Applying numerical continuation to the parameter dependence of solutions of the Schr\"odinger equation

In molecular reactions at the microscopic level the appearance of resonances has an important influence on the reactivity. It is important to predict when a bound state transitions into a resonance and how these transitions depend on various system parameters such as internuclear distances. The dynamics of such systems are described by the time-independent Schr\"odinger equation and the resonances are modeled by poles of the S-matrix. Using numerical continuation methods and bifurcation theory, techniques which find their roots in the study of dynamical systems, we are able to develop efficient and robust methods to study the transitions of bound states into resonances. By applying Keller's Pseudo-Arclength continuation, we can minimize the numerical complexity of our algorithm. As continuation methods generally assume smooth and well-behaving functions and the S-matrix is neither, special care has been taken to ensure accurate results. We have successfully applied our approach in a number of model problems involving the radial Schr\"odinger equation

Time-stepping and Krylov methods for large-scale instability problems

With the ever increasing computational power available and the development of high-performances computing, investigating the properties of realistic very large-scale nonlinear dynamical systems has been become reachable. It must be noted however that the memory capabilities of computers increase at a slower rate than their computational capabilities. Consequently, the traditional matrix-forming approaches wherein the Jacobian matrix of the system considered is explicitly assembled become rapidly intractable. Over the past two decades, so-called matrix-free approaches have emerged as an efficient alternative. The aim of this chapter is thus to provide an overview of well-grounded matrix-free methods for fixed points computations and linear stability analyses of very large-scale nonlinear dynamical systems.Comment: To appear in "Computational Modeling of Bifurcations and Instabilities in Fluid Mechanics", eds. A. Gelfgat, Springe

Efficient method for detection of periodic orbits in chaotic maps and flows

An algorithm for detecting unstable periodic orbits in chaotic systems [Phys. Rev. E, 60 (1999), pp. 6172-6175] which combines the set of stabilising transformations proposed by Schmelcher and Diakonos [Phys. Rev. Lett., 78 (1997), pp. 4733-4736] with a modified semi-implicit Euler iterative scheme and seeding with periodic orbits of neighbouring periods, has been shown to be highly efficient when applied to low-dimensional system. The difficulty in applying the algorithm to higher dimensional systems is mainly due to the fact that the number of stabilising transformations grows extremely fast with increasing system dimension. In this thesis, we construct stabilising transformations based on the knowledge of the stability matrices of already detected periodic orbits (used as seeds). The advantage of our approach is in a substantial reduction of the number of transformations, which increases the efficiency of the detection algorithm, especially in the case of high-dimensional systems. The performance of the new approach is illustrated by its application to the four-dimensional kicked double rotor map, a six-dimensional system of three coupled H\'enon maps and to the Kuramoto-Sivashinsky system in the weakly turbulent regime.Comment: PhD thesis, 119 pages. Due to restrictions on the size of files uploaded, some of the figures are of rather poor quality. If necessary a quality copy may be obtained (approximately 1MB in pdf) by emailing me at [email protected]