Orientador: Claudio Aguinaldo Buzzi / Banca: João Carlos da Rocha Medrado / Banca: Luciana de Fátima Martins / Resumo: Nesse trabalho são apresentados alguns resultados importantes sobre bifurcações de codimensão dois de campos de vetores. O resultado principal dessa dissertação e o teorema que d a o diagrama de bifurcação e os retratos de fase da Bifurcação de Bogdanov-Takens. Para a demonstracão são usadas algumas técnicas basicas de Sistemas Dinâmicos e Teoria das Singularidades, tais como Integrais Abelianas, desdobramentos de Sistemas Hamiltonianos, desdobramentos versais, Teorema de Preparação de Malgrange, entre outros. Outra importante bifurcação clássica apresentada e a Bifurca cão do tipo Hopf-Zero, quando a matriz Jacobiana possui um autovalor simples nulo e um par de autovalores imagin arios puros. Foram usadas algumas hipóteses que garantem propriedades de simetria do sistema, dentre elas, assumiuse que o sistema era revers vel. Assim como na Bifurcação de Bogdanov-Takens, foram apresentados o diagrama de bifurcao e os retratos de fase da Bifurcação Hopf-zero bifurcação reversível. As técnicas usadas para esse estudo foram a forma normal de Belitskii e o método do Blow-up polar. / Abstract: In this work is presented some important results about codimension two bifurcations of vector elds. The main result of this work is the theorem that gives the local bifurcation diagram and the phase portraits of the Bogdanov-Takens bifurcation. In order to give the proof, some classic tools in Dynamical System and Singularities Theory are used, such as Abelian Integral, versal deformation, Hamiltonian Systems, Malgrange Preparation Theorem, etc. Another classic bifurcation phenomena, known as the Hopf-Zero bifurcation, when the Jacobian matrix has a simple zero and a pair of purely imaginary eigenvalues, is presented. In here, is added the hypothesis that the system is reversible, which gives some symmetry in the problem. Like in Bogdanov-Takens bifurcation, the bifurcation diagram and the local phase portraits of the reversible Hopf-zero bifurcation were presented. The main techniques used are the Belitskii theory to nd a normal forms and the polar Blow-up method. / Mestre

Apresentaremos alguns resultados de rigidez e de bifurcação para soluções do problema de Yamabe em variedades produto com bordo. / We will discuss some rigidity and bifurcation results for solutions of the Yamabe problem in product manifolds with boundary.

Das Kernstueck der vorliegenden Arbeit ist die Betonung von kleinen Modulen als Schluesselkomponenten von biologischen Netzwerken. Unter den zahlreichen moeglichen Modulen scheinen besondere diejenigen interessant zu sein, welche die Rueckkopplungen realisieren und in regulatorischen Einheiten auftreten. Prozesse wie Genregulation, Differentiation oder Homeostasis benoetigen haeufig Autoregulation. Auf Grund dessen ist die detaillierte Kenntnis der dynamischen Eigenschaften von kleinen Modulen von groesserem Interesse. Es werden zwei biologische Systeme analysiert. Das erste beschaftigt sich mit dem Zellzyklus, das zweite Beispiel kommt aus der Immunologie und betrifft die Aktivierung von T-Zellen. Beide Modelle, d.h. ihre zugrundeliegende Netzwerke, lassen sich in Untereinheiten mit wohldefinierten Funktionen zerlegen. Diese Module entscheiden ueber das Verhalten des gesamten Netzwerkes. Mit anderen Worten, die von den Modulen getroffenen Entscheidungen, werden von dem gesamten System uebernommen. Bei der Analyse des Modells zum Zellzyklus wurde eine interessante Eigenschaft von gekoppelten Modulen deutlich, die wir dann getrennt behandelt haben. Seriell geschaltete Module mit positiver Rueckkopplung liefern ueberraschende Konstruktionsmoeglichkeiten fuer Systeme mit mehreren stabilen Gleichgewichtslagen. Obwohl nicht alle hier aufgestellten Hypothesen derzeit experimentell ueberpruefbar sind, es kann eine wichtige Aussage getroffen werden. Uebereinstimmende Strukturen und Mechanismen, die in verschiedenen