Browsing by Author "Fundinger, Danny Georg"

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Investigating dynamics by multilevel phase space discretization
(2006) Fundinger, Danny Georg; Levi, Paul (Prof. Dr.)
The subject of the thesis is the numerical investigation of dynamical systems. The aim is to provide approaches for the localization of several topological structures which are of vital importance for the global analysis of dynamical systems, namely, periodic orbits, the chain recurrent set, repellers, attractors and their domains of attraction as well as stable, unstable and connecting manifolds. The techniques introduced do not require any a priori knowledge about a system, and are also not restricted by the stability of the solution. Furthermore, they can generally be applied to a wide range of dynamical systems. Two theoretical concepts are considered to be at the center of the research - symbolic analysis and the RIM method. The underlying basic approach for both of them is multilevel phase space discretization. This means that a part of the phase space, the area of investigation, is subdivided in a finite number of sets. Then, instead of each point of the phase space, only these sets are subject of further analysis. The main target of every method proposed is to find those sets which contain parts of the solution and subdivide them into smaller parts until a desired accuracy is reached. In case of symbolic analysis, a directed graph is constructed which represents the structure of the state space for the investigated dynamical system. This graph is called the symbolic image of the focused system and can be seen as an approximation of the system flow. The theoretical background regarding the symbolic image graph as well as the constructive methods applied on it were already described in a series of works by G. Osipenko. In this work, strategies are introduced for a practical application. This requires the extension of the theoretical concepts and the development of appropriate algorithms and data structures. In practice, it turned out that these aspects are essential cornerstones for the usability of the discussed methods. Also some sophisticated tunings of the basic methods are proposed in order to extent the field of practical investigation. Although symbolic analysis can be seen as the main stimulation of this work, the investigation was not limited to it. Indeed, several shortcomings regarding the solution of some problems can be observed if the method is applied in practice. This led to the development of the RIM method. The core intention of the method is to solve the root finding problem. The standard approach toward this task is the application of an iteration scheme based on the Newton method. However, it has shown that such Newton schemes have several structural disadvantages which are especially crucial in the context of the fields of investigation which are relevant to this work. The RIM method proposes an alternative approach which does not require the application of any Newton-like method. Numerical case studies revealed that in several nontrivial scenarios the RIM method provides better results than both, symbolic analysis as well as Newton-based methods. Two applications of the RIM method for the investigation of dynamical systems are provided. One of them is the detection of periodic points. The other is the computation of stable manifolds. The proposed methods contribute not only to the direct investigation and simulation of specific dynamical processes but also to the research in the field of dynamical system theory in general. This is due to the fact that progress in theory depends to a large extent on the observation and investigation of phenomenons. These phenomenons can often only be revealed, analyzed and verified by numerical experiments. The presented numerical case studies give some concrete examples for the application of the methods. Hereby, the dynamical models are taken from different fields of scientific research, like geography, biology, meteorology, or physics.