We associate learning in living systems with the shaping of the velocity vector field of a dynamical system in response to external, generally random, stimuli. We consider various approaches to implement a system that is able to adapt the whole vector field, rather than just parts of it - a drawback of the most common current learning systems: artificial neural networks. This leads us to propose the mathematical concept of self-shaping dynamical systems. To begin, there is an empty phase space with no attractors, and thus a zero velocity vector field. Upon receiving the random stimulus, the vector field deforms and eventually becomes smooth and deterministic, despite the random nature of the applied force, while the phase space develops various geometrical objects. We consider the simplest of these - gradient self-shaping systems, whose vector field is the gradient of some energy function, which under certain conditions develops into the multi-dimensional probability density distribution of the input. We explain how self-shaping systems are relevant to artificial neural networks. Firstly, we show that they can potentially perform pattern recognition tasks typically implemented by Hopfield neural networks, but without any supervision and on-line, and without developing spurious minima in the phase space. Secondly, they can reconstruct the probability density distribution of input signals, like probabilistic neural networks, but without the need for new training patterns to have to enter the network as new hardware units. We therefore regard self-shaping systems as a generalisation of the neural network concept, achieved by abandoning the "rigid units - flexible couplings'' paradigm and making the vector field fully flexible and amenable to external force. It is not clear how such systems could be implemented in hardware, and so this new concept presents an engineering challenge. It could also become an alternative paradigm for the modelling of both living and learning systems. Mathematically it is interesting to find how a self shaping system could develop non-trivial objects in the phase space such as periodic orbits or chaotic attractors. We investigate how a delayed vector field could form such objects. We show that this method produces chaos in a class systems which have very simple dynamics in the non-delayed case. We also demonstrate the coexistence of bounded and unbounded solutions dependent on the initial conditions and the value of the delay. Finally, we speculate about how such a method could be used in global optimization.

Non-linearities and dynamic interactions between state variables are characteristic of complex social systems and processes. In this thesis, we present a new methodology to model these non-linearities and interactions from the large panel datasets available for some of these systems. We build macro-level statistical models that can verify theoretical predictions, and use polynomial basis functions so that each term in the model represents a specific mechanism. This bridges the existing gap between macro-level theories supported by statistical models and micro-level mechanistic models supported by behavioural evidence. We apply this methodology to two important problems in the social sciences, the demographic transition and the transition to democracy. The demographic transition is an important problem for economists and development scientists. Research has shown that economic growth reduces mortality and fertility rates, which reduction in turn results in faster economic growth. We build a non-linear dynamic model and show how this data-driven model extends existing mechanistic models. We also show policy applications for our models, especially in setting development targets for the Millennium Development Goals or the Sustainable Development Goals. The transition to democracy is an important problem for political scientists and sociologists. Research has shown that economic growth and overall human development transforms socio-cultural values and drives political institutions towards democracy. We model the interactions between the state variables and find that changes in institutional freedoms precedes changes in socio-cultural values. We show applications of our models in studying development traps. This thesis comprises the comprehensive summary and seven papers. Papers I and II describe two similar but complementary methodologies to build non-linear dynamic models from panel datasets. Papers III and IV deal with the demographic transition and policy applications. Papers V and VI describe the transition to democracy and applications. Paper VII describes an application to sustainable development.

It has been suggested that Patellofemoral Pain (PFP) may be the result of a coordinate state which exhibits less joint coordination variability. The ability to relate joint coordination variability to PFP pathology could have many clinical uses; however, evidence to support clinical application is lacking. Vector coding’s coupling angle variability (CAV) has been introduced as a possible analysis method to quantify joint coordination variability. The purpose of this study was to assess the clinical usefulness of CAV measures from a dynamical systems perspective. This involved establishing the precision limits of CAV measures when physiological conditions are held constant, altering control parameters of knee pain and population then determining if the observed changes in CAV were clinically meaningful. 20 female recreational runners with PFP and 21 healthy controls performed a treadmill acclimation protocol then ran at a self-selected pace for 15 minutes. 3-D kinematics, force plate kinetics, knee pain and perceived exertion were recorded each minute. CAV were calculated for six knee-ankle combinations for 2 sets of 5 non-consecutive stride cycles at each capture period. Data were selected for the PFP group at a high (=>3) and low (<=high-2) pain level in a non-exhausted state (<14). Healthy data were used from the 11th minute of the running. Levels of agreement were performed between the 2 sets of CAV measures for both populations, a paired t-test compared low to high pain CAV measures and independent t-tests compared populations at the high pain state. Several CAV measures showed a significant increase in value with an increase in pain and were significantly greater for the PFP group. None of the observed changes exceeded the precision limits of all CAV measures investigated. These results do not agree with previous claims that less variability is indicative of pathology but rather the opposite. This suggests that there might be an optimal amount of variability to maintain a healthy coordinate state with deviations in any direction being detrimental. However; due to the volatile nature of CAV measures, the clinical use of CAV is not recommended using current analysis methods since changes observed weren’t considered clinically meaningful.

