The SIR can be expressed either as a system of nonlinear ordinary differential equations or as a nonlinear Volterra integral equation. In general, neither of these can be solved in closed form. In this thesis, it is shown that if we assume S(t) is a finite multi-exponential, i.e. function of the form S(t) = a+ ∑nk=1 rke-σkt or a logistic function which is an infinite-multi-exponential, i.e. function of the form S(t) = c + a/b+ewt, then we can have closed form solution. Also we will formulate a method to determine R0 the basic reproductive rate of an infection.

The Carleman Embedding is a method that allows us to embed a finite dimensional system of nonlinear differential equations into a system of infinite dimensional linear differential equations. This technique works well when dealing with first-order nonlinear differential equations. However, for higher order nonlinear ordinary differential equations, it is difficult to use the Carleman Embedding method. This project will examine the Carleman Embedding and a variation of the method which is very convenient in applying to second order systems of nonlinear equations.

No description available.

In the past, analyses of lightcurve data have been applied to asteroids in order to determine their axis of rotation, rotation rate and other parameters. In recent decades, these analyses have begun to be applied in the domain of Earth orbiting spacecraft. Due to the complex geometry of spacecraft and the wide variety of parameters that can influence the way in which they reflect light, these analyses require more complex assumptions and a greater knowledge about the object being studied. Previous investigations have shown success in extracting attitude parameters from unresolved spacecraft using simulated data. This paper presents a focused attempt to derive attitude parameters using an Unscented Kalman Filter from both simulated and real data provided by Lockheed Martin Space. This thesis characterizes and presents the differences in performance between three simulated geometries in low, medium, and geostationary orbit in both cases where they are spinning about a constant axis and in cases in which they are tumbling. Additionally, this thesis hypothesizes and tests the idea that a predictable and extraneous angular velocity solution exists which is the reflection of the true solution about the plane defined by the sun and observation vectors. This thesis encountered multiple instances of this type solution appearing in simulation and provides an example as well as a visualization. Finally, this thesis demonstrates the ability to converge to a solution from real data although there were large discrepancies between the measurement model and the data. This thesis discusses the validity of these solutions and sources of error.

No description available.

The Ph.D. Thesis “Strong mixing measures and invariant sets in linear dynamics” has three differenced parts. Chapter 0 introduces the notation, definitions and the basic results that will be needed troughout the thesis. There is a first part consisting of Chapters 1 and 2, where we study the relation between the Frequent Hypercyclicity Criterion and the existence of strongly-mixing Borel probability measures. A third chapter, where we focus our attention on frequent hypercyclicity for translation C0-semigroups, and the last part corresponding to Chapters 4 and 5, where we study dynamical properties satisfied by autonomous and non-autonomous linear dynamical systems on certain invariant sets. In what follows, we give a brief description of each chapter: In Chapter 1, we construct strongly mixing Borel probability T-invariant measures with full support for operators on F-spaces which satisfy the Frequent Hypercyclicity Criterion. Moreover, we provide examples of operators that verify this criterion and we also show that this result can be improved in the case of chaotic unilateral backward shifts. The contents of this chapter have been published in [88] and [12]. In Chapter 2, we show that the Frequent Hypercyclicity Criterion for C0- semigroups, which was given by Mangino and Peris in [82], ensures the existence of invariant strongly mixing measures with full support. We will provide several examples, that range from birth-and-death models to the Black-Scholes equation, which illustrate these results. All the results of this chapter have been published in [86]. In Chapter 3, we focus our attention on one of the most important tests C0-semigroups, the translation semigroup. Inspired in the work of Bayart and Ruzsa in [22], where they characterize frequent hypercyclicity of weighted backward shifts we characterize frequently hypercyclic translation C0-semigroups on C ρ 0 (R) and L ρ p(R). Moreover, we first review some known results on the dynamics of the translation C0-semigroups. Later we state and prove a characterization of frequent hypercyclicity for weighted pseudo shifts in terms of the weights that will be used later to obtain a characterization of frequent hypercyclicity for translation C0-semigroups on C ρ 0 (R). Finally we study the case of L ρ p(R). We will also establish an analogy between the study of frequent hypercyclicity for the translation C0-semigroup in L ρ p(R) and the corresponding one for backward shifts on weighted sequence spaces. The contents of this chapter have been included in [81]. Chapter 4 is devoted to study hypercyclicity, Devaney chaos, topological mixing properties and strong mixing in the measure-theoretic sense for operators on topological vector spaces with invariant sets. More precisely, we establish links between the fact of satisfying any of our dynamical properties on certain invariant sets, and the corresponding property on the closed linear span of the invariant set, or on the union of the invariant sets. Viceversa, we give conditions on the operator (or C0-semigroup) to ensure that, when restricted to the invariant set, it satisfies certain dynamical property. Particular attention is given to the case of positive operators and semigroups on lattices, and the (invariant) positive cone. The contents of this chapter have been published in [85]. In the last chapter, motivated by the work of Balibrea and Oprocha [4], where they obtained several results about weak mixing and chaos for nonautonomous discrete systems on compact sets, we study mixing properties for nonautonomous linear dynamical systems that are induced by the corresponding dynamics on certain invariant sets. All the results of this chapter have been published in [87]. / Murillo Arcila, M. (2015). Strong mixing measures and invariant sets in linear dynamics [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48519

