Cílem této práce je se seznámení s možnostmi programu Matlab z hlediska detailní analýzy deterministických dynamických systémů. Jedná se především o analýzu časové posloupnosti a o nalezení Lyapunových exponentů. Dalším cílem je navrhnout algoritmus umožňující specifikovat chování systému na základě znalosti příslušných diferenciálních rovnic. To znamená, nalezení chaotických systémů.

This thesis deals with a complex analysis and prediction of foreign exchange markets. It uses advanced artificial intelligence methods, namely neural networks and chaos theory. It introduces unconventional approaches and methods of each of these areas, compares them and uses on a real problem. The core of this thesis is a comparison of several prediction models based on completely different principles and underlying theories. The outcome is then a selection of the most appropriate prediction model called NAR + H. The model is evaluated according to several criteria, the pros and cons are discussed and approximate expected profitability and risk are calculated. All analytical, prediction and partial algorithms are implemented in Matlab development environment and form a unified library of all used functions and scripts. It also may be considered as a secondary main outcome of the thesis.

Die sequenzielle Aktivität von Ortszellen im Hippocampus entspricht vielfach früheren Erlebnissen, was auf eine Rolle in Gedächtnisprozessen hinweist. Jüngere experimentelle Befunde zeigen allerdings, dass Zielorte in sequenzieller Aktivität überrepräsentiert sind. Dies legt eine Rolle dieser Aktivitätsmuster in der Verhaltensplanung nahe, wobei ein detailliertes Verständnis sowohl des Ursprungs als auch der Funktion von Ortszellsequenzen im Hippocampus bislang fehlt. Insbesondere ist nicht bekannt, welcher Mechanismus solche Sequenzen auf adaptive und konstruktive Weise generiert, wodurch effizientes Planen ermöglicht würde. Um der Beantwortung dieser Fragen näher zu kommen, stelle ich ein neu entwickeltes pulscodiertes Netzwerkmodell vor, in dem räumliches Lernen und die Generierung von Sequenzen untrennbar voneinander abhängig sind. Anhand von Simulationen zeige ich, dass dieses Modell die Erzeugung von noch nicht erlebten Sequenztrajektorien in bekannten Umgebungen erklärt, was deren Nutzen für flexible Pfadplanung hervorhebt. Zusätzlich stelle ich die Ergebnisse eines detaillierten Vergleichs zwischen simulierten neuronalen Pulsfolgen und experimentellen Daten auf der Ebene der Populationsdynamik vor. Diese Resultate zeigen, wie sequenzielle räumliche Repräsentationen durch die Interaktion zwischen lokaler oszillatorischer Dynamik und externen Einflüssen geprägt werden.:1. Introduction 2. Neurobiological and theoretical accounts of hippocampal function 3. A computational model of place-cell sequences for goal-finding 4. A statistical note on step size decoding in place-cell sequences 5. Summary and Discussion Bibliography / Hippocampal place-cell sequences observed during awake immobility often represent previous experience, suggesting a role in memory processes. However, recent reports of goals being overrepresented in sequential activity suggest a role in short-term planning, although a detailed understanding of the origins of hippocampal sequential activity and of its functional role is still lacking. In particular, it is unknown which mechanism could support efficient planning by generating place-cell sequences biased toward known goal locations, in an adaptive and constructive fashion. To address these questions, I propose a spiking network model of spatial learning and sequence generation as interdependent processes. Simulations show that this model explains the generation of never-experienced sequence trajectories in familiar environments and highlights their utility in flexible route planning. In addition, I report the results of a detailed comparison between simulated spike trains and experimental data, at the level of network dynamics. These results demonstrate how sequential spatial representations are shaped by the interaction between local oscillatory dynamics and external inputs.:1. Introduction 2. Neurobiological and theoretical accounts of hippocampal function 3. A computational model of place-cell sequences for goal-finding 4. A statistical note on step size decoding in place-cell sequences 5. Summary and Discussion Bibliography

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Conservation in the EU and all over the world aims at reducing biodiversity loss which has become a great issue in the last decades. However, despite existing efforts, Earth is assumed to face a sixth mass extinction. One major challenge for conservation is to reconcile the targets with conflicting interests, e.g. for food production in intensively used agricultural landscapes. Agriculture is an example of a coupled human-environment system that is approached in this thesis with the help of mathematical models from two directions. Firstly, the ecological subsystem is considered to find processes relevant for the effect of habitat connectivity on population abundances. Modelling theory predicts that the species-specific growth parameters (intrinsic growth rate and carrying capacity) indicate whether dispersal has a positive or negative effect on the total population size at equilibrium (r-K relationship). We use laboratory experiments in combination with a system of ordinary differential equations and deliver the first empirical evidence for a negative effect of dispersal on the population size in line with this theory. The result is of particular relevance for the design of dispersal corridors or stepping stones which are meant to increase connectivity between habitats. These measures might not be effective for biological