We examine a class of planar area preserving mappings and give a geometric condition that guarantees the existence of homoclinic points. To be more precise, let $f,g:R \to R$ be $C^1$ functions with domain all of $R$. Let $F:R^2 \to R^2$ denote a horizontal reflection in the curve $x=-f(y)$, and let $G:R^2 \to R^2$ denote a vertical reflection in the curve $y=g(x)$. We consider maps of the form $T=G \circ F$ and show that a simple geometric condition on the fixed point sets of $F$ and $G$ leads to the existence of a homoclinic point for $T$. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-09-17 14:22:35.72

We study evolutionary models of financial markets. In particular, we study an evolutionary market model with short-lived assets and an evolutionary model with long-lived assets. In the long-lived asset market, investors are allowed to use general dynamic investment strategies. We find sufficient conditions for the Kelly portfolio rule to dominate the market exponentially fast. Moreover, when investors use simple strategies but have incorrect beliefs, we show that the strategy which is "closer" to the Kelly rule cannot be driven out of the market. This means that this strategy will either dominate or at least survive, i.e., the relative market share does not converge to zero. In the market with short-lived assets, we study the dynamics when the states of the world are not identically distributed. This marks the first attempt to study the dynamics of the market when the probability of success changes according to the relative shares of investors. In this problem, we first study a skew product of the random dynamical system associates with the market dynamics. In particular, we compute the Lyapunov exponents of the skew product. This enables us to produce a "surviving" investment strategy, i.e., the investor who follows this rule will dominate the market or at least survive. All the mathematical tools in the thesis lie within the framework of random dynamical systems.

Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in answering this question. The rational case remains open, but it was already shown to be much more complex than the polynomial one. In this thesis, we review existing results on rational maps sharing a Julia set, and offer results of our own on the symmetry group of such maps. / Desde a década de oitenta, um enorme progresso foi feito no problema de determinar quais funções têm o mesmo conjunto de Julia. O caso polinomial foi completamente respondido em 1995, e mostrou-se que as simetrias do conjunto de Julia têm um papel central nessa questão. O caso racional permanece aberto, mas já se sabe que ele é muito mais complexo do que o polinomial. Nesta dissertação, nós revisamos resultados existentes sobre aplicações racionais com o mesmo conjunto de Julia e apresentamos nossos próprios resultados sobre o grupo de simetrias de tais aplicações.

Investigou-se a gênese de atividade irregular em neurônios de centros geradores de padrões através de modelos eletrofisiologicamente realistas. Para tanto, foram adotadas abordagens paralelas. Primeiramente, desenvolveram-se técnicas para determinar quais os mecanismos biofísicos subjacentes aos processos de codificação de informação nestas células. Também foi proposta uma nova metodologia híbrida (baseada em continuação numérica e em varreduras força bruta) para análise de bancos de dados de modelos neuronais, permitindo estendê-los e revelar instâncias de multiestabilidade entre regimes oscilatórios e quiescentes. Além disto, a fim de determinar a origem de comportamento complexo em modelos neuronais simplificados, empregaram-se métodos geométricos da teoria de sistemas dinâmicos. A partir da análise de mapas unidimensionais perturbados por ruído, foram discutidos possíveis cenários para o surgimento de caos em sistemas dinâmicos aleatórios. Finalmente mostrou-se que, levando em conta o ruído, uma classe de modelos de condutâncias reproduz padrões de disparo observados in vivo. Estas pertubações revelam a riqueza da dinâmica transiente, levando o sistema a visitar um arcabouço determinista complexo preexistente -- sem recorrer a ajustes finos de parâmetros ou a construções ad hoc para induzir comportamento caótico. / We investigated the origin of irregularities in the dynamics of central pattern generator neurons, through analyzing electrophysiologically realistic models. A number of parallel approaches were adopted for that purpose. Initially, we studied information coding processes in these cells and proposed a technique to determine the underlying biophysical mechanisms. We also developed a novel hybrid method (based on numerical continuation and brute force sweeps) to analyze neuronal model databases, extending them and unveiling instances of multistability between oscillatory and resting regimes. Furthermore, in order to determine the origin of irregular dynamics in simplified neuronal models, we employed geometrical methods from the theory of dynamical systems. The analysis of stochastically perturbed maps allowed us to discuss possible scenarios for the generation of chaotic behaviour in random dynamical systems. Finally we showed that, by taking noise into account, a class of conductance based models gives rise to firing patterns akin to the ones observed \\emph{in vivo}. These perturbations unveil the richness of the transient dynamics, inducing the system to populate a preexistent complex deterministic scaffolding -- without resorting to parameter fine-tuning or ad hoc constructions to induce chaotic activity.