biologischen Systemen vorkommen, bieten uns die Moeglichkeit einer Klassifizierung von biologischen Systemen bezueglich ihrer strukturellen Aehnlichkeiten. / The thesis emphasizes the importance of small modules as key components of biological networks. Especially, those which perform positive feedbacks seem to be involved in a number of regulatory units. Processes like gene regulation, differentiation and homeostasis often require autoregulation. Therefore, detailed knowledge of dynamics of small modules becomes nowadays an important subject of study. We analyze two biological systems: one regarding cell cycle regulation and one immunological example related to T-cell activation. Their underlying networks can be dissected into subunits with well defined functions. These modules decide about the behavior of the global network. In other words, they have decision taking function, which is inherited by the whole system. Stimulated by the cell cycle model and its interesting dynamics resulting from coupled modules, we analyzed the switching issue separately. Serial coupling of positive feedback circuits provides astonishing possibilities to construct systems with multiple stable steady states. Even though, in current stage, no exact experimental proof of all hypotheses is possible, one important observation can be made. Common structures and mechanisms found in different biological systems allow to classify biological systems with respect to their structural similarities.

Neuroprosthetics is at the intersection of neuroscience, biomedical engineering, and physics. A biocompatible neuroprosthesis contains artificial neurons exhibiting biophysically plausible dynamics. Hybrid systems analysis could be used to prototype such artificial neurons. Biohybrid systems are composed of artificial and living neurons coupled via real-time computing and dynamic clamp. Model neurons must be thoroughly tested before coupled with a living cell. We use bifurcation theory to identify hazardous regimes of activity that may compromise biocompatibility and to identify control strategies for regimes of activity desirable for functional behavior. We construct real-time artificial neurons for the analysis of hybrid systems and demonstrate a mechanism through which an artificial neuron could maintain duty cycle independent of variations in period.

Στην παρούσα εργασία εξετάζεται η δυναμική συμπεριφορά αμιγούς και απλού συναγωνισμού δύο μικροβιακών πληθυσμών που αναπτύσσονται σε δύο συζευγμένους χημοστάτες. Χρησιμοποιώντας το μοντέλο Andrews για τους ειδικούς ρυθμούς ανάπτυξης και συνθήκες βαθμίδας συγκέντρωσης στην τροφοδοσία, η μελέτη του συστήματος γίνεται με εφαρμογή μεθόδων της θεωρίας διακλαδώσεων. Εξετάζοντας δύο περιπτώσεις τροφοδοσίας, παρουσία μικροοργανισμών και απουσία, κατασκευάστηκαν δύο λειτουργικά διαγράμματα ως προς το βαθμό σύζευξης r και το λόγο των όγκων λ των δύο αντιδραστήρων και βρέθηκε το είδος ευστάθειας των υπαρχουσών καταστάσεων ισορροπίας. Παρουσία μικροοργανισμών στην τροφοδοσία παρατηρήθηκαν περιοχές συνύπαρξης των δύο πληθυσμών σε μόνιμη, περιοδική και οιονεί περιοδική κατάσταση, ενώ υπάρχουν ενδείξεις και για χαοτική συμπεριφορά. Υπό στείρα τροφοδοσία βρέθηκε ότι συνύπαρξη μπορεί να επιτευχθεί μόνο σε μόνιμη και περιοδική κατάσταση σε μία ευρεία περιοχή των παραμέτρων λειτουργίας λ και r. / The dynamic behavior of pure and simple competition of two microbial populations growing in two interconnected bioreactors is investigated. Using Andrews inhibitory model and gradient in feed concentration, the use of bifurcation theory allows an in-depth analysis of the stability change mechanisms occurring in the system, when the operating parameters of the degree of coupling and the volume ratio change. Regions of species coexistence in all steady, periodic and quasi-periodic states are observed, while there is substantial indication of chaotic behavior. Under clean feed conditions coexistence is only possible in steady and periodic states.

