Many seemingly simple systems can display extraordinarily complex dynamics which has been studied and uncovered through nonlinear dynamical theory. The leitmotif of this thesis is changing phase-space structures and their (linear or nonlinear) stabilities by adding control functions (which act on the system as external perturbations) to the relevant Hamiltonians. These phase-space structures may be periodic orbits, invariant tori or their stable and unstable manifolds. One-electron systems and diatomic molecules are fundamental and important staging ground for new discoveries in nonlinear dynamics. In past years, increasing emphasis and effort has been put on the control or manipulation of these systems. Recent developments of nonlinear dynamical tools can provide efficient ways of doing so. In the first subtopic of the thesis, we are adding a control function to restore tori at prescribed locations in phase space. In the remainder of the thesis, a control function with parameters is used to change the linear stability of the periodic orbits which govern the processes in question. In this thesis, we report our theoretical analyses on multiphoton ionization of Rydberg atoms exposed to strong microwave fields and the dissociation of diatomic molecules exposed to bichromatic lasers using nonlinear dynamical tools. This thesis is composed of three subtopics. In the first subtopic, we employ local control theory to reduce the stochastic ionization of hydrogen atom in a strong microwave field by adding a relatively small control term to the original Hamiltonian. In the second subtopic, we perform periodic orbit analysis to investigate multiphoton ionization driven by a bichromatic microwave field. Our results show quantitative and qualitative agreement with previous studies, and hence identify the mechanism through which short periodic orbits organize the dynamics in multiphoton ionization. In addition, we achieve substantial time savings with this approach. In the third subtopic we extend our periodic orbit analysis to the dissociation of diatomic molecules driven by a bichromatic laser. In this problem, our results based on periodic orbit analysis again show good agreement with previous work, and hence promise more potential applications of this approach in molecular physics.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2017-12-27T12:48:10Z No. of bitstreams: 2 Tese - Pablo Vandré Jacob Furlan - 2017.pdf: 3620430 bytes, checksum: 7275b5a734d392f78e2829268555ec68 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-12-28T09:41:38Z (GMT) No. of bitstreams: 2 Tese - Pablo Vandré Jacob Furlan - 2017.pdf: 3620430 bytes, checksum: 7275b5a734d392f78e2829268555ec68 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-12-28T09:41:38Z (GMT). No. of bitstreams: 2 Tese - Pablo Vandré Jacob Furlan - 2017.pdf: 3620430 bytes, checksum: 7275b5a734d392f78e2829268555ec68 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-11-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we establish a new method to calculate the index of curves in a neighborhood of a boundary and we show that the index of a trajectory of a vector field which intersects the boundary at two points is 1/2. Using this method we extended the index definition for discontinuous vector fields with a regular transition manifold and we calculate the index for closed curves that intersect the variety of transition = f−1(0), where f is a differentiable function, and is the union of the regions tangency, sewing, sliding and escaping. We also show that the index for solutions of the discontinuous vector field that are −closed of type 1 and intersect the boundary at 2-point is equal to 1. We also establish an index theory for discontinuous vector fields when the transition manifold is not regular in a point and we show that the index is given by the calculation in its regular regions and add ±1/2, depending on the dynamics at the non-regular point. We apply the theory of index developed in this work and we give quotas for the indices of continuous vector field and for polynomial vector fields on two zones. Finally, we demonstrate a version of the Poincaré-Hopf Theorem for discontinuous vector fields in compact manifolds. / Neste trabalho estabelecemos um novo método para calcular o índice de curvas numa vizinhança do bordo e mostramos que o índice de uma trajetória de um campo vetorial a qual intersecta o bordo em dois pontos é 12 . Utilizando este método estendemos a definição do índice para campos vetoriais descontínuos com variedade de transição regular e calculamos o índice para curvas fechadas que intersectam a variedade de transição = f−1(0), onde f é uma função diferenciável, e é a união das regiões de tangência, de deslize, escape ou costura. Mostramos também que o índice para soluções do campo vetorial descontínuo que são −fechadas do tipo 1 e intersectam o bordo em 2 pontos é igual a 1. Estabelecemos também uma teoria do índice para campos vetoriais descontínuos quando a variedade de transição não é regular em um ponto e mostramos que o índice é dado pelo cálculo em suas regiões regulares e somar ±1 2 , a depender da dinâmica no ponto não regular. Aplicamos a teoria do índice desenvolvida neste trabalho e damos cotas para índices de campos vetoriais contínuos e para campos vetoriais polinomiais por partes. Finalmente, demostramos uma versão do Teorema de Poincaré-Hopf para campos vetoriais descontínuos em variedades compactas.

