Orientador: Claudio Aguinaldo Buzzi / Banca: Miriam Garcia Manoel / Banca: Angela Maria Sitta / Banca: Parham Salehyan / Banca: Osvaldo Germano do Rocio / Resumo: Neste trabalho apresentamos um estudo dos campos de vetores lineares reversíveis e equivariantes. Tal estudo tem como base a Teoria de Representações de grupos de Lie compactos. Usaremos o fato de que a ascensão de um grupo de Lie compacto pode ser decomposta como soma direta de representações irredutíveis e de acordo com o Lema de Schur tais representações poderão ser de três tipos: R; C ou H. Daremos uma classificação das possíveis estruturas dos sistemas lineares reversíveis equivariantes baseado na teoria de representações citada acima e faremos um estudo dos autovalores para uma classe particular de funções Lreversíveis. Dessa forma temos um cenário bem claro da dinâmica de tais sistemas em cada uma dessas classes. / Abstract: In this work we present a study of the linear equivariant reversible vector fields. This study is based on the Theory of Representation of compact Lie groups. We use the fact that an action of a compact Lie group can be decomposed as a direct sum of irreducible representations, and according to Schur's Lemma these representations can be only of three types: R; C ou H. We give a classification of the possible structures of the linear equivariant reversible systems based on the Theory of Representations mentioned above and we study of the eigenvalues for a particular classes of Lreversible maps. In this way we have a very clear scenario about the dynamics of such systems in each one of these classes. / Mestre

És difícil dibuixar una frontera, dins la Teoria de Sistemas Dinàmics, entre lleis de conservació i simetries doncs, sovint, les seves característiques es confonen. Un clar exemple d'aquest fenómen el constitueixen els sistemes Hamiltonians i els sistemes reversibles.Breument, un sistema dinàmic es diu temps-reversible (o, per nosaltres, simplement reversible) si és invariant sota l'acció d'un difeomorfisme involutiu a l'espai i una inversió en el sentit del temps. és en aquest marc on cal situar aquesta memòria. Concretament, ens centrem en dos punts molt particulars: la Teoria de Formes Normals i el fenómen del trencament de separatrius, tots dos introduïts per Poincaré a la seva tesi (1890).Respecte al primer d'aquests punts, en aquesta tesi s'introdueix el concepte de Pseudo Forma Normal (breument PNF), inspirat en idees d'en Moser, i que permet transformar, sota certes hipotesis, un sistema analític en un d'equivalent d'aspecte el més simple possible. Aquesta PNF és una generalització de la coneguda Forma Normal de Birkhoff amb la qual coincideix si el sistema considerat és Hamiltonià o reversible. Com a conseqüència, s'obté, en determinats casos, l'equivalència local entre aquests dos tipus de sistemes. Aquesta PNF pot esdevenir una eina útil per estudiar la dinàmica d'un sistema analític a l'entorn d'un equilibri (un punt, una òrbita periòdica o un tor).El segon punt, l'escissió de separatrius, fa referència a l'intersecció transversal de varietats invariants procedent del trencament d'una certa connexió homoclínica a l'afegir al sistema una petita pertorbació. Un dels motius d'interés sobre aquest fenòmen és que és un dels principals causants de comportament estocàstic en sistemes Hamiltonians.Un problema relacionat amb aquest trencament de separatrius és el de mesurar-lo, sigui a partir del càlcul de l'angle amb el que es troben aquestes varietats per primer cop, per l'àrea que tanquen entre elles, etc. El mètode habitualment utilitzat per a estimar-lo és l'anomenat mètode de Poincaré-Melnikov. Malhauradament, si la pertorbació és ràpidament oscil-latòria els termes que proporciona aquest mètode són exponencialment petits en el paràmetre pertorbador, fet que dificulta el seu càlcul. En aquesta tesi s'ha demostrat, tal i com passa en el cas Hamiltonià, que en el cas d'un sistema reversible, respecte a una involució lineal, 2-dimensional i pertorbat de manera ràpidament periòdica i reversible, el mètode de Poincaré-Melnikov és correcte i dóna en primer ordre l'anomenada funció de Melnikov. / It is difficult, in the Theory of Dynamical Systems, to draw a boundary line between conservation laws and symmetries because often their effects on the dynamics are very similar. This is the case of the Hamiltonian and the Reversible systems.Briefly, a dynamical system is