conservation. A second population model, consisting of two coupled Ricker maps with a mate-finding Allee effect, is analyzed in order to examine the effect of bistability due to the Allee effect in combination with overcompensation in a spatial system. The interplay can cause complex population dynamics including multiple coexisting attractors, long transients and sudden population collapses. Essential extinction teaches us that not only small populations are prone to extinction but chaotic dynamics can drive a population extinct in a short period of time as well. By a comprehensive model analysis, we find that dispersal can prevent essential extinction of a population. In the context of conservation that is: habitat connectivity can promote rescue effects to save a population that exhibits an Allee effect. The two findings of the first part of this thesis have contrasting implications for conservation which shows that universal recommendations regarding habitat connectivity are impossible without knowledge of the specific system. Secondly, a model for the socio-economic subsystem is presented. Agri-environment schemes (AES) are payments that compensate farmers for forgone profits on the condition that they improve the ecological state of the agricultural system. However, classical economic models that describe the cost-effectiveness of AES often do not take the social network of farmers into account. Numerical simulations of the socio-economic model presented in this thesis suggest that social norms can hinder farmers from scheme participation. Moreover, social norms lead to multistability in farmers’ land-use decision behaviour. Informational campaigns potentially decrease the threshold towards more long-term scheme participation and might be a good tool to complement compensation payments if social norms affect land-use decisions. Finally, a coupled human-environment system is analyzed. An integrated economicecological model is studied to investigate the cost-effectiveness of AES if the species of concern exhibits an Allee effect. A numerical model analysis indicates large trade-offs between agricultural production and persistence probability. Moreover, conservation success strongly depends on the initial population size, meaning that conservation is well advised to start before the species is threatened. Spatial aggregation of habitat can promote rescue effects, suggesting land-sparing solutions for conservation. In that case,agglomeration bonuses may serve to increase the effectiveness of AES. Possible causes for population declines are diverse and can be a combination of human influences, e.g. due to habitat degradation and inherent ecosystem properties. That complicates the task of conservation. The models presented in this thesis simplify complex systems in order to extract processes relevant for biological conservation. The analysis of spatial effects and dynamical model complexity, e.g. due to Allee effects or a nonlinear utility function, allows us improve the understanding of coupled human-environment systems.

Human capability to recall episodic memories depends on how much time has passed since the memory was encoded. This dependency is described by a memory retrieval curve that reflects an interesting phenomenon referred to as a reminiscence bump - a tendency for older people to recall more memories formed during their young adulthood than in other periods of life. This phenomenon can be modelled with an attractor neural network, for example, the firing-rate Bayesian Confidence Propagation Neural Network (BCPNN) with incremental learning. In this work, the mechanisms underlying the reminiscence bump in the neural network model are systematically studied. The effects of synaptic plasticity, network architecture and other relevant parameters on the characteristics of the reminiscence bump are systematically investigated. The most influential factors turn out to be the magnitude of dopamine-linked plasticity at birth and the time constant of exponential plasticity decay with age that set the position of the bump. The other parameters mainly influence the general amplitude of the lifespan retrieval curve. Furthermore, the recency phenomenon, i.e. the tendency to remember the most recent memories, can also be parameterized by adding a constant to the exponentially decaying plasticity function representing the decrease in the level of dopamine neurotransmitters. / Människans förmåga att återkalla episodiska minnen beror på hur lång tid som gått sedan minnena inkodades. Detta beroende beskrivs av en sk glömskekurva vilken uppvisar ett intressant fenomen som kallas ”reminiscence bump”. Detta är en tendens hos äldre att återkalla fler minnen från ungdoms- och tidiga vuxenår än från andra perioder i livet. Detta fenomen kan modelleras med ett neuralt nätverk, sk attraktornät, t ex ett icke spikande Bayesian Confidence Propagation Neural Network (BCPNN) med inkrementell inlärning. I detta arbete studeras systematiskt mekanismerna bakom ”reminiscence bump” med hjälp av denna neuronnätsmodell. Exempelvis belyses betydelsen av synaptisk plasticitet, nätverksarkitektur och andra relavanta parameterar för uppkomsten av och karaktären hos detta fenomen. De mest inflytelserika faktorerna för bumpens position befanns var initial dopaminberoende plasticitet vid födseln samt tidskonstanten för plasticitetens avtagande med åldern. De andra parametrarna påverkade huvudsakligen den generella amplituden hos kurvan för ihågkomst under livet. Dessutom kan den s k nysseffekten (”recency effect”), dvs tendensen att bäst komma ihåg saker som hänt nyligen, också parametriseras av en konstant adderad till den annars exponentiellt avtagande plasticiteten, som kan representera densiteten av dopaminreceptorer.

