Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T09:35:17Z (GMT). No. of bitstreams: 1 AchireQuispe_JesusEnrique_D.pdf: 1199686 bytes, checksum: 2c263eb351ad3dfa30b13e0fecc5282b (MD5) Previous issue date: 2014 / Resumo: Recentemente, a Teoria de campos descontínuos (Non-Smooth Dynamic Systems) tem-se desenvolvido rapidamente, motivado principalmente pelas aplicações na física e nas engenharias, e também pela atraente beleza matemática. Neste trabalho, consideraremos campos de vetores suaves por partes, denominados campos de Filippov, e usamos o método convexo de Filippov para definir órbita solução deste tipo de campo. Assim, órbitas soluções passando por um ponto qualquer sempre existem. Há duas principais diferenças com o clássico caso diferenciável: a primeira é que as órbitas neste caso são curvas suaves por partes, enquanto que no caso diferenciável são curvas suaves. A segunda é que as órbitas soluções não tem a propriedade da unicidade, ou seja, podem existir duas ou mais órbitas passando pelo mesmo ponto. São esses fatos que fazem essa teoria um pouco diferente da teoria clássica de campos diferenciáveis. Estamos interessados em estudar qualitativamente os campos de Filippov, especialmente os que são genéricos e estruturalmente estáveis. Assim, nesta tese descrevemos propriedades genéricas necessárias para um campo de Filippov ser estruturalmente estável. Particularmente analisamos estabilidade estrutural local de singularidades tangenciais tais como o rabo de andorinha, a dobradobra,e dobra-cúspide, e adicionalmente pseudoequilíbrios e órbitas fechadas / Abstract: Recently, the Theory of Non-smooth Dynamic Systems has been developed, motivated mostly by their applications in physics and engineering, and also by its attractive mathematical beauty. In this work, we consider piecewise-smooth vector fields, called Filippov's vector fields, and we use the Filippov's convex method to define orbits solutions of this type of vector fields. Thus, orbit solution through any point always exists. But, there are two main differences with the classic differentiable case: the first is that orbits in this case are piecewise smooth curves while that in the differentiable case they are smooth curves. The second is that there is not uniqueness of solutions, this is, it may exist two or more than two orbits passing through a point. We are interested in to study qualitatively the Filippov's vector fields, especially those thatare generic and structurally stable. Thus, in this text we describe generic properties necessaryfor a vector field to be structurally stable. In particular, we analyze local structural stability attangential singularities, such as swallowtail-regular, fold-fold, fold-cusp, and additionally pseudoequilibriumsand closed orbits / Doutorado / Matematica / Doutor em Matemática

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In this Ph.D. thesis, we study regular and embedded solitons and generalized and degenerate Hopf bifurcations. These two areas of work are seperate and independent from each other. First, variational methods are employed to generate families of both regular and embedded solitary wave solutions for a generalized Pochhammer PDE and a generalized microstructure PDE that are currently of great interest. The technique for obtaining the embedded solitons incorporates several recent generalizations of the usual variational technique and is thus topical in itself. One unusual feature of the solitary waves derived here is that we are able to obtain them in analytical form (within the family of the trial functions). Thus, the residual is calculated, showing the accuracy of the resulting solitary waves. Given the importance of solitary wave solutions in wave dynamics and information propagation in nonlinear PDEs, as well as the fact that only the parameter regimes for the existence of solitary waves had previously been analyzed for the microstructure PDE considered here, the results obtained here are both new and timely.

A bifurcation analysis is used to investigate the complex dynamics of two heavily loaded single-machine-infinite-busbar power systems modeling the characteristics of the BOARDMAN generator with respect to the rest of the North-Western American Power System and the CHOLLA# generator with respect to the SOWARO station. In the BOARDMAN system, we show that there are three Hopf bifurcations at practical compensation values, while in the CHOLLA#4 system, we show that there is only one Hopf bifurcation. The results show that as the compensation level increases, the operating condition loses stability with a complex conjugate pair of eigenvalues of the Jacobian matrix crossing transversely from the left- to the right-half of the complex plane, signifying a Hopf bifurcation. As a result, the power system oscillates subsynchronously with a small limit-cycle attractor. As the compensation level increases, the limit cycle grows and then loses stability via a secondary Hopf bifurcation, resulting in the creation of a two-period quasiperiodic subsynchronous oscillation, a two-torus attractor. On further increases of the compensation level, the quasiperiodic attractor collides with its basin boundary, resulting in the destruction of the attractor and its basin boundary in a bluesky catastrophe. Consequently, there are no bounded motions. When a damper winding is placed either along the q-axis, or d-axis, or both axes of the BOARDMAN system and the machine saturation is considered in the CHOLLA#4 system, the study shows that, there is only one Hopf bifurcation and it occurs at a much lower level of compensation, indicating that the damper windings and the machine saturation destabilize the system by inducing subsynchronous resonance. Finally, we investigate the effect of linear and nonlinear controllers on mitigating subsynchronous resonance in the CHOLLA#4 system . The study shows that the linear controller increases the compensation level at which subsynchronous resonance occurs and the nonlinear controller does not affect the location and type of the Hopf bifurcation, but it reduces the amplitude of the limit cycle born as a result of the Hopf bifurcation. / Ph. D.

