Neste texto é feita uma discussão sobre alguns resultados que fornecem condições suficientes para que um difeomorfismo local, do espaço euclideano n-dimensional nele próprio, seja injetivo. Dentro deste cenário, são exploradas as contribuições destes resultados na tentativa de solucionar conhecidas conjecturas no meio científico como a Conjectura Jacobiana e a Conjectura de Ponto Fixo. Do ponto de vista dinâmico, existem relações entre injetividade global e estabilidade assintótica global. Neste sentido, os resultados também são contextualizados com respeito a importantes conjecturas de estabilidade assintótica: Conjectura de Markus-Yamabe e o Problema de LaSalle / We present some results which give suficient conditions for a local diffeomorphism from the n-dimensional Euclidean space into itself be globally injective. Within this context, we consider some partial results addressed to solve the well known Fixed Point Conjecture and Jacobian Conjecture. From the dynamical point of view, there are connections between global injectivity and global asymptotic stability. In this way, we present a solution of the Markus-Yamabe Conjecture and of the LaSalle Problem

Este trabalho é dedicado ao estudo do comportamento a longo prazo de uma equação de placas extensíveis acoplada a uma equação de calor do tipo hiperbólico. O problema corresponde a um modelo de termo-elasticidade baseado em teorias de calor do tipo não-Fourier. Considerando que efeitos de inércia de rotação estão presentes no modelo, mostramos que o efeito dissipativo do calor e suficiente para estabilizar exponencialmente o sistema, sem dissipações adicionais. Além disso, provamos que o sistema possui um atrator global de dimensão fractal finita e também atratores exponenciais. Nossos resultados generalizam e complementam diversos trabalhos existentes / This work is concerned with long-time dynamics of solutions of extensible plate equations with thermal memory. It corresponds to a model of thermoelasticity based on a theory of non-Fourier heat flux. By considering the case where rotational inertia is present we show that the thermal dissipation is sufficient to stabilize the system exponentially and guarantee the existence of a finite-dimensional global attractor. In addition the existence of an exponential attractor and some further properties are also considered. Our results complements several existing results

La cascade d’énergie qui a lieu dans les océans, depuis les grandes vers les petites échelles, est capitale pour comprendre leur dynamique et le mélange irréversible associé. Les attracteurs d’ondes internes font partie des mécanismes conduisant potentiellement à une telle cascade. Dans ce manuscrit, nous étudions expérimentalement les attracteurs d’ondes internes, dans une cuve trapézoïdale remplie d’un fluide stratifié linéairement en densité. Dans cette géométrie, les ondes peuvent être focalisées vers un cycle limite : l’attracteur. Nous montrons que la formation de l’attracteur est purement linéaire : des petites échelles sont donc créées grâce à la focalisation des ondes. Les principales caractéristiques de l’attracteur dépendent uniquement de la géométrie trapézoïdale de la cuve. A l’échelle de l’océan, nous montrons que les attracteurs d’ondes internes sont très probablement instables. En effet, ceux-ci sont sujets à une instabilité de résonance triadique, qui transfère de l’énergie depuis l’attracteur vers un couple d’ondes secondaires. Cette instabilité et ses principales caractéristiques sont décrites en fonction de la géométrie du bassin. Pour des expériences de longue durée, l’instabilité produit plusieurs paires d’ondes secondaires, créant une cascade d’instabilités triadiques et transférant l’énergie injectée à grandes échelles vers des échelles plus petites. Nous montrons, pour la première fois de façon expérimentale, de très fortes signatures de turbulence d’ondes internes. Au delà de cet état, la cascade atteint un régime de mélange partiel du fluide stratifié. Cet ultime régime apparait indépendant de la géométrie trapézoïdale du bassin, et donc, universel. Cette thèse est complétée par une étude sur la masse ajoutée et l’amortissement par émission d’ondes d’objets oscillant horizontalement dans un fluide stratifié en densité. Cela a des applications concernant la conversion de l’énergie des marées en ondes internes. / A question of paramount importance in the dynamics of oceans is related to the energy cascade from large to small scales and its contribution to mixing. Internal wave attractors may be one of the possible mechanisms responsible for such a cascade. In this manuscript, we study experimentally internal wave attractors in a trapezoidal test tank filled with linearly stratified fluid. In such a geometry, the waves can form closed loops called attractors. We show that the attractor formation is purely linear: small scales are thus created by wave focusing. The attractor characteristics are found to only depend on the trapezoidal geometry of the tank. At the ocean scale, we show that attractors are very likely to be unstable. Indeed, internal wave attractors are prone to a triadic resonance instability, which transfers energy from the attractor to a pair of secondary waves. This instability and its main characteristics are described as a function of the geometry of the basin. For long-term experiments, the instability produces several pairs of secondary waves, creating a cascade of triadic interactions and transferring energy from large-scale monochromatic input to multi-scale internal-wave motion. We reveal, for the first time, experimental convincing signatures of internal wave turbulence. Beyond this cascade, we have a mixing regime, which appears to be independent of the trapezoidal geometry and, thus, universal. This manuscript is completed by a study on added mass and wave damping coefficient of bodies oscillating horizontally in a stratified fluid, with applications to tidal conversion.