Η τοπολογική ταξινόμηση και μελέτη διανυσματικών πεδίων αποτελεί το κύριο θέμα αυτής της διατριβής. Στο Κεφάλαιο 1 δίνονται οι απαραίτητοι ορισμοί, καθώς και τα αποτελέσματα επί της ταξινόμησης διανυσματικών πεδίων σε μονοδιάστατες και δισδιάτατες πολλαπλότητες. Στο Κεφάλαιο 2 τεχνικές της Θεωρίας Κόμβων χρησιμοποιούνται προκειμένου να μελετηθεί η τοπολογική δομή ορισμένων παράξενων ελκυστών που εμφανίζονται στη διεθνή βιβλιογραφία. Στο Κεφάλαιο 3 αναπτύσσεται μία μέθοδος η οποία επιτρέπει την ολική τοπολογική ταξινόμηση διανυσματικών πεδίων σε ευκλείδειους χώρους οποιασδήποτε διάστασης. Η μέθοδος αυτή έπειτα εφαρμόζεται στην ταξινόμηση διανυσματικών πεδίων του R^2 και του R^3. Στο Κεφάλαιο 4 μελετάται ένα διανυσματικό πεδίο του R^3 αμετάβλητο από την D_2 ομάδα. Δίνεται η ολική του μελέτη, για διάφορες τιμές των παραμέτρων, και το μερικό του διάγραμμα διακλάδωσης. Αποδεικνύεται η ύπαρξη χάους και συνδέεται με τις συμμετρικές ιδιότητες του συστήματος, ενώ η μελέτη ολοκληρώνεται με τη συμπεριφορά του συστήματος στο άπειρο. / The topological classification and study of vector fields is the subject of this thesis. In Chapter 1 the necessary definitions are given, along with the known results on the classification of vector fields on 1-dimensional and 2-dimensional manifolds. In Chapter 2 methods of Knot Theory are used for the clarification of the topological study of some strange attractors found in the bibliography. In Chapter 3 a technique is developed, which can be used to classify globally vector fields defined on Euclidean spaces of any dimension. This technique is then used to classify some vector fields of R^2 and R^3. In the final Chapter 4 a vector field of R^3 is studied which is invariant under the D_2 symmetry group. We present its global phase portrait, for various parameter values, and its partial bifurcation diagram. The existence of chaos is proven and its connection to the symmetry properties of the attractor is discussed. We end its study presenting its behavior at infinity.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-02-29Bitstream added on 2014-06-13T20:55:43Z : No. of bitstreams: 1 arakawa_vat_me_sjrp.pdf: 795168 bytes, checksum: 1ce40af6d71942f94c4c2bb678ce986f (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Nesse trabalho são apresentados alguns resultados importantes sobre bifurcações de codimensão dois de campos de vetores. O resultado principal dessa dissertação e o teorema que d a o diagrama de bifurcação e os retratos de fase da Bifurcação de Bogdanov-Takens. Para a demonstracão são usadas algumas técnicas basicas de Sistemas Dinâmicos e Teoria das Singularidades, tais como Integrais Abelianas, desdobramentos de Sistemas Hamiltonianos, desdobramentos versais, Teorema de Preparação de Malgrange, entre outros. Outra importante bifurcação clássica apresentada e a Bifurca cão do tipo Hopf-Zero, quando a matriz Jacobiana possui um autovalor simples nulo e um par de autovalores imagin arios puros. Foram usadas algumas hipóteses que garantem propriedades de simetria do sistema, dentre elas, assumiuse que o sistema era revers vel. Assim como na Bifurcação de Bogdanov-Takens, foram apresentados o diagrama de bifurcao e os retratos de fase da Bifurcação Hopf-zero bifurcação reversível. As técnicas usadas para esse estudo foram a forma normal de Belitskii e o método do Blow-up polar. / In this work is presented some important results about codimension two bifurcations of vector elds. The main result of this work is the theorem that gives the local bifurcation diagram and the phase portraits of the Bogdanov-Takens bifurcation. In order to give the proof, some classic tools in Dynamical System and Singularities Theory are used, such as Abelian Integral, versal deformation, Hamiltonian Systems, Malgrange Preparation Theorem, etc. Another classic bifurcation phenomena, known as the Hopf-Zero bifurcation, when the Jacobian matrix has a simple zero and a pair of purely imaginary eigenvalues, is presented. In here, is added the hypothesis that the system is reversible, which gives some symmetry in the problem. Like in Bogdanov-Takens bifurcation, the bifurcation diagram and the local phase portraits of the reversible Hopf-zero bifurcation were presented. The main techniques used are the Belitskii theory to nd a normal forms and the polar Blow-up method.