Ban and Kalies [3] proposed an algorithmic approach to compute attractor- repeller pairs and weak Lyapunov functions based on a combinatorial multivalued mapping derived from an underlying dynamical system generated by a continuous map. We propose a more e cient way of computing a Lyapunov function for a Morse decomposition. This combined work with other authors, including Shaun Harker, Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes the process of nding a global Lyapunov function for Morse decomposition very e - cient. One of the them is to utilize highly memory-e cient data structures: succinct grid data structure and pointer grid data structures. Another technique is to utilize Dijkstra algorithm and Manhattan distance to calculate a distance potential, which is an essential step to compute a Lyapunov function. Finally, another major technique in achieving a signi cant improvement in e ciency is the utilization of the lattice structures of the attractors and attracting neighborhoods, as explained in [32]. The lattice structures have made it possible to let us incorporate only the join-irreducible attractor-repeller pairs in computing a Lyapunov function, rather than having to use all possible attractor-repeller pairs as was originally done in [3]. The distributive lattice structures of attractors and repellers in a dynamical system allow for general algebraic treatment of global gradient-like dynamics. The separation of these algebraic structures from underlying topological structure is the basis for the development of algorithms to manipulate those structures, [32, 31]. There has been much recent work on developing and implementing general compu- tational algorithms for global dynamics which are capable of computing attracting neighborhoods e ciently. We describe the lifting of sublattices of attractors, which are computationally less accessible, to lattices of forward invariant sets and attract- ing neighborhoods, which are computationally accessible. We provide necessary and su cient conditions for such a lift to exist, in a general setting. We also provide the algorithms to check whether such conditions are met or not and to construct the lift when they met. We illustrate the algorithms with some examples. For this, we have checked and veri ed these algorithms by implementing on some non-invertible dynamical systems including a nonlinear Leslie model. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

Matrix models are formulated to study the dynamics of the structured populations. We consider closed populations, that is, without migration, and populations with migration. The effects of specific patterns of migration, whether with constant or time-dependent terms, are explored within the context of how they manifest in model output, such as population size. Time functions, commonly known as relative sensitivities, are employed to rank the parameters of the models from most to least influential in the population size or abundance of individuals per group

O entendimento de quebra espontânea de simetria é um problema importante para o estudo de fenômenos na evolução de sistemas abertos, tanto em física quanto em química, como também na biologia. Aqui estudamos um método a mais para este tipo de análise, usando conceitos de sistemas dinâmicos com simetria. O sistema dinâmico escolhido é discreto, isto é, realizado por iteração de um difeomorfismo equivariante sob a ação de um grupo compacto, neste caso um grupo finito do tipo Dm. Especificamente, investigamos o comportamento de atratores caóticos sob a variação dos parâmetros. / The understanding of spontaneous symmetry breaking is an important problem in the study of phenomena in the evolution of open systems, in physics and chemistry as well as in biology. Here we study another method for this kind of analysis, using concepts from dynamical systems with symmetry. The chosen dynamical system is discrete, that is, realized by iteration of an equivariant diffeomorphism under the action of a compact group, in this case one of the finite groups of type Dm. Specifically, we investigate the behavior of chaotic attractors under variation of the parameters.