No / In structural mechanics there are several occasions where a linearized formulation of the original nonlinear problem reduces considerably the computational effort for the response analysis. In a broader sense, a linearized formulation can be viewed as a first-order expansion of the dynamic equilibrium of the system about a `static¿ configuration; yet caution should be exercised when identifying the `correct¿ static configuration. This paper uses as a case study the rocking response of a rigid block stepping on viscoelastic supports, whose non-linear dynamics is the subject of the companion paper, and elaborates on the challenge of identifying the most appropriate static configuration around which a first-order expansion will produce the most dependable results in each regime of motion. For the regime when the heel of the block separates, a revised set of linearized equations is presented, which is an improvement to the unconservative equations published previously in the literature. The associated eigenvalues demonstrate that the characteristics of the foundation do not affect the rocking motion of the block once the heel separates.

Tesis por compendio / [ES] La tesis "Teoría Combinatoria de Números, Recurrencia de Operadores y Dinámica Lineal" se sitúa dentro del estudio de la dinámica de operadores lineales, o Dinámica Lineal. El objetivo de este trabajo es estudiar múltiples nociones de recurrencia, que pueden presentar los sistemas dinámicos lineales, y que clasificaremos mediante la Teoría Combinatoria de Números. La Dinámica Lineal estudia las órbitas generadas por las iteraciones de una transformación lineal. Las propiedades más estudiadas en esta rama durante los últimos 30 años han sido la hiperciclicidad (existencia de órbitas densas) y el caos (con sus múltiples definiciones), siendo esta un área de investigación muy activa y obteniéndose un considerable número de resultados profundos e interesantes. Nosotros nos centraremos en la recurrencia, propiedad muy estudiada para sistemas dinámicos clásicos no lineales, pero prácticamente nueva en Dinámica Lineal pues no es hasta 2014, con el artículo de Costakis, Manoussos y Parissis titulado "Recurrent linear operators", cuando se empieza a estudiar esta noción de manera sistemática en el contexto de operadores actuando en espacios de Banach. La situación básica de la que parte nuestro estudio es la siguiente: "T : X ---> X" será un operador lineal y continuo actuando sobre un F-espacio "X" , aunque a veces necesitaremos que el espacio subyacente "X" sea un espacio de Fréchet, de Banach o de Hilbert. Dado un vector "x" y un entorno "U" de "x" estudiaremos el conjunto de retorno "N_T(x,U) = { n : T^n(x) está en U }" y dependiendo de su tamaño, observado mediante la Teoría Combinatoria de Números, diremos que el vector "x" presenta una propiedad de recurrencia u otra. La memoria de la tesis se ha realizado por compendio de artículos y consta de cuatro capítulos y un apéndice: 1. Adaptación de la "versión de autor" del artículo "Frequently recurrent operators. Journal of Functional Analysis, 283 (12) (2022), artículo núm. 109713, 36 páginas". En este se definen por primera vez las fuertes nociones de recurrencia reiterada, U-frecuente y frecuente, y sus propiedades básicas son estudiadas. Finalmente se generaliza el estudio mediante el concepto de F-recurrencia, que se conecta con la noción de F-hiperciclicidad. 2. Adaptación al formato de la tesis de la "versión de autor" revisada del artículo "Recurrence properties: An approach via invariant measures. Journal de Mathématiques Pures et Appliquées, 169 (2023), 155-188". En este se relaciona la recurrencia de operadores con la Teoría Ergódica y los sistemas dinámicos que conservan la medida. 3. Adaptación de la "versión de autor" del preprint "Questions in linear recurrence: From the T+T-problem to lineability". Se resuelve negativamente un problema abierto de 2014: Sea "T : X ---> X" un operador recurrente. ¿Es cierto que el operador "T+T" es recurrente en "X+X"? Para resolverlo introducimos la casi-rigidez, que será, para la recurrencia, la noción análoga a la propiedad débil-mezclante (topológica) para la transitividad/hiperciclicidad; y luego construimos operadores recurrentes pero no casi-rígidos en todo espacio de Banach infinito-dimensional y separable. 