Dynamics with choice is a generalization of discrete-time dynamics where instead of the same evolution operator at every time step there is a choice of operators to transform the current state of the system. Many real life processes studied in chemical physics, engineering, biology and medicine, from autocatalytic reaction systems to switched systems to cellular biochemical processes to malaria transmission in urban environments, exhibit the properties described by dynamics with choice. We study the long-term behavior in dynamics with choice. We prove very general results on the existence and properties of global compact attractors in dynamics with choice. In addition, we study the dynamics with restricted choice when the allowed sequences of operators correspond to subshifts of the full shift. One of practical consequences of our results is that when the parameters of a discrete-time system are not known exactly and/or are subject to change due to internal instability, or a strategy, or Nature's intervention, the long term behavior of the system may not be correctly described by a system with "averaged" values for the parameters. There may be a Gestalt effect.

A low-order discrete dynamical system (DDS) for finite-rate chemistry of H2-air combustion is derived in 3D. Fourier series with a single wavevector are employed to represent dependent variables of subgrid-scale (SGS) behaviors for applications to large-eddy simulation (LES). A Galerkin approximation is applied to the governing equations for comprising the DDS. Regime maps are employed to aid qualitative determination of useful values for bifurcation parameters of the DDS. Both isotropic and anisotropic assumptions are employed when constructing regime maps and studying bifurcation parameters sequences. For H2-air reactions, two reduced chemical mechanisms are studied via the DDS. As input to the DDS, physical quantities from experimental turbulent flow are used. Numerical solutions consisting of time series of velocities, species mass fractions, temperature, and the sum of mass fractions are analyzed. Numerical solutions are compared with experimental data at selected spatial locations within the experimental flame to check whether this model is suitable for an entire flame field. The comparisons show the DDS can mimic turbulent combustion behaviors in a qualitative sense, and the time-averaged computed results of some species are quantitatively close to experimental data.

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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.

O estudo da região de estabilidade é de extrema importância nas ciências, aplicações em engenharia e nos sistemas de controle não linear. Neste trabalho, uma caracterização completa da região de estabilidade e da fronteira da região de estabilidade de pontos fixos estáveis de uma classe ampla de sistemas dinâmicos discretos não lineares é desenvolvida. Os resultados deste trabalho estendem a caracterização da região de estabilidade já proposta na literatura para uma ampla classe de sistemas, modelados por difeomorfismos e que admitem a presença de órbitas periódicas e pontos fixos na fronteira da região de estabilidade. Caracterizações dinâmicas e topológicas são propostas para a fronteira da região de estabilidade. Além disso, são dadas condições necessárias e suficientes para que um ponto fixo ou órbita periódica pertença à fronteira da região de estabilidade. Exemplos numéricos, incluindo o modelo de uma rede neural simétrica com 2-neurônios, ilustram os resultados propostos neste trabalho. / The study of the stability region is very important in the sciences, engineering applications, and in nonlinear control systems. In this work, a complete characterization for both the stability region and the stability boundary of stable xed points of a nonlinear discrete dynamical systems is developed. The results of this work extend the characterization of the stability region already proposed in the literature for a larger class of systems, which are modeled by dieomorphisms and which admit the presence of periodic orbits and xed points on the stability boundary. Several dynamical and topological characterizations are proposed to the stability boundary. Moreover, several necessary and sucient conditions for xed points and periodic orbits to lie on the stability boundary are derived. Numerical examples, including the model of a symmetric neural network with 2-neurons, illustrate the results proposed in this work.