Considere um meio caracterizado por $N$ pontos cujas coordenadas são geradas aleatoriamente de maneira uniforme nas arestas unitárias de um hipercubo $d$-dimensional. Um caminhante parte de cada ponto deste meio desordenado e se movimenta obedecendo à regra determinista de ir para o ponto mais próximo que não tenha sido visitado nos últimos $mu$ passos. Este processo foi denominado de caminhada determinista do turista. Cada trajetória gerada por esta dinâmica possui uma parte inicial não-periódica de $t$ passos (transiente) e uma parte final periódica de $p$ passos (atrator). As probabilidades de vizinhança são expressas através da fórmula de Cox, que é parametrizada pela função beta incompleta normalizada $I_d = I_{1/4}[1/2,(d+1)/2]$. Enfati-zamos aqui que a distribuição relevante é $S_{mu,d}^{(N)}(t,p)$, a distribuição conjunta de $t$ e $p$, que tem como casos particulares as distribuições marginais, previamente estudadas. O objetivo deste estudo é obter analiticamente estas distribuições para a caminhada determinista do turista sem memória no espaço euclideano, no modelo de distâncias aleatórias (que corresponde ao limite $d ightarrow infty$) e no modelo de mapeamento aleatório (que é um caso limite das redes de Kauffman). As distribuições analíticas obtidas foram validadas através de experimentos numéricos. A distribuição conjunta de tempos de transiente e período de atratores, no limite termodinâmico para uma dimensionalidade arbitrária vale: $S_{1,d}^{(infty)}(t,p) = [Gamma(1+I_d^{-1}) cdot (t+I_d^{-1})/Gamma(t+p+I_d^{-1})] cdot delta_{p,2}$, onde $t=0,1,2,ldots,infty$; $Gamma(z)$ é a função gama e $delta_{i,j}$ é o delta de Kronecker. A caminhada determinista do turista sem memória no modelo de mapeamento aleatório produz uma distribuição de períodos não-trivial ($S_{0,rm}^{(N)}(p) propto p^{-1}$), que é obtida de $S_{0,rm}^{(N)}(t,p) = Gamma(N)/{Gamma[N+1-(t+p)]N^{t+p}}$, onde enfatizamos que o número de pontos explorados $n_e=t+p$ é a grandeza fundamental nos problemas considerados. / Consider a medium characterized by $N$ points whose coordinates are randomly generated by a uniform distribution along the unitary edges of a $d$-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go the nearest point which has not been visited in the preceding $mu$ steps. This process has been called the deterministic tourist walk. Each trajectory generated by this dynamics has an initial non-periodic part of $t$ steps (transient) and a final periodic part of $p$ steps (attractor). The neighborhood probabilities are given by the Cox formula, which is parameterized by the normalized incomplete beta function $I_d = I_{1/4}[1/2,(d+1)/2]$. Here we stress that the relevant distribution is the joint $t$ and $p$ distribution $S_{mu,d}^{(N)}(t,p)$, which has as particular cases, the marginal distributions previously studied. The objective of this study is to obtain analytically these distributions for the memoryless deterministic tourist walk in the euclidean space, random link model (which corresponds to $d ightarrow infty$ limit) and random map model (which is a limiting case of the Kauffman model). The obtained distributions have been validated by numerical experiments. The joint transient time and attractor period distribution in the thermodynamic limit for an arbitrary dimensionality is: $S_{1,d}^{(infty)}(t,p) = [Gamma(1+I_d^{-1}) cdot (t+I_d^{-1})/Gamma(t+p+I_d^{-1})] cdot delta_{p,2}$, where $t=0,1,2,ldots,infty$; $Gamma(z)$ is the gamma function and $delta_{i,j}$ is the Kronecker's delta. The memoryless deterministic tourist walk in the random map leads to a non-trivial cycle distribution ($S_{0,rm}^{(N)}(p) propto p^{-1}$), which is obtained from $S_{0,rm}^{(N)}(t,p) = Gamma(N)/{Gamma[N+1-(t+p)]N^{t+p}}$, where we stress that the number of explored points $n_e=t+p$ is the fundamental quantity in the considered problems.