[pt] Neste trabalho estendemos os resultados em (12) e (22), sobre a minimalidade de uma das folheações estável ou instável forte), para o caso de atratores robustamente transitivos parcialmente hiperbólico e com direção central unidimensional. No nosso contexto a hiperbolicidade parcial esta definida somente no atrator. Algumas consequências são obtidas tais como a verificação de que estes atratores são robustamente) classes homoclínicas, possuem robustamente) interior vazio e admitem uma decomposição espectral. Resultados similares ainda valem no caso de atratores genericamente transitivos. / [en] In this work we extend the results in (12) and (22) about the minimality of one of the strong foliations (stable or unstable), for the case of robustly transitive attractors that is partially hyperbolic with one dimensional center bundle. In our context the partial hyperbolicity is defined only in the attractor. Some consequences are obtained as the verification that these attractors are (robustly) homoclinic classes, have (robustly) empty interior and admit a spectral decomposition. Similar results still holds in the case of generically transitive attractors.

This thesis focuses on deterministic chaos and selected methods of chaos control. It briefly describes the matter of deterministic chaos and presents commonly used tools of analysis of dynamical systems exhibiting chaotic behavior. A list of frequently studied chaotic systems is presented and followed by a description of methods of chaos control and the optimization of these methods. The practical part is dedicated to the stabilization of two model systems and one real system with described methods.

Speech signal is being used in the meaning of nonlinear dynamic system. As such, it is transform to multidimensional phase space, where filtration method based on time series neighbors of analysed signal is used. For embedding phase space methods time delay and false nearest neighbors are applied.

In this article we investigate a pitchfork bifurcation of the local attractor of a simple capsizing model proposed by Thompson. Although this is a very simple system it has a very complicate dynamic. We try to reveal some properties of this dynamic with modern numerical methods. For this reason we approximate stable and unstable manifolds which connect the steady states to obtain a complete understanding of the topology in the phase space. We also consider approximations of the Lyapunov Exponents (resp. Floquet Exponents) which indicates the pitchfork bifurcation.

Recent findings suggest, that perception of acoustic stimuli in the mouse auditory cortex relies on categorization of object-based representations. Local neuronal populations in L2/3 of the mouse auditory cortex reportedly exhibit a limited number (1-3) of stable modes of response, each possibly evoked by multiple complex sounds of variable acoustic features. Stimulation using linear intensity mixing of sounds evoking different response modes revealed an attractor-like dynamic of the underlying representation. These modes of response were hypothesized to represent the neural correlate of perceptual categorization. We have developed an experimental protocol enabling chronic two-photon imaging of the previously described population coding under awake conditions. Using this protocol we acquired data suggesting that the pattern of population activity underlying a mode of response, is stable during a week-long timeframe. We have also recorded the neural activity of a local subpopulation of somatostatin-positive inhibitory interneurons (SST+ INs) during abrupt changes in cortical representation. Our preliminary results suggest that local SST+ INs exhibit maximal firing when the neural correlate of a mode of response is exhibited by the surrounding population of principal cells. In addition, we observed a...