Understanding pattern formation in nature is an important topic in applied mathematics. For more than three decades, the Ohta–Kawasaki energy has attracted considerable attention from applied mathematicians. This energy functional, which combines surface energy and electrostatic potential energy, captures the intricate patterns observed in various physical and biological systems. Despite its apparent simplicity, the Ohta–Kawasaki energy serves as a versatile framework for describing a wide range of pattern formation phenomena induced by competing interactions. In this dissertation, we aim to gain a better understanding of the important properties of the Ohta–Kawasaki energy, specifically its stationary points, global minimizers, and energy landscape. We explore these properties in the context of broad applications such as nuclear physics, block copolymers, and biological membranes. In order to investigate the complicated geometries in these applications, we utilize asymptotic analysis and numerical simulations. Firstly, we explore the stationary points of the Ohta–Kawasaki energy. Specifically, we study how a three-dimensional ball loses stability as the nonlocal coefficient increases in the binary case. Our approach combines numerical simulations and bifurcation analysis. We calculate the minimum energy path for the transition from a single ball to two separate balls, as well as the bifurcation branch orginating from the ball. In the context of nuclear physics, this bifurcation branch is known as the Bohr–Wheeler branch. Our simulations suggest that, unlike the previous understanding, all the stationary points on this bifurcation branch are unstable. Similar results are observed in two dimensions. This finding illustrates the unexpected mechanism governing the stability loss of balls and disks. Secondly, we explore the global minimizers of the Ohta–Kawasaki energy. We numerically compute the one-dimensional energy minimizers of relatively short patterns in the non-degenerate ternary case. Inspired by our numerical results, we propose an array of periodic candidates. We then show that our candidates can have lower energy than the previously conjectured global minimizer which is of the cyclic pattern. Our results are consistent with simulations based on other theories and physical experiments of triblock copolymers, in which noncyclic lamellar patterns have been found. This finding indicates that even in one dimension, the global minimizers of the Ohta–Kawasaki energy can exhibit unexpected richness. Lastly, we explore the energy landscape of the Ohta–Kawasaki energy. We propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model in the degenerate ternary case. Our phase-field simulations and asymptotic results suggest that the limit of the recovery sequence exhibits behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes. This finding reveals the intricate landscape of the Ohta–Kawasaki energy. In summary, this dissertation sheds light on three important aspects of the Ohta–Kawasaki energy: its stationary points, global minimizers, and energy landscape. Our findings are timely contributions to the ongoing research on pattern formation driven by energetic competition.

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.

Complex nonlinear and chaotic responses have been recently observed in various compliant ocean systems. These systems are characterized by a nonlinear mooring restoring force and a coupled fluid-structure interaction exciting force. A general class of ocean mooring system models is formulated by incorporating a variable mooring configuration and the exact form of the hydrodynamic excitation. The multi-degree of freedom system, subjected to combined parametric and external excitation, is shown to be complex, coupled and strongly nonlinear. Stability analysis by a Liapunov function approach reveals global system attraction which ensures that solutions remain bounded for small excitation. Construction of the system's Poincare map and stability analysis of the map's fixed points correspond to system stability of near resonance periodic orbits. Investigation of nonresonant solutions is done by a local variational approach. Tangent and period doubling bifurcations are identified by both local stability analysis techniques and are further investigated to reveal global bifurcations. Application of Melnikov's method to the perturbed averaged system provides an approximate criterion for the existence of transverse homoclinic orbits resulting in chaotic system dynamics. Further stability analysis of the subharmonic and ultraharmonic solutions reveals a cascade of period doubling which is shown to evolve to a strange attractor. Investigation of the bifurcation criteria obtained reveals a steady state superstructure in the bifurcation set. This superstructure identifies a similar bifurcation pattern of coexisting solutions in the sub, ultra and ultrasubharmonic domains. Within this structure strange attractors appear when a period doubling sequence is infinite and when abrupt changes in the size of an attractor occur near tangent bifurcations. Parametric analysis of system instabilities reveals the influence of the convective inertial force which can not be neglected for large response and the bias induced by the quadratic viscous drag is found to be a controlling mechanism even for moderate sea states. Thus, stability analyses of a nonlinear ocean mooring system by semi-analytical methods reveal the existence of bifurcations identifying complex periodic and aperiodic nonlinear phenomena. The results obtained apply to a variety of nonlinear ocean mooring and towing system configurations. Extensions and applications of this research are discussed. / Graduation date: 1992