4. Adaptación de la "versión de autor" revisada del preprint " Recurrent subspaces in Banach spaces". En este se estudia la propiedad de espaciabilidad (existencia de un subespacio vectorial cerrado y de dimensión infinita) para el conjunto de vectores recurrentes. - Apéndice. Para conseguir un carácter auto-contenido hemos añadido un apéndice con los resultados básicos de Teoría Combinatoria de Números que se han utilizado en los trabajos que componen la memoria. Siguiendo la normativa establecida por la Escuela de Doctorado también se incluye: - Introducción; - Discusión general de los resultados; - Conclusiones. / [CAT] La tesi "Teoria Combinatòria de Nombres, Recurrència d'Operadors i Dinàmica Lineal" se situa dins de l'estudi de la dinàmica d'operadors lineals, o simplement Dinàmica Lineal. L'objectiu d'aquest treball és estudiar múltiples nocions de recurrència, que poden presentar els sistemes dinàmics lineals, i que classificarem mitjançant la Teoria Combinatòria de Nombres. La Dinàmica Lineal estudia les òrbites generades per les iteracions d'una transformació lineal. Les propietats més estudiades en aquesta branca de les matemàtiques als darrers 30 anys han estat la hiperciclicitat (existència d'òrbites denses) i el caos (amb les seves múltiples definicions), sent aquesta una àrea de recerca molt activa i obtenint-se un considerable nombre de resultats profunds i interessants. Nosaltres ens centrarem en la recurrència, propietat molt estudiada per a sistemes dinàmics clàssics no lineals, però, pràcticament nova en Dinàmica Lineal doncs no és fins al 2014, amb l'article de Costakis, Manoussos i Parissis titulat "Recurrent linear operators", quan es comença a estudiar aquesta noció de manera sistemàtica en el context d'operadors actuant en espais de Banach. La situació bàsica de la qual parteix el nostre estudi és la següent: "T : X ---> X" serà un operador lineal i continu actuant sobre un F-espai "X", encara que de vegades necessitarem que l'espai subjacent X siga un espai de Fréchet, de Banach o de Hilbert. Llavors, donat un vector "x" i un entorn "U" de "x" estudiarem el conjunt de retorn "N_T(x,U) = { n : T^n(x) està en U }" i depenent de la seva mida, observada des del punt de vista de la Teoria Combinatòria de Nombres, direm que el vector "x" presenta una o altra propietat de recurrència. La memòria de la tesi s'ha realitzat per compendi d'articles i consta de quatre capítols i un apèndix: 1. Adaptació de la "versió d'autor" revisada de l'article "Frequently recurrent operators. Journal of Functional Analysis, 283 (12) (2022), article núm. 109713, 36 pàgines". En aquest es defineixen per primera vegada les nocions de recurrència reiterada, U-freqüent i freqüent, i les seves propietats bàsiques són estudiades. Finalment es generalitza l'estudi mitjançant el concepte de F-recurrència, que es connecta amb la noció de F-hiperciclicitat. 2. Adaptació al format de la tesi de la "versió d'autor" revisada de l'article "Recurrence properties: An approach via invariant measures. Journal de Mathématiques Pures et Appliquées, 169 (2023), 155-188". Es relaciona la recurrència d'operadors amb la Teoria Ergòdica i els sistemes dinàmics que conserven la mesura. 3. Adaptació de la "versió d'autor" del preprint "Questions in linear recurrence: From the T+T-problem to lineability". En aquest es resol un problema obert de l'any 2014: Siga "T : X ---> X" un operador recurrent. És cert que l'operador "T+T" és recurrent en "X+X"? Per resoldre'l introduïm la quasi-rigidesa, que serà, per a la recurrència, la noció anàloga a la propietat feble-barrejant (topològica) per a la transitivitat/hiperciclicitat; i després construïm operadors recurrents però no quasi-rígids en tot espai de Banach infinit-dimensional i separable. 4. Adaptació de la "versió d'autor" del preprint "Recurrent subspaces in Banach spaces". S'inclou l'estudi de la propietat d'espaiabilitat (existència d'un subespai vectorial tancat i de dimensió infinita) per al conjunt de vectors recurrents. - Apèndix:Per aconseguir un caràcter auto-contingut hem afegit un apèndix amb resultats bàsics de Teoria Combinatòria de Nombres que es donen per suposats en els treballs que componen la memòria. Seguint la normativa establerta per l'Escola de Doctorat també s'inclou: - Introducció; - Discussió general dels resultats; - Conclusions. / [EN] The thesis "Combinatorial Number Theory, Recurrence of Operators and Linear Dynamics" is part of the study of the dynamics of linear operators, simply called Linear Dynamics. The objective of this work is to study multiple notions of recurrence, that linear dynamical systems can present, and which will be classified through Combinatorial Number Theory. Linear Dynamics studies the orbits generated by the iterations of a linear transformation. The two most studied properties in this branch of mathematics during the last 30 years have been hypercyclicity (existence of dense orbits) and chaos (with its multiple definitions), being this a very active research area with a considerable number of exceptionally deep but also interesting results. We will focus on recurrence, a property widely studied in the classical setting of non-linear dynamical systems, but practically new with respect to Linear Dynamics since it was not until 2014, with the article by Costakis, Manoussos and Parissis entitled "Recurrent linear operators", when this notion started to be systematically studied in the context of operators acting on Banach spaces. The basic situation from which our study starts is the following: "T : X ---> X" will be a continuous linear operator acting on an F-space "X", although sometimes we will need the underlying space X to be a Fréchet, Banach or Hilbert space. Given a vector "x" and a neighbourhood "U" of "x" we will study the return set "N_T(x,U) = { n : T^n(x) is in U }" and depending on its size, observed from the Combinatorial Number Theory point of view, we will say that the vector "x" presents one property of recurrence or another. The thesis memoir is a compendium of articles and it has four chapters and one appendix: 1. Adaptation of the revised "author version" of article "Frequently recurrent operators. Journal of Functional Analysis, 283 (12) (2022), paper no. 109713, 36 pages". Here, the strong notions of reiterative, U-frequent and frequent recurrence are defined for the first time, and their basic properties are studied. The theory is finally generalized through the concept of F-recurrence, which is connected to the notion of F-hypercyclicity. 2. Adaptation of the revised "author version" of article "Recurrence properties: An approach via invariant measures. Journal de Mathématiques Pures et Appliquées, 169 (2023), 155-188". In this chapter the recurrence properties for linear operators are related to Ergodic Theory and measure preserving systems. 3. Adaptation of the revised "author version" of the preprint "Questions in linear recurrence: From the T+T-problem to lineability". We solve in the negative an open problem posed in 2014: Let "T : X ---> X" be a recurrent operator. Is it true that the operator "T+T" is recurrent on "X+X"? In order to do that we establish the analogous notion, for recurrence, to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and then we construct recurrent but not quasi-rigid operators on every separable infinite-dimensional Banach space. 4. Adaptation of the revised "author version" of the preprint "Recurrent subspaces in Banach spaces". In this chapter we study the spaceability (existence of an infinite-dimensional closed subspace) for the set of recurrent vectors. - Appendix. Looking for a self-contained text we have added an appendix with some of the basic Combinatorial Number Theory results that are taken for granted along the different chapters/articles forming this memoir. Following the regulations established by the Doctoral School the next sections are also included: - Introduction; - General discussion of the results; - Conclusions. / This thesis has been written at the “Institut Universitari de Matemàtica Pura i Aplicada” (IUMPA) of the “Universitat Politècnica de València” (UPV), during the period of enjoyment of a scholarship of the “Programa de Formación de Profesorado Universitario” granted by the “Ministerio de Ciencia, Innovación y Universidades”, reference number: FPU2019/04094. The research exposed has also been partially funded by the project “Dinámica de operadores” (MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00), thanks to which the author carried out a 3-month research stay in Lille, France (September-December 2021), that was supervised by Professor Sophie Grivaux; and also by the travel grant awarded by the “Fundació Ferran Sunyer i Balaguer” which allowed the author to carry out a 3-month research stay in Mons, Belgium (April-June 2023), supervised by Professor Karl Grosse-Erdmann. / López Martínez, A. (2023). Combinatorial Number Theory, Recurrence of Operators and Linear Dynamics [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/196101 / Compendio

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto, Strogatz, and others has led to a deep understanding of the emergent behavior of systems of such oscillators using traditional dynamical systems methods. In this project we outline the application of techniques from topological data analysis to understanding the dynamics of systems of coupled oscillators. This includes the examination of partitions, partial synchronization, and attractors. By looking for clustering in a data space consisting of the phase change of oscillators over a set of time delays we hope to reconstruct attractors and identify members of these clusters.

This is a study of the effect of harvesting on a system with two competing species. The system is a Ricker-type model that extends the work done by Luis, Elaydi, and Oliveira to include the effect of harvesting on the system. We look at the uniform bound of the system as well as the isoclines and perform a stability analysis of the equilibrium points. We also look at the effects of harvesting on the stability of the system by looking at the bifurcation of the system with respect to harvesting.