Les automates cellulaires forment une famille bien connue de modèles dynamiques discrets, introduits par S.Ulam et J. von Neumann dans les années 40. Ils ont été étudiés avec succès sous différents points de vue: modélisation, dynamique, ou encore complexité algorithmique. Dans ce travail, nous adoptons ce dernier point de vue pour étudier la famille des automates cellulaires coagulants, ceux dont l’état d’une cellule nepeut évoluer qu’en suivant une relation d’ordre prédéfinie sur l’ensemble de ses états. Nous étudions la complexité algorithmique de ces automates cellulaires de deux points de vue : la capacité de certains automates coagulants à simuler tous les autres automates cellulaires coagulants, appelée universalité intrinsèque, et la complexité temporelle de prédiction de l’évolution d’une cellule à partir d’une configuration finie, appelée complexité de prédiction. Nous montrons que malgré les sévères restrictions apportées par l’ordre sur les états,les automates cellulaires coagulants peuvent toujours exhiber des comportements de grande complexité.D’une part, nous démontrons qu’en dimension deux et supérieure il existe un automate cellulaire coagulants intrinsèquement universel pour les automates cellulaires coagulants en codant leurs états par des blocs de cellules ; cet automate cellulaire effectue au plus deux changements d’états par cellule. Ce résultat est minimal en dimension deux et peut être amélioré en passant à au plus un changement en dimensions supérieures.D’autre part, nous étudions la complexité algorithmique du problème de prédiction pour la famille des automates cellulaires totalistiques à deux états et voisinage de von Neumann en dimension deux. Dans cette famille de 32 automates, nous exhibons deux automates de complexité maximale dans le cas d’une mise à jour synchrone des cellules et nous montrons que dans le cas asynchrone cette complexité n’est atteinte qu’à partir de la dimension trois. Pour presque tous les autres automates de cette famille, nous montrons que leur complexité de prédiction est plus faible (sous l’hypothèse P 6≠NP). / Cellular automata are a well know family of discrete dynamic systems, defined by S. Ulam and J. von Neumannin the 40s. The have been successfully studied from the point of view of modeling, dynamics and computational complexity. In this work, we adopt this last point of view to study the family of freezing cellular automata, those where the state of a cell can only evolve following an order relation on the set of states. We study the complexity of these cellular automata from two points of view, the ability of some freezing cellular automata to simulate every other freezing cellular automata, called intrinsic universality, and the time complexity to predict the evolution of a cell starting from a given finite configuration, called prediction complexity. We show that despite the severe restriction of the ordering of states, freezing cellular automata can still exhibit highly complex behaviors.On the one hand, we show that in two or more dimensions there exists an intrinsically universal freezing cellular automaton, able to simulate any other freezing cellular automaton by encoding its states into blocks of cells, where each cell can change at most twice. This result is minimal in dimension two and can be even simplified to one change per cell in higher dimensions.On the other hand, we extensively study the computational complexity of the prediction problem for totalistic freezing cellular automata with two states and von Neumann neighborhood in dimension two. In this family of 32 cellular automata, we find two automata with the maximum complexity for classical synchronous cellular automata, while in the case of asynchronous evolution, the maximum complexity can only be achived in dimension three. For most of the other automata of this family, we show that they have a lower complexity (assuming P 6≠NP).

O estudo da região de estabilidade é de extrema importância nas ciências, aplicações em engenharia e nos sistemas de controle não linear. Neste trabalho, uma caracterização completa da região de estabilidade e da fronteira da região de estabilidade de pontos fixos estáveis de uma classe ampla de sistemas dinâmicos discretos não lineares é desenvolvida. Os resultados deste trabalho estendem a caracterização da região de estabilidade já proposta na literatura para uma ampla classe de sistemas, modelados por difeomorfismos e que admitem a presença de órbitas periódicas e pontos fixos na fronteira da região de estabilidade. Caracterizações dinâmicas e topológicas são propostas para a fronteira da região de estabilidade. Além disso, são dadas condições necessárias e suficientes para que um ponto fixo ou órbita periódica pertença à fronteira da região de estabilidade. Exemplos numéricos, incluindo o modelo de uma rede neural simétrica com 2-neurônios, ilustram os resultados propostos neste trabalho. / The study of the stability region is very important in the sciences, engineering applications, and in nonlinear control systems. In this work, a complete characterization for both the stability region and the stability boundary of stable xed points of a nonlinear discrete dynamical systems is developed. The results of this work extend the characterization of the stability region already proposed in the literature for a larger class of systems, which are modeled by dieomorphisms and which admit the presence of periodic orbits and xed points on the stability boundary. Several dynamical and topological characterizations are proposed to the stability boundary. Moreover, several necessary and sucient conditions for xed points and periodic orbits to lie on the stability boundary are derived. Numerical examples, including the model of a symmetric neural network with 2-neurons, illustrate the results proposed in this work.