Submitted by Helmut Patrocinio (hell.kenn@gmail.com) on 2017-12-01T23:43:39Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Heloisa_Onias_Dissertacao_2012.pdf: 9805428 bytes, checksum: 00e0f3bac6584320107351966c70da69 (MD5) / Approved for entry into archive by Ismael Pereira (ismael@neuro.ufrn.br) on 2017-12-04T12:33:01Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Heloisa_Onias_Dissertacao_2012.pdf: 9805428 bytes, checksum: 00e0f3bac6584320107351966c70da69 (MD5) / Made available in DSpace on 2017-12-04T12:33:37Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Heloisa_Onias_Dissertacao_2012.pdf: 9805428 bytes, checksum: 00e0f3bac6584320107351966c70da69 (MD5) Previous issue date: 2012-08 / O circuito RLD, formado por um resistor, um indutor e um diodo em s?rie, apresenta uma din?mica muito rica quando for?ado por uma tens?o externa harm?nica e vem sendo estudado h? d?cadas. Contudo, ainda existem t?picos em din?mica n?o-linear sendo estudados com variantes deste circuito. Varreduras nos par?metros de controle podem fazer com que esse sistema oscile eletronicamente entre regi?es peri?dicas e regi?es ca?ticas. O diodo ? o elemento n?o linear respons?vel pelo surgimento do caos. Utilizando um modelo de capacit?ncia n?o linear para descrever o comportamento do diodo, podemos escrever as equa??es para esse sistema e estudar a sua din?mica numericamente. Nosso principal objetivo foi o estudo de expoentes cr?ticos complexos em bifurca??es din?micas. Para isso, realizamos um estudo num?rico do circuito RLD for?ado senoidalmente utilizando como par?metros de controle a frequ?ncia e a amplitude da tens?o de entrada. Constru?mos, a partir das s?ries temporais da corrente total e da tens?o no diodo, diagramas de bifurca??o com diferentes cortes estrobosc?picos, que apresentam cascata de dobramento de per?odo, janelas peri?dicas e transi??o intermitente. Tamb?m realizamos estudos num?ricos do comportamento da m?dia na regi?o de transi??o caos-peri?dico na busca de encontrar um expoente cr?tico caracter?stico e oscilas??es na m?dia, elementos que j? foram observados no mapa log?stico. N?o foram poss?veis observar numericamente as oscila??es, mas observamos um decaimento exponencial com expoente cr?tico de aproximadamente 0,5. Montamos um sistema de controle, aquisi??o e tratamento de dados experimentais no qual ? poss?vel a realiza??o remota de experimentos simult?neos com dois circuitos diferentes. Obtivemos diagramas de bifurca??es experimentais nos quais observamos que o sistema apresentahisterese e alta sensibilidade ?s condi??es do experimento como, por exemplo, o passo de varredura do par?metro de controle. / The RLD circuit, formed by a resistor, an inductor and a diode in series, displays a very rich dynamics when forced by an external harmonic voltage, and it has being studied for decades. However, there are some topics in nonlinear dynamics that are still studied with variants of this circuit nowadays. Changes in the control parameters may cause electronic oscillations between regular and chaotic regions.The diode is the nonlinear element responsible for the appearance of chaos. Using a nonlinear capacitance model to describe the behavior of the diode, we can write the equations for this system and study its dynamics numerically. Our main objective was the study of critical exponents in complex dynamic bifurcations. For that, we did a numerical study of the RLD circuit forced sinusoidally using as control parameters the amplitude of the input voltage and the frequency. We made, from the time series obtained, bifurcation diagrams with different stroboscopic cuts, which have cascade of period-doubling, periodic windows and intermittent transition. We also did numerical studies of the average behavior in the periodic-chaos transition region searching for characteristic critical exponent and oscilas??es on average, elements that have been observed in the logistic map. It was not possible to observe the oscillations numerically, but we observed an exponential decay with critical exponent of approximately 0.5. We set up a system able to control, acquire and process experimental data making it possible to perform remote simultaneous experiments with two different circuits. We have obtained experimental diagrams bifurcations in which we observe that the system has hysteresis and high sensitivity to the conditions of the experiment such as the step of scanning the control parameter.