called time-reversible (or, simply, reversible) if it is invariant under the action of an involutive spatial diffeomorphism and a reversion in time's arrow. This is the frame where this work must be placed. Precisely, we focus our attention in two particular points: the Theory of Normal Forms and the phenomenon of the splitting of separatrices, both introduced by Poincare in his thesis (1890).Regarding the first one of this topics, we introduce the concept of Pseudo-normal Form (PNF in short). It comes from ideas of Moser and allows to transform, under suitable conditions, an analytic system around an equilibrium in another equivalent one having a quite simple form. This PNF is a generalization of the celebrated Birkhoff Normal Form and both coincide if the system is Hamiltonian or reversible. Consequently, the local equivalence between both types of systems is derived in some cases. This PNF can become a useful tool to study the dynamics of an analytic system in a neighborhood of an equilibrium (a fixed point, a periodic orbit or a torus).The second topic, the splitting of separatrices, is related to the transversal intersection of invariant manifolds derived from the splitting of a given homoclinic connection when some small perturbation is considered. One of the reasons that makes this phenomenon interesting is that it seems to be one of the main causes of the stochastic behavior in Hamiltonian systems.One problem related to this splitting of separatrices becomes to measure it, studying, for instance, some angle the form when they meet for the first time, the area of the first lobe, etc. The standard method to estimate this size is the celebrated Poincaré-Melnikov method. Unfortunately, if the perturbation oscillates rapidly the terms provided by this method are exponentially small in the perturbation parameter, and this fact makes this computation more involved. In this work we prove, like it happens in the Hamiltonian case, that in the case of a 2-dimensional analytic reversible system (reversible with respect to a linear spatial involution) perturbed by a rapidly periodic reversible perturbation, the Poincaré-Melnikov method works and it provides, at first order, the well known Melnikov Function.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-02Bitstream added on 2014-06-13T18:55:32Z : No. of bitstreams: 1 alves_mo_me_sjrp.pdf: 609574 bytes, checksum: 7280f95db92aacc87fc1116bf82914da (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho apresentamos um estudo dos campos de vetores lineares reversíveis e equivariantes. Tal estudo tem como base a Teoria de Representações de grupos de Lie compactos. Usaremos o fato de que a ascensão de um grupo de Lie compacto pode ser decomposta como soma direta de representações irredutíveis e de acordo com o Lema de Schur tais representações poderão ser de três tipos: R; C ou H. Daremos uma classificação das possíveis estruturas dos sistemas lineares reversíveis equivariantes baseado na teoria de representações citada acima e faremos um estudo dos autovalores para uma classe particular de funções Lreversíveis. Dessa forma temos um cenário bem claro da dinâmica de tais sistemas em cada uma dessas classes. / In this work we present a study of the linear equivariant reversible vector fields. This study is based on the Theory of Representation of compact Lie groups. We use the fact that an action of a compact Lie group can be decomposed as a direct sum of irreducible representations, and according to Schur's Lemma these representations can be only of three types: R; C ou H. We give a classification of the possible structures of the linear equivariant reversible systems based on the Theory of Representations mentioned above and we study of the eigenvalues for a particular classes of Lreversible maps. In this way we have a very clear scenario about the dynamics of such systems in each one of these classes.

In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, non-smooth M-wave solutions. While the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. In the second part of the thesis, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbits of their traveling-wave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. Of course, the convergence rate is not comparable to typical asymptotic series. However, asymptotic solutions for global behavior along a full homoclinic/heteroclinic orbit are currently not available.