Les modèles de pile de sable sont une sous-classe d'automates cellulaires. Bak et al. les ont introduit en 1987 comme une illustration de la notion intuitive d'auto-organisation critique.Le modèle de pile de sable Kadanoff est un système dynamique discret non-linéaire imagé par des grains cubiques se déplaçant de colonne parfaitement empilée en colonne parfaitement empilée. Pour un paramètre p fixé, une règle d'éboulement est appliquée jusqu'à atteindre une configuration stable, appelée point fixe : si la différence de hauteur entre deux colonnes consécutives est strictement supérieure à p, alors p grains chutent de la colonne de gauche, un retombant sur chacune des p colonnes adjacentes sur la droite.A partir d'une règle locale simple, décrire et comprendre le comportement macroscopique des piles de sable s'avère très rapidement compliqué. La difficulté consiste en la prise en compte simultanée des modalités discrète et continue du système : vue de loin, une pile de sable s'écoule comme un liquide ; mais de près, lorsque l'on s'attache à décrire exactement une configuration, les effets de la dynamique discrète doivent être pris en compte. Si par exemple nous ajoutons un unique grain à une configuration stable, celui-ci déclenche une avalanche qui ne modifie que la couche supérieure de la pile, mais dont la taille est très difficile à prédire car sensible au moindre changement sur la configuration.En analogie avec un sablier, nous nous intéressons en particulier à la séquence des points fixes atteints par l'ajout répété d'un nombre fini de grains à une même position, et à l'émergence de structures étonnamment régulières.Après avoir établi une conjecture sur l'émergence de motifs de vague sur les points fixes, nous nous pencherons dans un premier temps sur une procédure inductive de calcul des points fixes. Chaque étape de l'induction correspond au calcul d'une avalanche provoquée par l'ajout d'un nouveau grain, et nous en proposerons une description simple. Cette étude sera prolongée par la définition de trace des avalanches sur une colonne i, qui capture dans un mot d'un alphabet fini l'information nécessaire à la reconstitution du point fixe pour les colonnes à la droite de l'indice i. Des liens entre les traces à des indices successifs seront alors exploités, liens qui permettent de conclure l'émergence de traces régulières, pour lesquelles la reconstitution du point fixe implique la formation des motifs de vague observés. Cette première approche est concluante pour le plus petit paramètre conjecturé jusqu'ici, p=2.L'étude du cas général que nous proposons passe par la construction d'un nouveau système mêlant différentes représentations des points fixes, qui sera analysé par l'association d'arguments d'algèbre linéaire et combinatoires (liés respectivement aux modalités continue et discrète des piles de sable). Ce résultat d'émergence de régularités dans un système dynamique discret fait appel à des techniques nouvelles, dont la compréhension d'un élément de preuve reste en particulier à raffiner, ce qui permet d'envisager un cadre plus général d'appréhension de la notion d'émergence. / Sandpile models are a subclass of Cellular Automata. Bak et al. introduced them in 1987 for they exemplify the intuitive notion of Self-Organized Criticality.The Kadanoff sandpile model is a non-linear discrete dynamical system illustrating the evolution of cubic sand grains from nicely packed columns to nicely packed columns. For a fixed parameter p, a rule is applied until reaching a stable configuration, called a fixed point : if the height difference between two consecutive columns is strictly greater than p, then p grains fall from the left column, one landing on each of the p adjacent columns on the right.From a simple local rule, to describe and understand the macroscopic behavior of sandpiles is very quickly challenging. The difficulty consists in the simultaneous study of continuous and discrete aspects of the system: on a large scale, a sandpile flows like a liquid; but on a small scale, when we want to describe exactly the shape of a fixed point, the effects of the discrete dynamic must be taken into account. If for example we add a single grain on a stabilized sandpile, it triggers an avalanche that roughly changes only the upper layer of the configuration, but which size is hard to predict because it is sensitive to the tiniest change of the configuration.In analogy with an hourglass, we are particularly interested in the sequence of fixed points reached after adding a finite number of grains on one position, with the aim of explaining the emergence of surprisingly regular patterns.After conjecturing the emergence of wave patterns on fixed points, we firstly consider an inductive procedure for computing fixed points. Each step of the induction corresponds to the computation of an avalanche triggered by the addition of a new grain, for which we propose a simple description. This study is carried on with the definition of the trace of avalanches on a column i, which catches in a word among a finite alphabet enough information in order to reconstruct the fixed point on the right of index i. Links between traces on successive columns are then investigated, links allowing to conclude the emergence of regular traces, whose fixed point's reconstruction involves the appearance and maintain of the wave patterns observed. This first approach is conclusive for the smallest conjectured parameter so far, p=2.The study of the general case goes through the design of a new system meddling in different representations of fixed points, which will be analyzed via an association of arguments of linear algebra and combinatorics (respectively corresponding to the continuous and discrete modalities of sandpiles). This result stating the emergence of regularities in a discrete dynamical system put new technics into light, for which the comprehension of a particular point in the proof remains to be increased. This motivates the consideration of a more general frame of work tackling the notion of emergence.

This paper is a bachelor thesis in Applied Mathematics at Linnaeus University. The goal of this thesis is to find a structure in sets of discrete quadratic dynamical systems modulo a number, with a main focus on 1-, and 2-periodic points in sets of systems modulo an odd prime. The amount of 1-, and 2-periodic points in such sets is numerically investigated and is proven directly to be p, respectively p-1. Furthermore, the sets are visualized using a diagram, where some apparent structures are noticed, and later explained. Finally, through numerical investigations, an expression for the amount of 2-periodic points in a system modulo a composite number is also perceived, and expressed as a conjecture. Proving the conjecture should be done in a future project.