polarity genes in the D. melanogaster. Once a mathematical model has been built, it can be used to run simulations of the biological system under study. Comparison of simulated dynamics with experimental measurements can help refine the model or provide insight into qualitative properties of the systems dynamical behavior. Within this work, we propose an efficient algorithm to describe the phase space, in particular to compute the number and length of all limit cycles of linear systems over a general finite field. This research has been partially supported by NIH Grant Nr. RO1GM068947-01. / Ph. D.

Doctor of Philosophy / Department of Mathematics / Alexander G. Ramm / Several methods for a stable solution to the equation $F(u)=f$ have been developed. Here $F:H\to H$ is an operator in a Hilbert space $H$, and we assume that noisy data $f_\delta$, $\|f_\delta-f\|\le \delta$, are given in place of the exact data $f$. When $F$ is a linear bounded operator, two versions of the Dynamical Systems Method (DSM) with stopping rules of Discrepancy Principle type are proposed and justified mathematically. When $F$ is a non-linear monotone operator, various versions of the DSM are studied. A Discrepancy Principle for solving the equation is formulated and justified. Several versions of the DSM for solving the equation are formulated. These methods consist of a Newton-type method, a gradient-type method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation $F(u)=f$ is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to the equation $F(u)=f$ is proved. This dissertation consists of six chapters which are based on joint papers by the author and his advisor Prof. Alexander G. Ramm. These papers are published in different journals. The first two chapters deal with equations with linear and bounded operators and the last four chapters deal with non-linear equations with monotone operators.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Biofilms are aggregations of bacteria that can thrive wherever there is a watersurface or water-interface. Sometimes they can be beneficial; for example, biofilms are used in water and waste-water treatment. The filter used to remove contaminants acts as a scaffold for microbial attachment and growth. However, biofilms could have bad effects, especially on a persons health. They can cause chronic diseases and serious infections. The importance of biofilms in industrial and medical settings, is the main reason of the mathematical studies performed up to now, concerning biofilms. Biofilms have been mathematical modelling targets over the last 30 years. The complex structure and growth of biofilms make them difficult to study. Biofilm formation is a multi-stage process and occurs in even the most unlikely of environmental conditions. Models of biofilms vary from the discrete to the continuous; accounting for one-species to multi-species and from one-scale to multi-scale models. A model may even have both discrete and continuous parts. The implication of these differences is that the tools used to model biofilms differ; we present and review some of these models. The aim in this thesis is to model the early initiation of biofilm formation. This stage involves bacterial movement towards a surface and the attachment to the boundary which seeds a biofilm. We use a diffusion equation to describe a bacterial random walk and appropriate boundary conditions to model surface attachment. An analytical solution is obtained which gives the bacterial density as a function of position and time. The model is also analysed for stability. Independent of this model, we also give a reaction diffusion equation for the distribution of sensing molecules, accounting for production by the bacteria and natural degradation. The last model we present is of Keller-Segel type, which couples the dynamics of bacterial movement to that of the sensing molecules. In this case, bacteria perform a biased random walk towards the sensing molecules. The most important part of this chapter is the derivation of the boundary conditions. The adhesion of bacteria to a surface is presented by zero-Dirichlet boundary conditions, while the equation describing sensing molecules at the interface needed particular conditions to be set. Bacteria at the boundary also produce sensing molecules, which may then diffuse and degrade. In order to obtain an equation that includes all these features we assumed that mass is conserved. We conclude with a numerical simulation. / AFRIKAANSE OPSOMMING: Biofilms is die samedromming van bakterieë wat kan floreer waar daar ’n wateroppervlakte of watertussenvlak is. Soms kan hulle voordelig wees, soos byvoorbeeld, biofilms word gebruik in water en afvalwater behandeling. Die filter wat gebruik word om smetstowwe te verwyder, dien as ’n steier vir mikrobiese