Considere um meio desordenado constituído por $N$ pontos cujas coordenadas são geradas aleatoriamente de maneira uniforme e independente nas arestas unitárias de um hipercubo $d$-dimensional. As probabilidades de vizinhança entre os pares de pontos deste meio são expressas através da fórmula de Cox. Um caminhante parte de um dado ponto deste meio desordenado e se movimenta obedecendo à regra determinista de ir para o ponto mais próximo que não tenha sido visitado nos últimos $\\mu$ passos. Este processo foi denominado de caminhada determinista do turista. Cada trajetória gerada por esta dinâmica possui uma parte inicial não-periódica de $t$ passos (transiente) e uma parte final periódica de $p$ passos (atrator). Neste trabalho, obtemos analiticamente algumas distribuições estatísticas para a caminhada determinista do turista com memória $\\mu$ arbitrária em sistemas unidimensionais e com memória $\\mu=2$ no modelo Random Link (que corresponde ao limite $d ightarrow 1$). Estes resultados nos permitiram compreender o papel da memória no comportamento exploratório do turista e explicar a equivalência não-trivial entre o modelo Random Link e o modelo Random Map (que é um caso limite das redes de Kauffman). Enfatizamos que o número de pontos explorados pelo turista é a grandeza fundamental nos problemas considerados. As distribuições analíticas obtidas foram validadas através de experimentos numéricos. Também obtivemos uma dedução alternativa para a fórmula de Cox, apresentando os resultados finais em termos de distribuições estatísticas elementares. / Consider a medium characterized by $N$ points whose coordinates are randomly and independently generated by a uniform distribution along the unitary edges of a $d$-dimensional hypercube. The neighborhood probabilities between any pair of points in this medium are given by the Cox formula. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go the nearest point which has not been visited in the preceding $\\mu$ steps. This process has been called the deterministic tourist walk. Each trajectory generated by this dynamics has an initial non-periodic part of $t$ steps (transient) and a final periodic part of $p$ steps (attractor). In this work, we obtain analytically some statistical distributions for the deterministic tourist walk with arbitrary memory $\\mu$ in one-dimensional systems and with memory $\\mu=2$ in the random link model (which corresponds to $d ightarrow 1$ limit). These results enable us to understand the main role played by the memory on the tourist\'s exploratory behavior and explain the non-trivial equivalence between the random link model and the random map model (which is a limiting case of the Kauffman model). We stress that the number of explored points is the fundamental quantity in the considered problems. The obtained distributions have been validated by numerical experiments. We also obtain an alternative derivation for the Cox formula, writing the final results in terms of known statistical distributions.