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T09:33:59Z (GMT). No. of bitstreams: 1 Souza_WenderJosede_D.pdf: 1230622 bytes, checksum: 578f86e5fe4ff35247fcfb0fb04975b8 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho consideramos alguns aspectos da teoria qualitativa de sistemas dinâmicos suaves por partes. Nosso principal objetivo é estudar uma classe de tais sistemas, onde o conjunto de descontinuidade é dado por uma hipersuperfície ? e além disso, assumimos que em cada região determinada por ? o campo de vetores definido é um sistema Hamiltoniano. Apresentamos estudos relacionados à regularização de campos de vetores suaves por partes em Rn que preservam volume nas componentes suaves. Abordamos também singularidades de funções suaves por partes, onde formas normais e seus desdobramentos são apresentados. Por fim estudamos bifurcações de campos de vetores Hamiltonianos refrativos / Abstract: In this work, we consider some aspects of the qualitative theory of non smooth dynamical systems in Rn. Our main goal is to study a class of such systems where the discontinuity set is concentrated in a hypersurface ? and moreover, we assume that in each region determined by ? the vector field is a Hamiltonian system. We present studies related to the regularization of piecewise vector fields in Rn that are volume preserving on each smooth components. We also analyze singularities of piecewise smooth functions where normal forms and their unfolding are presented. Finally, we study bifurcations of refractive Hamiltonian vector fields / Doutorado / Matematica / Doutor em Matemática

This thesis addresses chemotherapy-induced peripheral neuropathy (CIPN)- a form of pain sensation and a prevalent dose-limiting side-effect of several chemotherapy agents such as vincristine, paclitaxel, and oxaliplatin. These agents are used for treating various cancers such as leukemia, brain tumor, lung cancer. Peripheral neuropathy is a numbing, tingling, and burning sensation felt in the palms and feet, which occurs due to damage to neurons (nerve cells). Prolonged CIPN can impact the quality of life of cancer patients. Occasionally, severe CIPN can result in termination of chemotherapy treatment altogether. Currently, there are no established strategies for treating CIPN due to a lack of understanding of its mechanisms. Moreover, different patients react differently to the same treatment; a subgroup of patient population may never experience CIPN, while another may experience severe CIPN for the same dose. In addition, there are no established strategies for predicting CIPN either. This thesis addresses both prediction and mechanisms of CIPN.<br><br>The following paragraphs reflect the organization of this thesis. Each paragraph introduces a research problem, the approaches taken to investigate it, and states the key results.<br><br>First, a metabolomics-based approach was used to investigate CIPN prediction. Blood samples of pediatric leukemic cancer patients who underwent treatment with a chemotherapy agent - vincristine were provided. These blood samples were analyzed at different treatment time points using mass spectrometry to obtain the metabolite profiles. Machine learning was then employed to identify specific metabolites that can predict overall susceptibility to peripheral neuropathy in those patients at specific treatment time points. Subsequently, selected metabolites were used to train machine learning models to predict neuropathy susceptibility. Finally, the models were deployed into an open-source interactive tool- <i>VIPNp</i>- that can be used by researchers to predict CIPN in new pediatric leukemic cancer patients.