verbinding en groei. Biofilms kan ook egter slegte gevolge he, veral op ’n persoon se gesondheid. Hulle kan slepende siektes en ernstige infeksies veroorsaak. Die belangrikheid van biofilms in industriële en mediese omgewings, is die hoof rede vir die wiskundige studies wat tot dusver uitgevoer is met betrekking tot biofilms. Biofilms is oor die afgelope 30 jaar al ’n teiken vir wiskundige modellering. Die komplekse struktuur en groei van biofilms maak dit moeilik om hul te bestudeer. Biofilm formasie is ’n multi-fase proses, en gebeur selfs in die mees onwaarskynlikste omgewings. Modelle wat biofilms beskryf wissel van die diskreet tot die kontinu, inkorporeer een of meer spesies, en strek van eentot multi-skaal modelle. ’n Model kan ook oor beide diskreet en kontinue komponente besit. Dit beteken dat die tegnieke wat gebruik word om biofilms te modelleer ook verskil. In hierdie proefskrif verskaf ons ’n oorsig van sommige van hierdie modelle. Die doel in hierdie proefskrif is om die vroeë aanvang van biofilm ontwikkeling te modeleer. Hierdie fase behels ’n bakteriële beweging na ’n oppervlak toe en die aanvanklike aanhegsel wat sal ontkiem in ’n biofilm. Ons gebruik ’n diffusievergelyking om ’n bakteriële kanslopie te beskryf, met geskikte randvoorwaardes. ’n Analities oplossing is verkry wat die bakteriële bevolkingsdigtheid beskryf as ’n funksie van tyd en posisie. Die model is ook onleed om te toets vir stabiliteit. Onafhanklik van die model, gee ons ook ’n reaksiediffusievergelyking vir die beweging van waarnemings-molekules, wat insluit produksie deur die bakterieë en natuurlike afbreking. Die laaste model wat ten toon gestel word is ’n Keller-Segel tipe model, wat die bakteriese en waarnemings-molekule dinamika koppel. In hierdie geval, neem die bakterieë ’n sydige kanslopie agter die waarnemings molekules aan. Die belangrikste deel van hierdie hoofstuk is die afleiding van die randvoorwaardes. Die klewerigheid van die bakterieë tot die oppervlak word vvorgestel deur nul-Dirichlet randvoorwaardes, terwyl die vergelyking wat waarnemingsmolekule gedrag by die koppelvlak beskryf bepaalde voorwaardes nodig het. Bakterieë op die grensvlak produseer ook waarnemings-molekules wat diffundeer en afbreek. Om te verseker dat al hierdie eienskappe omvat is in ’n vergelyking is die aanname gemaak dat massa behoud bly. Ter afsluiting is numeriese simulasie van die model gedoen.

Thesis (MSc) -- University of Stellenbosch, 2000. / ENGLISH ABSTRACT: A discrete dynamical system is a pair (X, cf;) comprising a non-empty set X and a map cf; : X ---+ X. A study is made of the effect of repeated application of cf; on X, whereby points and subsets of X are classified according to their behaviour under iteration. These subsets include the JULIA and FATOU sets of the map and the sets of periodic and preperiodic points, and many interesting questions arise in the study of their properties. Such questions have been extensively studied in the case of complex dynamics, but much recent work has focussed on non-archimedean dynamical systems, when X is projective space over some field equipped with a non-archimedean metric. This work has uncovered many parallels to complex dynamics alongside more striking differences. In this thesis, various aspects of the theory of non-archimedean dynamics are presented, with particular reference to JULIA and FATOU sets and the relationship between good reduction of a map and the empty JULIA set. We also discuss questions of the finiteness of the sets of periodic points in special contexts. / AFRIKAANSE OPSOMMING: 'n Paar (X, <jJ) bestaande uit 'n nie-leë versameling X tesame met 'n afbeelding <jJ: X -+ X vorm 'n diskrete dinamiese sisteem. In die bestudering van so 'n sisteem lê die klem op die uitwerking op elemente van X van herhaalde toepassing van <jJ op die versameling. Elemente en subversamelings van X word geklasifiseer volgens dinamiese kriteria en op hierdie wyse ontstaan die JULIA en FATOU versamelings van die afbeelding en die versamelings van periodiese en preperiodiese punte. Interessante vrae oor die eienskappe van hierdie versamelings kom na vore. In die geval van komplekse dinamika is sulke vrae reeds deeglik bestudeer, maar onlangse werk is op nie-archimediese dinamiese sisteme gedoen, waar X 'n projektiewe ruimte is oor 'n liggaam wat met 'n nie-archimediese norm toegerus is. Hierdie werk het baie ooreenkomste maar ook treffende verskille met die komplekse dinamika uitgewys. In hierdie tesis word daar ondersoek oor verskeie aspekte van die teorie van nie-archimediese dinamika ingestel, in besonder met betrekking tot die JULIA en FATOU versamelings en die verband tussen goeie reduksie van 'n afbeelding en die leë JULIA versameling. Vrae oor die eindigheid van versamelings van periodiese punte in spesiale kontekste word ook aangebied.