Orientador: Laurence Duarte Colvara / Banca: Carlos Roberto Minussi / Banca: Wagner Peron Ferreira / Resumo: Este trabalho apresenta um estudo sobre a relação entre os principais tipos de bifurcações que ocorrem em sistemas elétricos de potência e em quais ocasiões elas podem ocorrer em máquinas síncronas com ou sem RAT (Regulador Automático de Tensão). Para explorar tais fenômenos, primeiramente o sistema é modelado, sendo utilizado para o caso MBI (Máquina - Barramento In nito) o modelo um eixo e, então, a matriz de estado é calculada para a análise dos autovalores. Para os sistemas multimáquinas estudados, são incluídos dois enrolamentos amortecedores nos eixos d ¡ q. São então apresentados os métodos de análise de estabilidade transitória convencionais, amplamente utilizados, conhecidos como método Tradicional e Método Direto. As condições para a ocorrência de bifurcações são analisadas utilizando os coe - cientes linearizados do modelo de He ron-Phillips para o caso MBI, onde é mostrado que se espera perder a estabilidade para o sistema com regulador automático de tensão através de uma bifurcação de Hopf e para o caso sem RAT através de uma bifurcação Sela-Nó. Por m, é analisado o ciclo-limite para o caso de uma máquina - barramento in nito e para sistemas multimáquinas através do modelo não-linear. A região de estabilidade é analisada no plano de fase, sendo mostrada a necessidade de incluir a variação de uxo no enrolamento de campo para uma análise correta da estabilidade. É também mostrado que este ciclo-limite pode reduzir a fronteira de estabilidade calculada pelo método convencional. / Abstract: The aim of this study is the relation among main types of bifurcations that occur in electrical power systems and the circumstances they can happen with the synchronous machines considered with or without AVR (Automatic Voltage Regulator). To explore such phenomena, the system is rst modeled with the synchronous machines described by the one axis model for the MIB (Machine - In nite Bus) case , and so the state matrix is computed for the analysis of its eigenvalues. For multimachine systems case two windings dampers are included in d-q axes. The conditions for the occurrence of bifurcations are analyzed using the coe cients of the He ron-Phillips model for MIB case, where it is shown that one expects the system with automatic voltage regulator lose synchronism through a Hopf bifurcation and for the case without RAT through a Saddle- Node Bifurcation. Finally, the nonlinear model is accounted for in order to consider the limit-cycle for the case of one machine - in nite bus case as well as for multimachine system. Since internal voltage a ects the boundary of the stability region it must be considered. Then the phase portrait does not su ce and the trajectories must to be observed in a sub space de ned with the internal voltage. It is also shown that this limit-cycle can reduce the boundary of stability calculated by means of the direct method. / Mestre

Submitted by DIANA YOVANI RODRÍGUEZ VILLENA null (dayaniss_23@hotmail.com) on 2017-10-09T19:16:47Z No. of bitstreams: 1 Dissertação Diana.pdf: 1051753 bytes, checksum: df5c2679c43a774ec3d6809c69271fd4 (MD5) / Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-10-09T19:43:34Z (GMT) No. of bitstreams: 1 rodriguezvillena_dy_me_sjrp.pdf: 1051753 bytes, checksum: df5c2679c43a774ec3d6809c69271fd4 (MD5) / Made available in DSpace on 2017-10-09T19:43:34Z (GMT). No. of bitstreams: 1 rodriguezvillena_dy_me_sjrp.pdf: 1051753 bytes, checksum: df5c2679c43a774ec3d6809c69271fd4 (MD5) Previous issue date: 2017-08-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho, estudamos a teoria de bifurcação e algumas das suas aplicações. Apresentamos alguns resultados básicos e definimos o conceito de ponto de bifurcação. Logo, estudamos a teoria do grau topológico. Em seguida, enunciamos dois teoremas importantes que são os teoremas de Krasnoselski e de Rabinowitz. Finalmente apresentamos um exemplo e duas aplicações do teorema de Rabinowitz nas quais os valores característicos com que lidamos são simples, no exemplo se consegue provar que a segunda alternativa do teorema ocorre, a primeira aplicação é um problema de autovalores não lineares de Sturm-Liouville para uma E.D.O de segunda ordem na qual se prova que a primeira alternativa do teorema de Rabinowitz é válida e a segunda aplicação é um problema de autovalores para uma equação diferencial parcial quase-linear a qual se prova que também ocorre a primeira alternativa do teorema. / In this work, we study bifurcation theory and its applications. We present some basic results and define the concept of bifurcation point. Then we study the theory of topological degree. Next we state two important theorems that are Krasnoselski's theorem and Rabinowitz's theorem. Finally we present an example and two applications of Rabinowitz theorem in which the characteristic values we deal with are simple, in an example we can prove that the second item of theorem occurs and the first application is a nonlinear Sturm-Liouville eigenvalue problem for a second order ordinary differential equation were we prove that the first alternative of Rabinowitz's theorem holds and the second application is an eigenvalue problem for a quasilinear elliptic partial differential equation where we prove that the first alternative of the theorem also holds.