We investigate chaotic ionization of highly excited hydrogen atom in crossed electric and magnetic fields (Rydberg atom) and intra-molecular relaxation in planar carbonyl sulfide (OCS) molecule. The underlying theoretical framework of our studies is dynamical systems theory and periodic orbit theory. These theories offer formulae to compute expectation values of observables in chaotic systems with best accuracy available in given circumstances, however they require to have a good control and reliable numerical tools to compute unstable periodic orbits. We have developed such methods of computation and partitioning of the phase space of hydrogen atom in crossed at right angles electric and magnetic fields, represented by a two degree of freedom (dof) Hamiltonian system. We discuss extensions to a 3-dof setting by developing the methodology to compute unstable invariant tori, and applying it to the planar OCS, represented by a 3-dof Hamiltonian. We find such tori important in explaining anomalous relaxation rates in chemical reactions. Their potential application in Transition State Theory is discussed.

Nous nous intéressons à des équations différentielles stochastiques obtenues en perturbant par un bruit blanc des équations différentielles ordinaires admettant N orbites périodiques asymptotiquement stables. Nous construisons une chaîne de Markov à temps discret et espace d’états continu appelée application de Poincaré aléatoire qui hérite du comportement métastable du système. Nous montrons que ce processus admet exactement N valeurs propres qui sont exponentiellement proches de 1 et nous donnons des expressions pour ces valeurs propres et les fonctions propres associées en termes de fonctions committeurs dans les voisinages des orbites périodiques. Nous montrons également que ces valeurs propres sont bien séparées du reste du spectre. Chacune de ces valeurs propres exponentiellement proche de 1 est également reliée à un temps d’atteinte de ces voisinages. De plus, les N valeurs propres exponentiellement proches de 1 et fonctions propres à gauche et à droite associées peuvent être respectivement approchées par des valeurs propres principales, des distributions quasi-stationnaires, et des fonctions propres principales à droite de processus tués quand ils atteignent ces voisinages. Les preuves reposent sur une représentation de type Feynman–Kac pour les fonctions propres, la transformée harmonique de Doob, la théorie spectrale des opérateurs compacts et une propriété de type équilibré détaillé satisfaite par les fonctions committeurs. / We consider stochastic differential equations, obtained by adding weak Gaussian white noise to ordinary differential equations admitting N asymptotically stable periodic orbits. We construct a discrete-time,continuous-space Markov chain, called a random Poincaré map, which encodes the metastable behaviour of the system. We show that this process admits exactly N eigenvalues which are exponentially close to 1,and provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committorfunctions of neighbourhoods of periodic orbits. We also provide a bound for the remaining part of the spectrum. The eigenvalues that are exponentially close to 1 and the right and left eigenfunctions are well-approximated by principal eigenvalues, quasistationary distributions, and principal right eigenfunctions of processes killed upon hitting some of these neighbourhoods. Each eigenvalue that is exponentially close to 1is also related to the mean exit time from some metastable neighborhood of the periodic orbits. The proofsrely on Feynman–Kac-type representation formulas for eigenfunctions, Doob’s h-transform, spectral theory of compact operators, and a recently discovered detailed balance property satisfied by committor functions.

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