<br><br>Second, the focus was shifted to the pathophysiology of pain and the pain-sensing neuron; specifically: (i) investigating pain sensation mutations and the dynamics of the pain-sensing neuron, and (ii) exploring chemotherapy-induced peripheral neuropathy mechanisms. <br><br>While pain is a common experience, genetic mutations in individuals can alter their experience of pain, if any at all (certain mutations yield individuals insensitive to pain). Despite its ubiquity, we do not have a complete understanding of the onset and/or mechanisms of pain sensation. Pain sensation can be broadly classified into three types: (i) nociceptive, (ii) neuropathic, and (iii) inflammatory. Nociceptive pain arises due to a noxious external stimulus (e.g., upon touching a hot object). Neuropathic pain (which is felt as a side-effect of the aforementioned chemotherapy agents) is the numbing and tingling sensation due to nerve damage. Inflammatory pain occurs due to damage to internal tissues. Pain in any form can be characterized in terms of electrical signaling by the pain-sensing neuron. Signal transmission regarding pain occurs through generation of an electrical signal called the action potential- a peak in neuron membrane potential. Excessive firing of action potentials by a pain-sensing neuron indicates pain of a specific form and intensity. In order to investigate this electrical signaling, a mathematical modeling approach was employed. The neuron membrane was assumed to be an electrical circuit and the potential across the membrane was modeled in terms of the sodium and potassium ions flowing across it via voltage-gated sodium and potassium channels, respectively. Generation of a single action potential followed by a resting state corresponds to a normal state, whereas periodic firing of action potentials (an oscillatory state) corresponds to pain of some form and intensity <i>in vitro</i>. Therefore, an investigation into the switch from a resting state to an oscillatory state was proposed. A bifurcation theory approach (generally useful in exploring changes in qualitative behavior of a model) was used to explore possible bifurcations to explain this switch. Firstly, genetic mutations that can shift the pain sensation threshold in the pain-sensing neuron were explored. The detected bifurcation points were found to be sensitive to specific sodium channels’ model parameters, implying sodium channels’ sensitivity towards the pain sensation threshold. This was corroborated by experimental evidence in existing literature. Secondly, a theoretical analysis was performed to explore all possible bifurcations that can explain the dynamics of this pain-sensing neuron model. The mathematical modeling simulations revealed a mixture of small amplitude and large amplitude membrane potential oscillations (mixed-mode oscillations) for specific parameter values. The onset and disappearance of the oscillations were investigated. Model simulations further demonstrated that the mixed-mode oscillations solutions belonged to Farey sequences. Furthermore, regions of bistability- where stable steady state and periodic solutions coexisted- were explored. Additionally, chaotic behavior was observed for specific model parameters.<br><br>Finally, this thesis investigated the role of voltage-gated ion channels in inducing CIPN using the same mathematical model. Repetitive firing of action potentials in the absence of any external stimulus was used as an indicator of peripheral neuropathy. Bifurcation analysis revealed that specific sodium and potassium conductances can induce repetitive firing without any external stimulus. The findings were supplemented by recording the firing rate of a sensory neuron culture. Next, a chemotherapy agent - paclitaxel, was introduced in the model to investigate its potential effects on the firing behavior. It was seen that a medium dose of paclitaxel increased repetitive firing. This was supported by the firing rate recordings of the sensory neuron culture.<br><br>In summary, this thesis presents a prediction strategy for CIPN. Moreover, it presents a bifurcation theory-based framework that can be used to investigate pain sensation, in particular, genetic mutations related to pain sensation and chemotherapy-induced peripheral neuropathy. This framework can be used to find potential voltage-gated ion channels that can be targeted to control the pain sensation threshold in individuals, and can be extended to investigate various degeneracies in CIPN mechanisms to find therapeutic cures for it.