[EN] This thesis addresses the treatment of perturbed problems with splitting methods. After motivating these problems in Chapter 1, we give a thorough introduction in Chapter 2, which includes the objectives, several basic techniques and already existing methods. In Chapter 3, we consider the numerical integration of non-autonomous separable parabolic equations using high order splitting methods with complex coefficients (methods with real coefficients of order greater than two necessarily have negative coefficients). We propose to consider a class of methods that allows us to evaluate all time dependent operators at real values of the time, leading to schemes which are stable and simple to implement. If the system can be considered as the perturbation of an exactly solvable problem and the flow of the dominant part is advanced using real coefficients, it is possible to build highly efficient methods for these problems. We show the performance of this class of methods for several numerical examples and present some new improved schemes. In Chapter 4, we propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum A = D+epsilon*B of a sparse and efficiently exponentiable matrix D with sparse exponential exp(D) and a dense matrix epsilon*B which is of small norm in comparison with D. The predominant algorithm is based on scaling the large matrix A by a small number 2^(-s) , which is then exponentiated by efficient Padé or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbation matrix B in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought. In Chapter 5, we consider the numerical integration of the perturbed Hill's equation. Parametric resonances can appear and this property is of great interest in many different physical applications. Usually, the Hill's equations originate from a Hamiltonian function and the fundamental matrix solution is a symplectic matrix. This is a very important property to be preserved by the numerical integrators. In this chapter we present new sixth-and eighth-order symplectic exponential integrators that are tailored to the Hill's equation. The methods are based on an efficient symplectic approximation to the exponential of high dimensional coupled autonomous harmonic oscillators and yield accurate results for oscillatory problems at a low computational cost. Several numerical examples illustrate the performance of the new methods. Conclusions and pointers to further research are detailed in Chapter 6. / [ES] Esta tesis aborda el tratamiento de problemas perturbados con métodos de escisión (splitting). Tras motivar el origen de este tipo de problemas en el capítulo 1, introducimos los objetivos, varias técnicas básicas y métodos existentes en capítulo 2. En el capítulo 3 consideramos la integración numérica de ecuaciones no autónomas separables y parabólicas usando métodos de splitting de orden mayor que dos usando coeficientes complejos (métodos con coeficientes reales de orden mayor de dos necesariamente tienen coeficientes negativos). Proponemos una clase de métodos que permite evaluar todos los operadores con dependencia temporal en valores reales del tiempo lo cual genera esquemas estables y fáciles de implementar. Si el sistema se puede considerar como una perturbación de un problema resoluble de forma exacta y si el flujo de la parte dominante se avanza usando coeficientes reales, es posible construir métodos altamente eficientes para este tipo de problemas. Demostramos la eficiencia de estos métodos en varios ejemplos numéricos. En el capítulo 4 proponemos métodos de splitting para el cálculo de la exponencial de matrices perturbadas que se pueden escribir como suma A = D + epsilon*B de una matriz dispersa y eficientemente exponenciable con exponencial dispersa exp(D) y una matriz densa epsilon*B de noma pequeña. El algoritmo predominante se basa en escalar la matriz grande con un número pequeño 2^(-s) para poder exponenciar el resultado con métodos eficientes de Padé o Taylor y finalmente obtener la aproximación a la exponencial elevando al cuadrado repetidamente. En este contexto, el coste computacional proviene de las multiplicaciones de matrices densas y presentamos una cuadratura modificada aprovechando la estructura perturbada para reducir el número de productos. Resultados teóricos sobre errores locales y propagación de error para métodos de splitting son complementados con experimentos numéricos y muestran una clara mejora sobre métodos existentes a precisión media. En el capítulo 5, consideramos la integración numérica de la ecuación de Hill perturbada. Resonancias paramétricas pueden aparecer y esta propiedad es de gran interés en muchas aplicaciones físicas. Habitualmente, las ecuaciones de Hill provienen de una función hamiltoniana y la solución fundamental es una matriz simpléctica, una propiedad muy importante que preservar con los integradores numéricos. Presentamos nuevos integradores simplécticos exponenciales de orden seis y ocho tallados a la ecuación de Hills. Estos métodos se basan en una aproximación simpléctica eficiente a la exponencial de osciladores armónicos acoplados de dimensión alta y dan lugar a resultados precisos para problemas oscilatorios a un coste computacional bajo y varios ejemplos numéricos ilustran su rendimiento. Conclusiones e indicadores para futuros estudios se detallan en el capítulo 6. / [CA] La present tesi està enfocada al tractament de problemes perturbats utilitzant, entre altres, mètodes d'escisió (splitting). Comencem motivant l'oritge d'aquest tipus de problems al capítol 1, i a continuació introduïm el objectius, diferents tècniques bàsiques i alguns mètodes existents al capítol 2. Al capítol 3, consideram la integració numèrica d'equacions no autònomes separables i parabòliques utilitzant mètodes d'splitting d'ordre major que dos