Podemos considerar a caminhada determinista do turista como um processo do tipo dinâmico, que ocorre sobre uma rede composta por N pontos. Os pontos são gerados de maneira aleatória, no espaço euclidiano d dimensional. Um caminhante, partindo de um ponto qualquer do meio desordenado, se movimenta seguindo uma regra determinista de ir para o ponto mais próximo que não tenha sido visitado nos últimos ?= µ - 1 passos. Cada uma das trajetórias geradas através dessa dinâmica possui uma parte inicial não periódica de t passos, denominada transiente, e uma parte final, periódica, de p passos, denominada atrator. Devido ao custo computacional de memória, só é possível simular sistemas com N ? O(103) e µ << N. Neste estudo uma nova implementação na estrutura de armazenamento de dados, no modelo numérico do turista, nos permitiu obter algumas distribuições estatísticas para a caminhada, com valores de memória µ ? O(N). Com estes resultados verificamos a eficiência da estrutura proposta e avançamos no conhecimento acerca do comportamento do turista em caminhadas com memória da ordem de N. Também neste trabalho, obtivemos resultados numéricos interessantes, que serviram para explicar a formação de atratores com determinados períodos na caminhada determinista do turista unidimensional, bem como a não formação de atratores com períodos 2µ+1, 2µ+2 e 2µ+3.não são constituídos. Também neste trabalho, uma nova implementação na estrutura de armazenamento de dados, no modelo numérico do turista, nos permitiu obter algumas distribuições estatísticas para a caminhada, com valores de memória ? muito acima do que se tinha alcançado anteriormente. Com estes resultados verificamos a eficiência da estrutura proposta, e avançamos o conhecimento a cerca do comportamento do turista em sistema da ordem de N. / We may consider the deterministic tourist walk as a dynamic process performed over a landscape of N points. These points are randomly spread on a d dimensional euclidean space. A walker leaves from any point of that landscape and moves according to the deterministic rule of going to the nearest point that has not been visited in the last ?= µ - 1 steps. Each trajectory generated by this dynamics has an initial non-periodic part of t steps, called transient, and a final periodic one of p steps, called attractor. Due to computational costs of memory usage, it is possible to simulate only small sistems, with N ? O(103) and µ << N. In this work, we propose a new implementation of the structure for data storage. The numerical model of the tourist walk, allowed us to obtain some statistical distributions for the walk with a memory value µ ? O(N). Moreover, in this study we obtain interesting and useful numerical results to explain the presence of some specific attractors in deterministic walk in one-dimensional space and the absence of attractors with periods 2µ+1, 2µ+2 and 2µ+3. are not made. In this work, we propose a new implementation of the structure for storing data, the numerical model of the tourist, has allowed us to obtain some statistical distributions for the walk with a memory value ? over and above what had been achieved previously. With these results, we verifed the efficiency of the HL structure proposed, and advance knowledge about the behavior of the tourist walk in the order of N.

Cette thèse est consacrée à l’étude théorique et numérique de plusieurs équations aux dérivées partielles non linéaires qui apparaissent dans la modélisation de la séparation de phase et des micro-systèmes électro-mécaniques (MSEM). Dans la première partie, nous étudions des modèles d’ordre élevé en séparation de phase pour lesquels nous obtenons le caractère bien posé et la dissipativité, ainsi que l’existence de l’attracteur global et, dans certains cas, des simulations numériques. De manière plus précise, nous considérons dans cette première partie des modèles de type Allen-Cahn et Cahn-Hilliard d’ordre élevé avec un potentiel régulier et des modèles de type Allen-Cahn d’ordre élevé avec un potentiel logarithmique. En outre, nous étudions des modèles anisotropes d’ordre élevé et des généralisations d’ordre élevé de l’équation de Cahn-Hilliard avec des applications en biologie, traitement d’images, etc. Nous étudions également la relaxation hyperbolique d’équations de Cahn-Hilliard anisotropes d’ordre élevé. Dans la seconde partie, nous proposons des schémas semi-discrets semi-implicites et implicites et totalement discrétisés afin de résoudre l’équation aux dérivées partielles non linéaire décrivant à la fois les effets élastiques et électrostatiques de condensateurs MSEM. Nous faisons une analyse théorique de ces schémas et de la convergence sous certaines conditions. De plus, plusieurs simulations numériques illustrent et appuient les résultats théoriques. / This thesis is devoted to the theoretical and numerical study of several nonlinear partial differential equations, which occur in the mathematical modeling of phase separation and micro-electromechanical system (MEMS). In the first part, we study higher-order phase separation models for which we obtain well-posedness and dissipativity results, together with the existence of global attractors and, in certain cases, numerical simulations. More precisely, we consider in this first part higher-order Allen-Cahn and Cahn-Hilliard equations with a regular potential and higher-order Allen-Cahn equation with a logarithmic potential. Moreover, we study higher-order anisotropic models and higher-order generalized Cahn-Hilliard equations, which have applications in biology, image processing, etc. We also consider the hyperbolic relaxation of higher-order anisotropic Cahn-Hilliard equations. In the second part, we develop semi-implicit and implicit semi-discrete, as well as fully discrete, schemes for solving the nonlinear partial differential equation, which describes both the elastic and electrostatic effects in an idealized MEMS capacitor. We analyze theoretically the stability of these schemes and the convergence under certain assumptions. Furthermore, several numerical simulations illustrate and support the theoretical results.