Hierarchical clustering is a family of machine learning methods that has many applications, amongst other data science and data mining. This thesis belongs to the research area of brain-like computing and introduces a novel approach to hierarchical clustering using a brain-like recurrent neural network. Attractor networks can cluster samples by converging to the same network state. We modulate the network behaviour by varying a parameter in the activity propagation rule such that the granularity of the resulting clustering is changed. A hierarchical clustering is then created by combining multiple levels of granularity. The method is developed for two different datasets and evaluated on a variety of clustering metrics. Its performance is compared to standard clustering algorithms and the structure and composition of the clustering is inspected. We show that the method can produce clusterings for different levels of granularity and new data without retraining. As a novel clustering method, it is relevant to machine learning applications. As a model for hierarchical recall in a memory model, it is relevant to computational neuroscience and neuromorphic computing. / Hierarkiskt klusterarbete är en grupp av maskininlärningsmetoder som har många tillämpningar, bland annat datavetenskap och datagrävning. Denna avhandling tillhör forskningsområdet för hjärnlikt databehandling och introducerar ett nytt tillvägagångssätt för hierarkiskt klusterarbete med hjälp av ett hjärnlikt återkommande neuronnätverk. Attraktornätverk kan klustra prover genom att konvergera till samma nätverksstadium. Vi modulerar nätverkets beteende genom att variera en parameter i regeln för aktivitetspropagering så att granulariteten i det resulterande klusterarbetet förändras. Ett hierarkiskt klusterarbete skapas sedan genom att kombinera flera nivåer av granularitet. Metoden utvecklas för två olika datasets och utvärderas med hjälp av olika klustringsmått. Dess prestanda jämförs med standard klusteringsalgoritmer och strukturen och sammansättningen av klusterarbetet inspekteras. Vi visar att metoden kan producera klusterarbeten för olika nivåer av granularitet och nya data utan omträning. Som en ny klusteringsmetod är den relevant för maskininlärningsapplikationer. Som en modell för hierarkisk återkallelse i en minnesmodell är den relevant för beräkningsneurovetenskap och neuromorfisk databehandling.

Ce rapport de thèse est consacré à l'étude de modèles de champ de phase de type Caginalp. Nous considérons ici, deux modèles : le premier étant une généralisation du modèle de champ de phase de Caginalp basée sur une généralisation de la loi de Maxwell-Cattaneo et le second une généralisation provenant de la théorie de la conduction de chaleur introduite par Chen et Gurtin. L'étude du premier modèle est faite aussi bien dans un domaine borné (avec un potentiel régulier puis dans le cas d'un potentiel non régulier), que dans un domaine non borné, en l'occurrence R3. Le second modèle est un problème de champ de phase avec un couplage (linéaire et non linéaire). Tout d'abord, l'existence, l'unicité et la régularité des solutions sont analysées aux moyens d'arguments classiques. Ensuite, l'existence d'ensembles bornés absorbants et compacts attractifs est établie, assurant ainsi l'existence de l'attracteur global. Enfin, dans certains cas, l'existence d'attracteurs exponentiels, ainsi que le comportement spatial des solutions lorsque le domaine spatial est un cylindre semi-infini tri-dimensionnel, sont analysés. / This thesis report is dedicated to the study of Caginalp type phase-field Models. Here, we consider two models: the first one being a generalization of the field phase Caginalp based on a generalization of the Maxwell-Cattaneo law and the second one coming from the theory of heat conduction involving two temperatures. We study the first model in bounded (with regular and irregular potentials) and unbounded (i.e. R3) domains. The second model is a phase-field one with coupling term (linear and nonlinear). Firstly, the existence, uniqueness and regularity of solutions are analyzed by means of classical arguments. Secondly, the existence of bounded absorbing sets and attractive compact is established. Such results ensures the existence of the global attractor. Finally, in some cases, the existence of exponential attractors, as well as the spatial behavior of solutions when the spatial domain is a three-dimensional semi-infinite cylinder, are analyzed.