utilitzant coeficients complexos (mètodes amb coeficients reials d'ordre major que dos necesariament tenen coeficients negatius). Proposem una clase de mètodes que permeten evaluar tots els operadors amb dependència temporal explícita amb valors reials del temps. Esta forma de procedir genera esquemes estables i fàcils d'implementar. Si el sistema es pot considerar com una perturbació d'un problema exactament resoluble, i la part dominant s'avança utilitzant coeficients reials, es posible construir mètodes altament eficients per aquest tipus de problemes Demostrem la eficiència d'estos mètodes per a diferents exemples numèrics. Al capítol 4, proposem mètodes d'splitting per al càcul de la exponencial de matrius pertorbades que es poden escriure com suma A = D + epsilon*B (una matriu que es pot exponenciar fàcilment i eficientemente, com es el cas d'algunes matrius disperses exp(D), i una matriu densa epsilon*B de norma menuda). L'algorisme predominant es basa en escalar la matriu gran amb un nombre menut 2^(-s) per a poder exponenciar el resultat amb mètodes eficients de Padé o Taylor i finalment obtindre la aproximació a la exponencial elevant al quadrat repetidament. En este context, el cost computacional prové de les multiplicacions de matrius denses i presentem una quadratura modificada aprofitant la estructura de matriu pertorbada per reduir el nombre de productes. Resultats teòrics sobre errors locals i propagació d'error per a mètodes d'splitting son analitzats i corroborats amb experiments numèrics, mostrant una clara millora respecte a mètodes existens quan es busca una precisió moderada. Al capítol 5, considerem la integració numèrica de l'ecuació de Hill pertorbada. En este tipus d'equacions poden apareixer resonàncies paramètriques i esta propietat es de gran interés en moltes aplicacions físiques. Habitualment, les equacions de Hill provenen d'una función hamiltoniana i la solució fonamental es una matriu simplèctica, siguent esta una propietat molt important a preservar pels integradors numèrics. Presentams nous integradors simplèctics exponencials d'orden sis i huit construits especialmente per resoldre l'ecuació de Hill. Estos mètodes es basen en una aproxmiació simplèctica eficient a la exponencial d'osciladors harmònics acoplats de dimensió alta i donen lloc a resultats precisos per a problemas oscilatoris a un cost computacional baix. La eficiencia dels mètodes s'il.lustra en diferents exemples numèrics. Conclusions i indicadors per a futurs estudis es detallen al capítol 6. / Seydaoglu, M. (2016). Splitting methods for autonomous and non-autonomous perturbed equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/71358

The research work reported in the thesis deals with the development of dynamic covalent systems and their applications in evaluation and screening of protein-ligands and enzyme inhibitors, as well as in synthetic methodologies. The thesis is divided into four parts as described below. In part one, synthetic methodologies to access 3-functionalized phthalides and 3-thioisoindolinones using the concept of cascade reactions are demonstrated. Efficient syntheses of the target products are designed and performed in one-pot process under mild reaction conditions.  In part two, phosphine-catalyzed disulfide metathesis for the generation of dynamic carbohydrate system in aqueous solution is demonstrated. In the presence of biological target (Concanavalin A), the optimal dynamic ligand is successfully identified in situ by the 1H STD-NMR spectroscopy. In part three, lipase-catalyzed resolutions of dynamic reversible systems using reversible cyanohydrin and hemithioacetal reactions in one-pot processes are demonstrated. The dynamic systems are generated under thermodynamic control in organic solution and subsequently resolved by lipase-mediated resolution under kinetic control. The resolution processes resulted in the lipase-selected substrates with high structural and stereochemical specificities. In the last part, dynamic fragment-based strategy is presented using β-galactosidase as a model target enzyme. Based on our previous study, the best dynamic inhibitor of β-galactosidase was identified using 1H STD-NMR technique from dynamic hemithioacetal systems. The structure of the dynamic inhibitor is tailored by fragment linking and optimization processes. The designed inhibitor structures are then synthesized and tested for inhibition activities against β-galactosidase. / QC 20110526

The goal of this dissertation is characterize the stability of equilibrium solutions of Reversible Systems of second-order resonance. To this end, we provide de nitions and basic properties relevant to the Reversible systems; we obtain the normal form of the linearized system and from the Poincar e-Dulac method, we will write the Normal Form of third order of the system studies. For last, we treat the necessary and su cient conditions for the stability of the trivial solution of a reversible system at 1:1 resonance analyzing two cases: the case in which the system matrix is Diagonalizable and the case where the matrix is non-diagonalizable. / O objetivo dessa disserta c~ao e a caracteriza c~ao da estabilidade de solu c~oes de equil brio de Sistemas Revers veis com resson^ancia de segunda ordem. Para tanto, fornecemos de ni c~oes e propriedades b asicas pertinentes aos Sistemas Revers veis; obteremos a forma normal do sistema linearizado e, a partir do m etodo de Poincar e-Dulac, escreveremos a forma normal de terceira ordem do sistema em estudo. Por m, trataremos das condi c~oes necess arias e/ou su cientes a estabilidade de uma solu c~ao nula de um Sistema Revers vel com resson^ancia de segunda ordem analisando dois casos: o caso em que a matriz do sistema e diagonaliz avel e o caso em que a matriz e n~ao-diagonaliz avel